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http://arxiv.org/abs/1704.08257v2 | {
"authors": [
"Ludovic Berthier",
"Patrick Charbonneau",
"Daniele Coslovich",
"Andrea Ninarello",
"Misaki Ozawa",
"Sho Yaida"
],
"categories": [
"cond-mat.stat-mech",
"cond-mat.soft"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170426180001",
"title": "Configurational entropy measurements in extremely supercooled liquids that break the glass ceiling"
} |
|
Sunspots rotation and magnetic transients associated with flares in NOAA AR 11429 ^* * Supported by the National Natural Science Foundation of China.Jianchuan Zheng1 Zhiliang Yang1 Jianpeng Guo1, 2 Kaiming Guo1 Hui Huang1 Xuan Song1 Weixing Wan2, 3 December 30, 2023 ====================================================================================================================================================== Hancheng Guo^* Department of Mathematics, Faculty of Science and Technology University of Macau Macau, 999078, P. R. ChinaJie XiongDepartment of Mathematics, Faculty of Science and Technology University of Macau Macau, 999078, P. R. China(Communicated by the associate editor name) 1.0In this paper, we study the generalized mean-field stochastic control problem when the usual stochastic maximum principle (SMP) is not applicable due to the singularity of the Hamiltonian function. In this case, we derive a second order SMP. We introduce the adjoint process by the generalized mean-field backward stochastic differential equation. The keys in the proofs are the expansion of the cost functional in terms of a perturbation parameter, and the use of the range theorem for vector-valued measures. § INTRODUCTION We consider the following optimal stochastic control problem of mean-field type with the state equation{[ dX_t=b(t,X_t,P_ X_t,v_t) dt +σ(t,X_t,P_ X_t) dB_t,; X_0=x, ].and the cost functionalJ(v)=𝔼{∫^T_0 h(t,X_t,P_ X_t,v_t)dt+Φ(X_T,P_ X_T)}, where P_ξ denotes the law of the random variable ξ.The agent wishes to minimize his cost functional, namely, an admissible control u∈𝒰 is said to be optimal ifJ(u)=min_v∈𝒰J(v).where 𝒰 is the set of all admissible controls to be defined later in Section 3. About stochastic maximum principle (SMP), some pioneering works have been done byPontryagin et al. <cit.>. They obtained Pontryagin's maximum principle by using “spike variation”.Kushner (<cit.>, <cit.>) studied the SMP in the framework when the diffusion coefficient does not depend on the control variable, and the cost functional consists of terminal cost only. Haussmann <cit.> gave a version of SMP when the diffusion of the state does not depend on the control variable. Arkin and Saksonov <cit.>, Bensoussan<cit.> and Bismut <cit.>, proved different versions of SMP under various setups. An SMP was obtained by Peng <cit.> in 1990. In that paper, first and second order variational inequalities are introduced, when the control domain need not to be convex, and the diffusion coefficient contains the control variable.Pardoux and Peng <cit.> introduced non-linear backward stochastic differential equations (BSDE) in 1990. They showed that under appropriate assumptions, BSDE admits an unique adapted solution, and the associated comparison theorem holds. Buckdahn et al <cit.> obtained mean-field BSDE in a natural way as the limit of some high dimensional system of forward and backward stochastic differential equations. Li <cit.> studied SMP for mean-filed controls when the domain of the control is assumed to be convex. Under some additional assumptions, both necessary and sufficient conditions for the optimality of a control were proved. Buckdahn et al <cit.> studied generalized mean-field stochastic differential equations and the associated partial differential equations (PDEs). “Generalized" means that the coefficients depend on both the state process and its law. They proved that under appropriate regularity conditions on the coefficients, the SDE has a unique classical solution.Buckdahn et al. <cit.> obtained SMP for generalized mean-field system in 2016. Sometimes, the Hamiltonian function becomes constant in the control variable, as we will see in the next example, which makes the aforementioned SMP not applicable.Consider the control problem with state equation:{[dX^v_t = v_tdt+{(X^v_t-1)+𝔼[(X^v_t-1)]}dB_t,v∈U:={-1,0,1},; X^v_0 =1, ].and cost functional:J(v)=1/2𝔼{(X^v_T-1)+𝔼̃[(X^v_T-1)]}^2.For the control u_t≡ 0, X^u_t≡ 1 is the unique solution of (<ref>). It is clear that J(u)=0, and hence, u is an optimal control. On the other hand, the first order adjoint processes satisfy the following equation:{[dp_t = {q_t+𝔼[q_t] }dt-q_tdB_t; p_T =0. ]. Clearly (p_t,q_t)≡ (0,0) is the solution. ThereforeH(t,X^u_t,P_X^u_t,p_t,q_t,v)≡ 0, v∈U.which makes the SMP useless in charactering the optimal control u_t=0.Now, we discuss singular optimal stochastic controls defined as follows.An admissible control ũ(·) is singular on region V if V⊂ U is of positive measure and for a.e. t∈ [0,T] and v∈ V, we have for any v∈ V, H(t,X^ũ_t,P_X^ũ_t,p_t^ ũ,q_t^ ũ,ũ_t)=H(t,X^ũ_t,P_X^ũ_t,p_t^ ũ,q_t^ ũ,v),a.s.As we have seen in last example, the SMP is not very useful under singular control. Our goal is to derive further necessary condition for optimality. We shall call the original SMP as the first order SMP while the one we will derive as the secondorder one.For second-order SMP of singular control problems, Bell <cit.>, Gabasov <cit.>, Kazemi-Dehkordi <cit.>, Krener <cit.>, Mizukami and Wu <cit.> devoted themselves to the deterministic case. Lu <cit.> interested in second order necessary conditions forstochastic evolution system. Tang <cit.> studied the singular optimal control problem for stochastic system with state equation{[ dX_t=b(t,X_t,v_t) dt +σ(t,X_t) dB_t,; X_0=x. ].and the cost functionalJ(v)=𝔼{∫^T_0 h(t,X_t,v_t)dt+Φ(X_T)},By applying spike variation and vector-value measure theory, a second-order maximum principle is presented which involves the second-order adjoint process. In this paper, we study the case when the state equation and the cost functional are in generalized mean-field form. The rest of this paper is organized as follows: In Section 2, we introduce the preliminaries about the generalized mean-field BSDEs. In Section 3, we set up the formulation of the singular optimal stochastic control problem and state the main result of the paper. Section 4 is devoted to the study of the impact of the control actions on the state and the cost functional by using Taylor's expansion. In that section, we also present some estimations about the state. In Section 5, the method in Section 4 is reused for the expansion of the cost functional with respect to the control variable.Sections 6 is devoted to the proof of the second order stochastic maximum principle. § PRELIMINARIES In this section, for the convenience of the reader, we state some results of Buckdahn et al. <cit.> without proofs. Let 𝒫_2(ℝ^n)be the collection of all square integrable probability measures over (ℝ^n,ℬ(ℝ^n)), endowed with the 2-Wasserstein metric W_2, which is defined asW_2(P_μ,P_ν)=inf{(𝔼[|μ'-ν'|^2])^1/2},for all μ',ν' ∈ L^2(ℱ_0;ℝ^d) with P_μ'=P_μ,P_ν'=P_ν. Denote by L^2(ℱ;ℝ^n) the collection of all ℝ^n-valued square integrable random variables. The following definition is taken from Cardaliaguet <cit.>.A function f: 𝒫_2(ℝ^n)⟶ℝ is said to be differentiable in μ∈𝒫_2(ℝ^n) if, the function f̃: L^2(ℱ;ℝ^n)⟶ℝ given by f̃(𝔳)=f(P_𝔳) is differentiable (in Fréchet sense) at 𝔳_0, defined by P_𝔳_0=μ, i.e. there exists a linear continuous mapping Df̃(𝔳_0):L^2( ℱ ;ℝ^n)⟶ℝ, such thatf̃(𝔳_0+η)-f̃(𝔳_0)=Df̃(𝔳_0)(η)+o(|η|_L^2),with |η|_L^2⟶ 0 for η∈ L^2( ℱ ;ℝ^n).According to the Riesz representation theorem, there exists a unique random variable θ_0∈ L^2( ℱ ;ℝ^n) such that D f̃(𝔳_0)(η)=(θ_0,η)_L^2=𝔼[θ_0η], for all η∈L^2( ℱ ;ℝ^n). In <cit.> it has been proved that there is a Borel function h_0:ℝ^n⟶ℝ^n such that θ_0=h_0(𝔳_0) a.s. Then,f(P_𝔳)-f(P_𝔳_0) =𝔼[h_0(𝔳_0)(𝔳-𝔳_0)] +o(|𝔳-𝔳_0|_L^2),𝔳∈ L^2( ℱ ;ℝ^n). We call ∂_μ f( P_𝔳_0,y):=h_0(y),y∈ℝ^n, the derivative of f: 𝒫_2(ℝ^n)⟶ℝ^n at P_𝔳_0. Note that∂_μ f( P_𝔳_0,y) is P_𝔳_0(dy)-a.s. uniquely determined. For mean-field type SDE and BSDE, we introduce the following notations. Let (Ω',ℱ',P') be a copy of the probability space (Ω,ℱ,P). For each random variable ξ over (Ω,ℱ,P) we denote by ξ' a copy of ξ defined over (Ω',ℱ',P'). 𝔼'[·]=∫_Ω'(·)dP' acts only over the variablesω'.We say that f∈ C^1,1_b(𝒫_2(ℝ^d))(continuously differentiable over 𝒫_2(ℝ^d) with Lipschitz-continuous bounded derivative), if for all 𝔳∈ L^2(ℱ,ℝ^d), there exists a P_𝔳-modification of ∂_μf(P_𝔳,·), again denote by ∂_μf(P_𝔳,·), such that ∂_μf:𝒫_2(ℝ^d)×ℝ^d⟶ℝ^d is bounded and Lipschitz continuous, i.e., there is a real constant C such thati)|∂_μf(μ ,x)|≤ C,∀μ∈𝒫_2( ℝ^d), x∈ℝ^d, ii)|∂_μf(μ ,x)-∂_μf(μ' ,x')|≤ C(W_2(μ,μ')+|x-x'|), ∀μ, μ' ∈𝒫_2( ℝ^d), x,x'∈ℝ^d;we call this function ∂_μf the derivative of f. Given f∈ C^1,1_b(𝒫_2(ℝ^d)), and y∈ℝ^d, the question of the differentiability of its components (∂_μf)_j(· , y): 𝒫_2(ℝ^d)→ℝ,1≤ j ≤ d, raises. This can be discussed in the same way as the first order derivative ∂_μf above. If (∂_μf)_j(· , y): 𝒫_2(ℝ^d)→ℝ belongs to C^1,1_b(𝒫_2(ℝ^d)), we have that its derivative ∂_μ((∂_μf)_j(· , y))(· , ·): 𝒫_2(ℝ^d)×ℝ^d→ℝ^d is a Lipschitz-continuous function. Then∂^2_μf(μ, x,y):=(∂_μ((∂_μf)_j(· , y))(μ , x))_1≤ j ≤ d,(μ ,x,y)∈𝒫_2(ℝ^d)×ℝ^d×ℝ^d,defines a function ∂^2_μf:𝒫_2(ℝ^d)×ℝ^d×ℝ^d→ℝ^d⊗ℝ^d. We say that f∈ C^2,1_b(𝒫_2(ℝ^d)), if f∈ C^1,1_b(𝒫_2(ℝ^d)) and i) (∂_μf)_j(· , y)∈ C^1,1_b(𝒫_2(ℝ^d)), for all y∈ℝ^d,1≤ j ≤ d, and ∂^2_μf: 𝒫_2(ℝ^d)×ℝ^d×ℝ^d→ℝ^d⊗ℝ^d is bounded and Lipschitz-continuous;ii)(∂_μf)(μ , ·): ℝ^d→ℝ^d is differentiable for every μ∈𝒫_2(ℝ^d), and its derivative ∂_y ∂_μf:𝒫_2(ℝ^d)×ℝ^d→ℝ^d⊗ℝ^d is bounded and Lipschitz-continuous.For twice continuously differentiable functions h:ℝ^d→ℝ and g:ℝ→ℝ with bounded derivatives. Consider f(P_𝔳):=g(𝔼[h(𝔳)]), 𝔳∈ L^2(ℱ;ℝ^d). Then, given any 𝔳_0∈L^2(ℱ;ℝ^d), f̃(𝔳):=f(P_𝔳)=g(𝔼[h(𝔳)]) is Fréchet differentiable in 𝔳_0, andf̃(𝔳_0+η)-f̃(𝔳_0) = ∫^1_0 g'(𝔼[h(𝔳_0+sη)])𝔼[h'(𝔳_0+sη)η]ds =g'(𝔼[h(𝔳_0)])𝔼[h'(𝔳_0)η]+o(|η|_L^2) =𝔼[g'(𝔼[h(𝔳_0)])h'(𝔳_0)η]+o(|η|_L^2).So, Df̃(𝔳_0)(η)=𝔼[g'(𝔼[h(𝔳_0)])h'(𝔳_0)η], η∈ L^2(ℱ;ℝ^d), i.e.,∂_μf(P_𝔳_0,y)=g'(𝔼[h(𝔳_0)])(∂_y h)(y), y∈ℝ^d.Similarly, we see that∂_μ^2f(P_𝔳_0,x,y)=g”(𝔼[h(𝔳_0)])(∂_xh)(x)×(∂_yh)(y),and∂_y∂_μf(P_𝔳_0,y) =g'(𝔼[h(𝔳_0)])(∂_y^2h)(y). Let us now consider a complete probability space (Ω,ℱ,P) on which we define a d-dimensional Brownian motion B=(B^1,⋯,B^d)=(B_t)_t∈ [0,T], where T≥ 0 denotes an arbitrarily fixed time horizon. We make the following assumptions: There is a sub-σ-field ℱ_0⊂ℱ such thati) the Brownian motion B is independent of ℱ_0, andii) ℱ_0 is “rich enough", i.e., 𝒫_2(ℝ^d)={ P_𝔳, 𝔳∈ L^2(ℱ_0; ℝ^d) }.By 𝔽=(ℱ_t)_t∈ [0,T] we denote the filtration generated by B, completed and augmented by ℱ_0.Given deterministic Lipschitz functions σ: ℝ^d×𝒫_2(ℝ^d)⟶ℝ^d× d andb: ℝ^d×𝒫_2(ℝ^d)⟶ℝ^d, we consider for the initial state (t,x)∈ [0,T]×ℝ^d and ξ∈ L^2(ℱ_t;ℝ^d) the stochastic differential equations (SDEs)X^t,ξ_s=ξ+∫^s_t σ(X^t,ξ_r,P_X^t,ξ_r)dB_r+∫^s_t σ(X^t,ξ_r,P_X^t,ξ_r)dr, s∈[t,T],andX^t,x,ξ_s=x+∫^s_t σ(X^t,x,ξ_r,P_X^t,ξ_r)dB_r+∫^s_t σ(X^t,x,ξ_r,P_X^t,ξ_r)dr, s∈[t,T].It is well-known that under the assumptions above both SDEs have unique solutions in 𝒮^2([t,T];ℝ^d), which is the space of 𝔽-adapted continuous processes Y=(Y_s)_s∈[t,T] with 𝔼[sup_s∈ [t,T]|Y_s|^2]≤∞.The couple of coefficients (σ,b) belongs to C^1,1_b(ℝ^d×𝒫_2(ℝ^d)⟶ℝ^d× d×ℝ^d), i.e., the components σ_i,j,b_j, 1≤ i,j ≤ d, satisfy the following conditions:i) σ_i,j(x,·), b_j(x,·) belong to C^1,1_b( 𝒫_2(ℝ^d)), for all x∈ℝ^dii) σ_i,j(·,μ), b_j(·,μ) belong to C^1_b(ℝ^d), for all μ∈𝒫_2(ℝ^d)iii) The derivatives ∂_xσ_i,j, ∂_xb_j:ℝ^d×𝒫_2(ℝ^d)⟶ℝ^d, ∂_μσ_i,j, ∂_μb_j:ℝ^d×𝒫_2(ℝ^d)×ℝ^d⟶ℝ^d,are bounded and Lipschitz continuous.The couple of coefficient (σ,b) belongs to C^2,1_b(ℝ^d×𝒫_2(ℝ^d)⟶ℝ^d× d×ℝ^d), i.e., (σ,b)∈ C^1,1_b(ℝ^d×𝒫_2(ℝ^d)⟶ℝ^d× d×ℝ^d) and the components σ_i,j,b_j, 1≤ i,j ≤ d, satisfies the following conditions:i) ∂_x_kσ_i,j(·,·), ∂_x_k b_j(·,·) belong to C^1,1_b( ℝ^d ×𝒫_2(ℝ^d)), for all 1≤ k ≤ d;ii) ∂_μσ_i,j(·,·,·), ∂_μb_j(·, ·,·) belong to C^1,1_b(ℝ^d ×𝒫_2(ℝ^d)×ℝ^d), for all μ∈𝒫_2(ℝ^d)iii) All the derivatives of σ_i,j, b_j, up to order 2 are bounded and Lipschitz continuous. The following theorem is taken from <cit.>. It gives the Itô formula related to a probability measure.Let Φ∈ C^2,1_b(ℝ^d×𝒫_2(ℝ^d)). Then, under Hypothesis <ref>, for all 0≤ t ≤ s ≤ T, x∈ℝ^d, ξ∈ L^2(ℱ_t;ℝ^d) the Itô formula is satisfied as follow:Φ(X^t,x,P_ξ_s, P_X^t,ξ_s)-Φ(x,P_ξ)= ∫^s_t(∑^d_i=1∂_x_iΦ(X^t,x,P_ξ_r, P_X^t,ξ_r)b_i(X^t,x,P_ξ_r, P_X^t,ξ_r) +1/2∑^d_i,j,k=1∂^2_x_i,x_jΦ(X^t,x,P_ξ_r, P_X^t,ξ_r)(σ_i,kσ_j,k)(X^t,x,P_ξ_r, P_X^t,ξ_r) +𝔼'[∑^d_i=1(∂_μΦ)_i(X^t,x,P_ξ_r, P_X^t,ξ_r,(X^t,ξ_r)')b_i((X^t,ξ_r)', P_X^t,ξ_r) +1/2∑^d_i,j,k=1∂_y_i((∂_μΦ)_j(X^t,x,P_ξ_r, P_X^t,ξ_r,(X^t,ξ_r)')(σ_i,kσ_j,k)((X^t,ξ_r)', P_X^t,ξ_r)])dr +∫^s_t ∑^d_i,j=1∂_x_iΦ(X^t,x,P_ξ_r, P_X^t,ξ_r)σ_i,j(X^t,x,P_ξ_r, P_X^t,ξ_r)dB^j_r,s∈ [t,T].For simplicity, we will make use of the following notations concerning matrices. We denote by ℝ^n× d the space of real matrices of n× d-type, and by ℝ^n× n_d the linear space of the vectors of matrices M=(M_1,⋯, M_d), with M_i∈ℝ^n× n, 1≤ i ≤ d. Given any α, β∈ℝ^n,L, S ∈ℝ^n× d,γ∈ℝ^d andM, N ∈ℝ^n× n_d, we introduce the following notation: αβ=∑^n_i=1α_i β_i ∈ℝ, α×β=(α_i β_j)_1≤ i,j≤ n; LS=∑^d_i=1 L_i S_i ∈ℝ, where L=(L_1,⋯, L_d), S=(S_1,⋯, S _d); ML=∑^d_i=1M_i L_i ∈ℝ^n; Mαγ=∑^d_i=1(M_i α) γ_i ∈ℝ^n; MN=∑^d_i=1M_i N_i ∈ℝ^n× n;For mean-field type SDE and BSDE, we have still to introduce some notations. Let (Ω̃,ℱ̃,P̃), (Ω̅,ℱ̅,P̅) be two copies of the probability space (Ω,ℱ,P). For any random variable ξ over (Ω,ℱ,P), we denote by ξ̃ and ξ̅ its copies on Ω̃ and Ω̅, respectively, which means that they have the same law as ξ, but defined over(Ω̃,ℱ̃,P̃) and (Ω̅,ℱ̅,P̅). Ẽ[·]=∫_Ω̃(·)dP̃ and 𝔼̅[·]=∫_Ω̅(·)dP̅ act only over the variables fromω̃ and ω̅, respectively. § FORMULATION OF THE SINGULAR OPTIMAL STOCHASTIC CONTROL PROBLEM AND THE MAIN RESULT In this section, weformulateour generalized mean-field optimal control problem and state the main result of this article. Let (Ω,ℱ,P) be a probability space with filtration ℱ_t. Suppose that B_tis a Brownian motion on (Ω,ℱ,P), where ℱ is the filtration generated by B_t,augmented by all P-null sets. Let 𝒰 denote the admissible control set consisting of ℱ_t-adapted process u_t, take values in U, such that sup_0≤ t≤ T𝔼|u_t|^8 < ∞, where U is a subset of ℝ^k. Let b:[0,T]×ℝ^n× 𝒫_2(ℝ^n) × U ⟶ℝ^n, σ:[0,T]×ℝ^n× 𝒫_2(ℝ^n) ⟶ℝ^n× d,h:[0,T]×ℝ^n× 𝒫_2(ℝ^n) × U ⟶ℝ, and Φ:ℝ^n×𝒫_2(ℝ^n) ⟶ℝ. The state equation and the cost functional are defined by (<ref>) and (<ref>). Throughout this paper, we make the following assumptions on the coefficients: (1) The functions b, σ, h ,Φ are differentiable with respect to (x,μ, v). b,σ satisfy Lipschitz condition with respect to (x,μ,v).(2) The first-order derivatives with respect to (x,μ) of b, σ are Lipschitz continuous and bounded.(3)The first-order derivativeswith respect to (x,μ) of h, Φ are Lipschitz continuous and bounded by C(1+|x|+|v| ).(4) The second-order derivatives with respect to (x,μ) of b, σ, h ,Φ are continuous and bounded. All the second-order derivatives are Borel measurable with respect to (t,x,μ,v). Suppose that u is an optimal control and X^u is the associated trajectory. We are to find the necessary conditions satisfied by u. Firstly, we introduce the following abbreviations: b(t):=b(t,X^u_t, P_X^u_t,u_t), b_x(t):=b_x(t,X^u_t,P_X^u_t,u_t),b̃(t):=b(t,X̃^u_t, P_X̃^u_t,ũ_t),b̃_x(t):=b_x(t,X̃^u_t,P_X̃^u_t,ũ_t),b_xx(t):=b_xx(t,X^u_t,P_X^u_t,u_t), b_μ(t):=b_μ(t,X^u_t,P_X^u_t,X̃^u_t, u_t),b̃_μ(t):=b_μ(t,X̃^u_t,P_X̃^u_t,X^u_t, ũ_t), b_μμ(t):=b_μμ(t,X^u_t,P_X^u_t,X̃^u_t, X̅^u_t, u_t),b̃̅̃_μμ(t):=b_μμ(t,X̃^u_t,P_X̃^u_t,X̅^u_t, X^u_t, ũ_t),b_xμ(t):=b_x μ(t,X^u_t,P_X^u_t,X̃^u_t, u_t),b_yμ(t):=b_y μ(t,X^u_t,P_X^u_t,X̃^u_t, u_t),△ b(t;v):=b(t,X^u_t,P_X^u_t,v)-b(t),△ b_x(t;v):=b_x(t,X^u_t,P_X^u_t,v)-b_x(t),△ b_μ(t;v):=b_μ(t,X^u_t,P_X^u_t,X̃^u_t, v)-b_μ(t),Similar shorthand notations for the second-order derivatives and those about σ, h can also be introduced.Consider the first order adjont process{[-dp_t= { b_x(t)p_t+σ_x(t)q_t+h_x(t);+𝔼̃[b̃_μ(t)p̃_t+σ̃_μ(t)q̃_t+h̃_μ(t)] } dt-q_tdB_t,;p_T= Φ_x(X^u_T,P_X^u_T)+𝔼̃[Φ_μ(X̃^u_T,P_X̃^u_T,X^u_T)]. ].According to Theorem 3.1 <cit.>, this BSDE admit a unique adapted solution. We also denote the solution as (p^u_t, q^u_t). Define the Hamiltonian as follows:H(t,x,μ ,p,q,v)=pb(t,x,μ,v)+qσ(t,x,μ)+h(t,x,μ,v) The following first-order SMP is obtained as a special case of <cit.>. Let Hypothesis <ref> hold. Suppose that X^u_t is the associated trajectory of the optimal control u, and (p,q) is the solution to the mean-field backward stochastic differential equation (MFBSDE) (<ref>). Then, there is a subset I_0 ⊂ [0,T] which is of full measure such that∀ t∈ I_0,H(t,X^u_t,P_X^u_t,p_t,q_t,u_t)=inf_v∈𝒰H(t,X^u_t,P_X^u_t,p_t,q_t,v), a.s.. As we pointed out in the introduction, the aim of this article is to derive another SMP when the Hamiltonian functionabove becomes singular, and hence, the SMP above is not suitable for characterizing of the optimal control u_t. To this end, we define the second-order adjoint process as follows:{[ dP_t=-{b^*_x(t)P_t+P_tb_x(t)+𝔼̃[b̃_μ^*(t)]P_t +P_t𝔼̃[b̃_μ(t) ];+σ^*_x(t)P_tσ_x(t) +𝔼̃[σ̃^*_μ(t)]P_t𝔼̃[σ̃_μ(t) ];+σ^*_x(t)P_t𝔼̃[σ̃_μ(t) ]+𝔼̃[σ̃^*_μ(t)]P_t σ_x(t);+σ^*_x(t)Q_t+P_tσ_x(t)+𝔼̃[σ̃_μ^*(t)]Q_t +Q_t𝔼̃[σ̃_μ(t) ];+H_xx(t)+𝔼̃𝔼̅[H̃̅̃_μμ(t) ]+𝔼̃[H̃_yμ(t) ]+2𝔼̃[H̃_xμ(t) ]}dt; +Q_tdB_t,;P_T= 0. ].By changing the terminal condition p_T, we can always eliminate the terminal cost when deducing the variational inequality. In fact, the terminal condition P_T=0 is due to the assumption that Φ≡ 0. Without this assumption, we only need to setP_T = Φ_xx(X^u_T,P_X^u_T)+2𝔼̃[Φ_xμ(X^u_T,P_X^u_T,X̃^u_T)]+𝔼̅𝔼̃[Φ_μμ(X^u_T,P_X^u_T,X̃^u_T,X̅^u_T)]+𝔼̃[Φ_yμ(X^u_T,P_X^u_T,X̃^u_T)].Without loss of generality, we assume the terminal cost Φ≡ 0 in the following sections. Finally, we present our main result in this article.Assume that Hypothesis <ref> hold. Let (X^u_·, u_·) be an optimal pair and let u_· be singular on the control region V. Suppose that (P, Q) is the unique adapted solution of equation (<ref>). Then, there is a full measure subset I_0⊂ [0,T] such that at each t∈ I_0, (X^u_·, u_·) satisfies, not only the first-order stochastic maximum principle, but also the following inequality △ H_x(t;v)△ b(t;v)+𝔼̃[△ H_μ(t;v)△b̃(t;v)] +△b^*(t;v) P_t△ b(t;v)≥ 0, ∀ v∈ U, a.s..§ QUANTITATIVE ANALYSIS OF THE IMPACT OF CONTROL ACTIONS ON THE STATEIn this section, we expand the state process according to different orders of the perturbation parameter d(u,v), a distance between the optimal control u and its perturbation v. Under Hypothesis <ref> on the coefficients, we have,𝔼sup_0≤ t ≤ T|X^v_t|^8 ≤ K(1+𝔼|∫^T_0 |v_s|ds|^8 ), By the state equation (<ref>), for τ∈[0,T] we have,𝔼sup_0≤ t ≤τ|X^v_t|^8≤ K 𝔼(|x|^8 +sup_0≤ t ≤τ|∫^t_0 b(s,X^v_s,P_ X^v_s,v_s) ds|^8 +(∫^τ_0 |σ(s,X^v_s,P_ X^v_s)|^2 ds)^4)≤ K(|x|^8+ 𝔼∫^τ_0sup_0≤ s ≤ r| X^v_s|^8 dr+ 𝔼|∫^T_0 |v_s| ds|^8)From Gronwall's inequality, we then have the desired result.For v_i ∈𝒰, i=1,2, we defineI(v_1,v_2)={ t∈ [0,T] |P({ω : v_1(t)≠ v_2(t)} )>0 }and d(v_1,v_2)=|I(v_1,v_2)| is the Lebesgue measure of I(v_1,v_2). Then, (𝒰,d) is a metric space.Given the optimal pair (X^u_·, u_·), we now proceed to the perturbation X^v of X^u. Let X^v,1_t = ∫^t_0 { b_x(s)X^v,1_s+𝔼̃[b_μ(s) X̃^v,1_s]+△ b(s,v) }ds +∫^t_0{σ_x(s)X^v,1_s+𝔼̃[σ_μ(s) X̃^v,1_s]}dB_s:= ∫^t_0b^1(s,v) ds+∫^t_0 σ^1(s,v)dB_s,and X^v,2_t = ∫^t_0{ b_x(s)X^v,2_s+𝔼̃[b_μ(s) X̃^v,2_s]+△ b_x(s,v)X^v,1_s+𝔼̃[△ b_μ(s,v)X̃^v,1_s]+1/2 b_xx(s)X^v,1_s× X^v,1_s+ 𝔼̃[ b_x μ(s)X^v,1_s×X̃^v,1_s] +1/2𝔼̃[ b_y μ(s)X̃^v,1_s×X̃^v,1_s]+1/2𝔼̅𝔼̃[ b_μμ(s)X̃^v,1_s×X̅^v,1_s] }ds +∫^t_0 {σ_x(s)X^v,2_s+𝔼̃[σ_μ(s) X̃^v,2_s] +1/2σ_xx(s)X^v,1_s× X^v,1_s+𝔼̃[ σ_x μ(s)X^v,1_s×X̃^v,1_s] +1/2𝔼̃[ σ_y μ(s)X̃^v,1_s×X̃^v,1_s]+1/2𝔼̅𝔼̃[ σ_μμ(s)X̃^v,1_s×X̅^v,1_s] }dB_s:= ∫^t_0b^2(s,v) ds+∫^t_0 σ^2(s,v)dB_s DenoteX^v*_.:=X^*_.(v)=X^u_.-X^v,1_.-X^v,2_.-X^v_..The following lemmas give the estimation of their orders according to parameter d(v,u).Assume that Hypothesis <ref> holds. Then, there exists a K> 0, such that for any v(·), u(·)∈𝒰, we have 𝔼sup_0≤ t ≤ T|X^v,1_t|^2 ≤ Kd^2(v,u), 𝔼sup_0≤ t ≤ T|X^v,2_t|^2 ≤ Kd^4(v,u). For any τ∈[0,T], denote g_1(τ)=𝔼sup_0≤ t ≤τ|X^v,1_t|^2,g_2(τ)=𝔼sup_0≤ t ≤τ|X^v,2_t|^2.By Hypothesis 3.1 and the Burkholder-Davis-Gundy inequality, we haveg_1(τ)≤ K(∫^τ_0 g_1(s) ds+𝔼|∫^T_0|△ b(s;v)|ds|^2),andg_2(τ) ≤ K(∫^τ_0 g_2(s) ds+[g_1(T)]^2 +𝔼|∫^T_0|△ b_x(s;v)|ds|^4+𝔼𝔼̃|∫^T_0|△ b_μ(s;v)|ds|^4).The application of Grownwall's inequality allows to obtain that𝔼sup_0≤ t ≤ T|X^v,1_t|^2 ≤K(𝔼|∫^T_0|△ b(s;v)|ds|^2), 𝔼sup_0≤ t ≤ T|X^v,2_t|^2 ≤K(𝔼|∫^T_0|△ b(s;v)|ds|^4+𝔼|∫^T_0 |△ b_x(s;v)|ds|^4 +𝔼𝔼̃|∫^T_0|△ b_μ(s;v)|ds |^4)Notice that the first-order derivative b_x is bounded. Then, (<ref>) implies the following estimate𝔼sup_0≤ t≤ T|X^v,2_t|^2 ≤ K(𝔼|∫^T_0|△ b(s;v)|ds|^4+d^4(u,v)).According to assumption about v and u, then,𝔼sup_0≤ t≤ T|X^v,1_t|^2 ≤ Kd^2(v, u), 𝔼sup_0≤ t≤ T|X^v,2_t|^2 ≤ Kd^4(v, u). In fact, by Minkowski's inequality, we have𝔼|∫^T_0|△ b(s;v)|ds|^4≤|∫^T_0 (𝔼|△ b(s;v)|^4)^1/4ds|^4= |∫_I(u,v)(𝔼|△ b(s;v)|^4)^1/4ds|^4≤ Kd^4(u,v). The following lemma gives the order of X^v*_t.Assume Hypothesis <ref> holds. For v(·)∈𝒰 and Borel subset I_ρ⊂ [0,T] with Lebesgue measure |I_ρ|, define v̂_t=v_t1_I_ρ(t)+u_t1_[0,T]∖ I_ρ(t), X^v̂*_t:=X^*(t,v̂).Then we have 𝔼sup_0≤ t ≤ T|X^v̂*_t|^2=o(|I_ρ|^4) when|I_ρ|→ 0. We introduce the following notations first△ b_xx(s;λη;v):=b_xx(s,X^u_s+λη X^v,12_s, P_X^u_s+λη X^v,12_s,v_s)-b_xx(s) △ b_μμ(s;λη;v):=b_μμ(s,X^u_s, P_X^u_s+λη X^v,12_s,X̃^u_s+ληX̃^v,12_s,X̅^u_s+ληX̅^v,12_s,v_s)-b_xx(s), △ b_μ y(s;λη;v)=b_μ y(s,X^u_s ,P_X^u_s+λη X^v,12_s,X̃^u_s+ληX̃^v,12_s,v_s)-b_μ y(s) △ b_xμ(s;λ;v):=b_xμ(s,X^u_s, P_X^u_s+λ X^v,12_s,X̃^u_s+λX̃^v,12_s,v_s)-b_xμ(s),where X^v,12_·:=X^v,1_·+X^v,2_·. Similarly notations can be introduced with b replaced by σ.We now proceed to estimating X^*_.(v̂) defined by (<ref>). By (<ref>), (<ref>) and (<ref>), we havedX^v̂*_t=α(t)dt+β(t)dB_t,whereα(t)=b(t,X^v̂,P_X^v̂,v̂_t)-[b(t) +b^1(t,v̂)+b^2(t,v̂)],β(t)=σ(t,X^v̂_t,P_X^v̂_t)-[σ(t) +σ^1(t,v̂)+σ^2(t,v̂)].We can represent α(t) as follows. α(t) = b(t,X^v̂_t,P_X^v̂_t,v̂_t)-[b(t) +{ b_x(t)X^v̂,1_t+𝔼̃[b_μ(t) X̃^v̂,1_t]+△ b(t,v̂)} +{ b_x(t)X^v̂,2_t+𝔼̃[b_μ(t) X̃^v̂,2_t]+△ b_x(t,v̂)X^v̂,1_t+𝔼̃[△ b_μ(t,v̂)X̃^v̂,1_t]+1/2 b_xx(t)X^v̂,1_t× X^v̂,1_t+ 𝔼̃[ b_x μ(t)X^v̂,1_t×X̃^v̂,1_t] +1/2𝔼̃[ b_y μ(t)X̃^v̂,1_t×X̃^v̂,1_t]+1/2𝔼̅𝔼̃[ b_μμ(t)X̃^v̂,1_t×X̅^v̂,1_t] }]. DenoteA(t;v̂) = 1/2 b_xx(t)(X^v̂,2_t× X^v̂,2_t+2X^v̂,1_t × X^v̂,2_t) +1/2𝔼̃[ b_μ y(t)(X̃^v̂,2_t×X̃^v̂,2_t+2X̃^v̂,1_t ×X̃^v̂,2_t)] +1/2𝔼̃𝔼̅[ b_μμ(t)(X̃^v̂,2_t×X̅^v̂,2_t+X̃^v̂,1_t ×X̅^v̂,2_t+X̅^v̂,1_t ×X̃^v̂,2_t)] +△ b_x (t;v̂)X^v̂,2_t+𝔼̃[△ b_μ (t;v̂)X̃^v̂,2_t] +𝔼̃[b_xμ(t)(X^v̂,2_t×X̃^v̂,2_t+X^v̂,1_t ×X̃^v̂,2_t+X̃^v̂,1_t × X^v̂,2_t)] +∫^1_0 ∫^1_0 λ[△ b_xx(t;λη;v̂)d λ dη X^v̂,12_t × X^v̂,12_t ] +∫^1_0 ∫^1_0𝔼̅𝔼̃[λ (△ b_μμ(t;λη;v̂) d λ dηX̃^v̂,12_t ×X̅^v̂,12_t] +∫^1_0 ∫^1_0𝔼̃[λ (△ b_μ y(t;λη;v̂)d λ dηX̃^v̂,12_t ×X̃^v̂,12_t] +𝔼̃[∫^1_0 △ b_xμ(t;λ;v̂)dλ X^v̂,12_t×X̃^v̂,12_t].It is easy to show thatα(t) =b(t,X^v̂_t,P_X^v̂_t,v̂_t)-[b(t,X^u_t,P_X^u_t+X^v̂,12_t,v̂_t) + b_x(t,X^u_t,P_X^u_t+X^v̂,12_t,v̂_t)X^v̂,12_t+∫^1_0 ∫^1_0 λ b_xx(t,X^u_t+λη X^v̂,12_t,P_X^u_t+X^v̂,12_t,v̂_t)d λ dη X^v̂,12_t× X^v̂,12_t] +A(t;v̂)=b(t,X^v̂_t,P_X^v̂_t,v̂_t)-b(t,X^u_t+X^v̂,12_t,P_X^u_t+X^v̂,12_t,v̂_t)+A(t;v̂). Simularly, by setting B(t;v̂) := 1/2σ_xx(t)(X^v̂,2_t× X^v̂,2_t+2X^v̂,1_t × X^v̂,2_t) +1/2𝔼̃[ σ_μ y(t)(X̃^v̂,2_t×X̃^v̂,2_t+2X̃^v̂,1_t ×X̃^v̂,2_t)] +1/2𝔼̃𝔼̅[ σ_μμ(t)(X̃^v̂,2_t×X̅^v̂,2_t+X̃^v̂,1_t×X̅^v̂,2_t+X̅^v̂,1_t ×X̃^v̂,2_t)] +𝔼̃[σ_xμ(t)(X^v̂,2_t×X̃^v̂,2_t+X^v̂,1_t ×X̃^v̂,2_t+X̃^v̂,1_t × X^v̂,2_t)] +∫^1_0 ∫^1_0 λ[△σ_xx(t;λη;v̂)d λ dη X^v̂,12_t × X^v̂,12_t ] +∫^1_0 ∫^1_0𝔼̅𝔼̃[λ△σ_μμ(t;λη;v̂) d λ dηX̃^v̂,12_t ×X̅^v̂,12_t] +∫^1_0 ∫^1_0𝔼̃[λ△σ_μ y(t;λη;v̂)d λ dηX̃^v̂,12_t ×X̃^v̂,12_t] +𝔼̃[∫^1_0 (△σ_xμ(t;λ;v̂)dλ X^v̂,12_t×X̃^v̂,12_t)dλ X^v̂,12_t×X̃^v̂,12_t],we have β(t) = σ(t,X^v̂_t,P_X^v̂_t)-[σ(t,X^u_t,P_X^u_t+X^v̂,12_t)+ σ_x(t,X^u_t,P_X^u_t+X^v̂,12_t)X^v̂,12_t+∫^1_0 ∫^1_0 λσ_xx(t,X^u_t+λη X^v̂,12_t,P_X^u_t+X^v̂,12_t)d λ dη X^v̂,12_t× X^v̂,12_t] +B(t;v̂)= σ(t,X^v̂_t,P_X^v̂_t)-σ(t,X^u_t+X^v̂,12_t,P_X^u_t+X^v̂,12_t)+B(t;v̂).According to Hypothesis <ref>, we have|b(t,X^v̂_t,P_X^v̂_t,v̂_t)-b(t,X^u_t+X^v̂,12_t,P_X^u_t+X^v̂,12_t,v̂_t)|≤K(|X^v̂*_t|+W_2(P_X^v̂_t,P_X^u_t+X^v̂,12_t)). Note that W_2(P_X^v̂_s, P_X^u_s+ X^v̂,12_s)^2≤ 𝔼|X^v̂_s-X^u_s-X^v̂,12_s|^2=𝔼|X^v̂*_s|^2. ByBurkholder-Davis-Gundy inequality, for τ∈[0,T], we obtain the following estimation𝔼sup_0 ≤ t ≤τ|X^v̂*_t|^2≤ ∫ ^τ_0 K 𝔼sup_0≤ r ≤s |X^v̂*_r|^2 ds+𝔼∫^T_0 |A(s;v̂)|^2ds+𝔼∫^T_0 |B(s;v̂)|^2ds. According to Gronwall's inequality, we have 𝔼sup_0 ≤ t ≤ T|X^v̂*_t|^2≤K (𝔼∫^T_0 |A(s;v̂)|^2ds +𝔼∫^T_0 |B(s;v̂)|^2ds).About A(s;v̂) we have𝔼∫^T_0|A(s;v̂)|^2ds≤ K 𝔼( sup_0≤ s ≤ t|X^v̂,2_s|^4+sup_0≤ s ≤ t|X^v̂,1_s|^2sup_0≤ s ≤ t|X^v̂,2_s|^2 +sup_0≤ s ≤ t|X^v̂,2_s|^2∫^T_0[|△ b_x(s;v̂)|^2 +|△ b_μ(s;v̂)|^2]ds +sup_0≤ s ≤ t|X^v̂,1_s|^4∫^T_0∫^1_0 ∫^1_0 |λ△ b_xx(s;λη;v̂)|^2 d λ dη ds+sup_0≤ s ≤ t|X^v̂,1_s|^4∫^T_0∫^1_0 ∫^1_0𝔼̅𝔼̃|λ△ b_μμ(s;λη;v̂)|^2· d λ dη ds+sup_0≤ s ≤ t|X^v̂,1_s|^4∫^T_0∫^1_0 ∫^1_0𝔼̃|λ△ b_μ y(s;λη;v̂)|^2d λ dη ds +sup_0≤ s ≤ t|X^v̂,1_s|^4𝔼̃∫^T_0∫^1_0 |△ b_xμ(s;λ;v̂)|^2dλ ds).Note that𝔼|∫^T_0|△ b(s;v̂)|ds|^8≤ K|I_ρ|^8,similar estimates hold with b replaced by b_x and b_μ. Since X^v̂,12_t→ 0 as |I_ρ|→ 0, so we also have△ b_xx(s;λη;v̂)→ 0,replace b_xx by b_μμ, b_μ y and b_xμ, we can get the similar result when |I_ρ|→ 0. According to estimation of X^v̂,1_·,X^v̂,2_· in Lemma <ref>, we obtain𝔼∫^T_0 |A(s;v̂)|^2ds ≤ K(√(𝔼|∫^T_0|△ b(s;v̂)|ds|^8)+√(𝔼|∫^T_0|△ b_x(s;v̂)|ds|^8) +√(𝔼𝔼̃|∫^T_0|△ b_μ(s;v̂)|ds|^8))×(√(𝔼|∫^T_0|△ b(s;v̂)|ds|^8) +√(𝔼|∫^T_0|△ b_x(s;v̂)|ds|^8)+√(𝔼𝔼̃|∫^T_0|△ b_μ(s;v̂)|ds|^8) + √(𝔼∫^T_0|△ b(s;v̂)|^8ds)+√(𝔼∫^T_0|△ b_x(s;v̂)|^4ds) +√(𝔼𝔼̃∫^T_0|△ b_μ(s;v̂)|^4ds)+√(𝔼∫^T_0 ∫^1_0∫^1_0| λ△ b_xx(s;λη;v̂)|^4d λ dη ds) +√(𝔼𝔼̃𝔼̅∫^T_0 ∫^1_0∫^1_0| λ△ b_μμ(s;λη;v̂)|^4d λ dη ds) +√(𝔼𝔼̃𝔼̅∫^T_0 ∫^1_0∫^1_0 |λ△ b_μ y(s;λη;v̂)|^4d λ dη ds) +√(𝔼𝔼̃∫^T_0 ∫^1_0 |λ△ b_xμ(s;λ;v̂)|^4d λ ds))=o(|I_ρ|^4),|I_ρ|→ 0.Similarly 𝔼∫^T_0 |B(s;v̂)|^2ds=o(|I_ρ|^4),|I_ρ|→ 0.Finally, by (<ref>) we have the desire result.§ EXPANSION OF THE COST FUNCTIONAL WITH RESPECT TO CONTROL VARIABLEIn this section, we use the method of Lemma <ref> again to study the expansion of the cost functional according to different order of the purtubation.Assume that Hypothesis <ref> holds. Define J^*(v_·) = J(v_·)-J(u_·)-𝔼∫^t_0 { h_x(s)X^v,1_s+𝔼̃[h_μ(s) X̃^v,1_s]+△ h(s,v) +h_x(s)X^v,2_s+𝔼̃[h_μ(s) X̃^v,2_s]+△ h_x(s,v)X^v,1_s+𝔼̃[△ h_μ(s,v)X̃^v,1_s]+1/2 h_xx(s)X^v,1_s× X^v,1_s+ 𝔼̃[ h_x μ(s)X^v,1_s×X̃^v,1_s] +1/2𝔼̃[ h_y μ(s)X̃^v,1_s×X̃^v,1_s]+1/2𝔼̅𝔼̃[ h_μμ(s)X̃^v,1_s×X̅^v,1_s] }ds. Recall that v̂ is defined by (<ref>) by I_ρ. Then,when|I_ρ|→ 0, we haveJ^*(v̂_·)=o(|I_ρ|^2). Denote Y^v_t=∫^t_0 h(s,X^v_s,P_X^v_s,v_s)ds. By (<ref>) and Lemma 4.3, we haveY^v_t=Y^u_t+Y^v,1_t+Y^v,2_t+Y^v*_t,where Y^v_t = ∫^t_0 h(s,X^v_s,P_X^v_s,v_s)ds, Y^v,1_t = ∫^t_0 { h_x(s)X^v,1_s+𝔼̃[h_μ(s) X̃^v,1_s]+△ h(s,v)} ds,Y^v,2_t =∫^t_0{ h_x(s)X^v,2_s+𝔼̃[h_μ(s) X̃^v,2_s]+△ h_x(s,v)X^v,1_s+𝔼̃[△ h_μ(s,v)X̃^v,1_s]+1/2 h_xx(s)X^v,1_s× X^v,1_s+ 𝔼̃[ h_x μ(s)X^v,1_s×X̃^v,1_s] +1/2𝔼̃[ h_y μ(s)X̃^v,1_s×X̃^v,1_s]+1/2𝔼̅𝔼̃[ h_μμ(s)X̃^v,1_s×X̅^v,1_s] }ds. Then, J(v_·)-J(u_·)=𝔼Y^v_T-𝔼Y^u_T, and hence, J^*(v̂_·)=𝔼Y^v_T-𝔼Y^u_T-𝔼Y^v,1_T-𝔼Y^v,2_T.Using the same method in Lemma <ref>, we complete the proof.Now, we proceed to deriving the expansion of the perturbed cost function.X^v,12_t = ∫^t_0 { b_x(s)X^v,1_s+𝔼̃[b_μ(s) X̃^v,1_s]+△ b(s,v) +b_x(s)X^v,2_s+𝔼̃[b_μ(s) X̃^v,2_s]+△ b_x(s,v)X^v,1_s+𝔼̃[△ b_μ(s,v)X̃^v,1_s]+1/2 b_xx(s)X^v,1_s× X^v,1_s+ 𝔼̃[ b_x μ(s)X^v,1_s×X̃^v,1_s] +1/2𝔼̃[ b_y μ(s)X̃^v,1_s×X̃^v,1_s]+1/2𝔼̅𝔼̃[ b_μμ(s)X̃^v,1_s×X̅^v,1_s] }ds +∫^t_0 {σ_x(s)X^v,1_s+𝔼̃[σ_μ(s) X̃^v,1_s] + σ_x(s)X^v,2_s+𝔼̃[σ_μ(s) X̃^v,2_s] +1/2σ_xx(s)X^v,1_s× X^v,1_s+𝔼̃[ σ_x μ(s)X^v,1_s×X̃^v,1_s] +1/2𝔼̃[ σ_y μ(s)X̃^v,1_s×X̃^v,1_s]+1/2𝔼̅𝔼̃[ σ_μμ(s)X̃^v,1_s×X̅^v,1_s] }dB_s. Recalling that p_t is given by (<ref>) and applying Itô's formula to p_t X^v,12_t, we obtain𝔼∫^T_0 { h_x(s)X^v,12_s + 𝔼̃[h_μ(s) X̃^v,12_s] }ds = 𝔼 ∫^T_0p(s){△ b(s;v)+△ b_x(s;v)X^v,1_s+𝔼̃[△ b_μ(s;v)X̃^v,1_s] +1/2 b_xx(s)X^v,1_s× X^v,1_s+ 𝔼̃[ b_x μ(s)X^v,1_s×X̃^v,1_s] +1/2𝔼̃[ b_y μ(s)X̃^v,1_s×X̃^v,1_s]+1/2𝔼̅𝔼̃[ b_μμ(s)X̃^v,1_s×X̅^v,1_s]}ds +𝔼∫^T_0q(s){1/2σ_xx(s)X^v,1_s× X^v,1_s+𝔼̃[ σ_x μ(s)X^v,1_s×X̃^v,1_s] +1/2𝔼̃[ σ_y μ(s)X̃^v,1_s×X̃^v,1_s]+1/2𝔼̅𝔼̃[ σ_μμ(s)X̃^v,1_s×X̅^v,1_s]}ds.Hence,J(v_·)-J(u_·)= 𝔼∫^T_0 △ H(s;v)ds+𝔼∫^T_0 △ H_x(s;v)X^v,1_sds +𝔼𝔼̃∫^T_0△ H_μ(s;v)X̃^v,1_sds +1/2𝔼∫^T_0 Trace{ H_xx(s)X^v,1_s X^*v,1_s} ds + 𝔼𝔼̃∫^T_0 Trace{ H_xμ(s)X^v,1_sX̃^*v,1_s} ds +1/2𝔼𝔼̃∫^T_0 Trace{H_yμ(s)X̃^v,1_s X̃^*v,1_s }ds +1/2𝔼𝔼̃𝔼̅∫^T_0 Trace{H_μμ(s)X̃^v,1_s X̅^*v,1_s} ds +J^*(v(·)).Now we apply the range theorem for vector-valued measures due to <cit.>,to deduce the variational inequality.Recall that v̂_t is defined by (<ref>). According to Lemma 4.1 <cit.>, for any 0<ρ<1, we can choose a suitable I_ρ⊂ [0,T] such that |I_ρ|=ρ T ,ρ∫^T_0△ b(s;v)ds=∫_I_ρ△ b(s;v)ds+η^*,𝔼|η^*|^2=o(ρ^4),ρ∫^T_0𝔼 [△ H(s;v) ]ds=∫_ I_ρ𝔼[△ H(s;v)]ds+o(ρ^2),andρ∫^T_0𝔼{△ H_x(s;v)X^v,1_s+𝔼̃[ △ H_μ(s;v)X̃^v,1_s]+△b^*(s;v) P_sX^v,1_s }ds= ∫_ I_ρ𝔼{△ H_x(s;v)X^v,1_s+𝔼̃[ △ H_μ(s;v)X̃^v,1_s]+△b^*(s;v) P_sX^v,1_s }ds +o(ρ^2).For the I_ρ above, t∈[0,T], we also haveρ∫^t_0△ b(s;v)ds=∫_I_ρ∩[0,t]△ b(s;v)ds+η_t^*, sup_0≤ t≤ T𝔼|η^*_t|^2=o(ρ^4).The proof of the above lemma is essentially the same asLemma 4.1<cit.>. For the convenience of readers, we present the proof here. Let ϕ_i(·)∈ L^2(Ω; L^2(0,T;ℝ^l_i)), l_i≥ 1, i=1,⋯,k. Suppose sup_0≤ t ≤ T𝔼|ϕ_1(t)|^2< ∞ .Given 0<ρ<1 and set δ=ρ^5, then there exists a n>0, we can find a process ϕ^ρ_i(·) in the form ofϕ^ρ_i(t)=∑_j=0^nξ_i^jI_[t_j,t_j+1)(t),1≤ i≤ k,with 0=t_0<t_1<⋯<t_n+1=T, max|t_i+1-t_i|< δ, ξ^j_i being ℱ_t_j-measurable, s.t.sup_0≤ t ≤ T𝔼|ϕ_1(t)-ϕ^ρ_1(t)|^2< δ. ∑^k_i=2𝔼(∫^T_0|ϕ_i(t)-ϕ^ρ_i(t)|^2dt)< δ.Note that we can always choose the partition {t_j}_0 ≤ j ≤ n+1 independent of i=1,⋯, k. Now letting G=⋃^n_j=0[t_j,t_j+ρ(t_j+1-t_j)).It's easy to see |G|=ρ T. Thus (<ref>), (<ref>), and(<ref>) are proved by taking ϕ_i suitably. For any s∈[0,T], we can always find a m≥ 0, s.t. s∈ [t_m,t_m+1). Then we have sup_0≤ s ≤ T𝔼|∫^T_0I_{t≤ s}(1-I_G(t)/ρ)ϕ^ρ_1(t) dt|^2≤ sup_0≤ s ≤ T𝔼|∑^m-1_j=0ξ^j_1[(t_j+1-t_j)-ρ(t_j+1-t_j)/ρ]+ξ^m_1[(s-t_m)-(s-t_m)/ρ]|^2 +sup_0≤ s ≤ T𝔼|∑^m-1_j=0ξ^j_1[(t_j+1-t_j)-ρ(t_j+1-t_j)/ρ] +ξ^m_1[(s-t_m)-ρ(t_m+1-t_m)/ρ]|^2≤Kδ^2 ρ^-2+ Kδ^2,where I_G(t) is the indicator function of G. Thus, for each 1≤ i≤ k,sup_0≤ s≤ T𝔼|∫^T_0 I_{t≤ s}(1-I_G(t)/ρ)ϕ_1(t) dt|^2≤ sup_0≤ s≤ T𝔼|∫^T_0I_{t≤ s}(1-I_G(t)/ρ)ϕ_1(t)-I_{t≤ s}(1-I_G(t)/ρ)ϕ^ρ_1(t) dt|^2 +Kδ^2ρ^-2≤ 𝔼|∫^T_0|(1-I_G(t)/ρ)[ϕ_1(t)-ϕ^ρ_1(t)]| dt|^2+Kδ^2ρ^-2≤ ρ^-2Tδ+Kδ^2ρ^-2.Set ϕ_1(t)=△ b(t,v), then letting ρ→ 0, we finish the proof.For any ρ∈[0,1], there exists a subset I_ρ of [0,T], such thatlim_ρ→ 0+sup_0≤ t ≤ T𝔼[|X^v̂,1_t-ρ X^v,1_t/ρ|^2]=0.where v̂ is defined by (<ref>) with I_ρ given here.By (<ref>) and lemma <ref>, we haveX^v̂,1_t-ρ X^v,1_t= ∫^t_0 b_x(s)(X^v̂,1_t-ρ X^v,1_t)ds+∫^t_0𝔼̃[b_μ(s)(X̃^v̂,1_t-ρX̃^v,1_t)]ds +∫^t_0(△ b(s;v̂)-ρ△ b(s;v))ds +∫^t_0 σ_x(s)(X^v̂,1_t-ρ X^v,1_t)ds+∫^t_0𝔼̃[σ_μ(s)(X̃^v̂,1_t-ρX̃^v,1_t)]dB_s.Thus, we can get𝔼| X^v̂,1_t-ρ X^v̂,1_t |^2 ≤ K∫^t_0 𝔼| X^v̂,1_t-ρ X^v̂,1_t |^2ds+sup_0≤ t≤ T𝔼[∫^t_0(△ b(s;v̂)-ρ△ b(s;v))ds]^2.Notice that Ksup_0≤ t≤ T𝔼1/ρ^2[∫^t_0(△ b(s;v̂)-ρ△ b(s;v))ds]^2= K sup_0≤ t ≤ T1/ρ^2𝔼|η^*_t|^2=o(1). By Gronwall's inequality, we get (<ref>). Further more, inspired by <cit.>, we also have the following lemma holds. For any ρ∈[0,1], there exists a set I_ρ∈ [0,T] and a matrix value process Φ(t), s.t.X^v̂,1_t is represented by the followingX^v̂,1_t=Φ(t)∫^t_0Φ^-1(s)△ b(s;v̂)dt+A^*_t,wherelim_ρ→ 0+sup_0≤ t ≤ T𝔼[|A^*_t/ρ|^2]=0. Let Φ(t) be the unique solution of the following matrix value SDE:Φ(t)=I+∫^t_0{b_x(s)+𝔼̃[b_μ(s)]}Φ(s)ds+∫^t_0{σ_x(s)+𝔼̃[σ_μ(s)]}Φ(s)dB_s,and setΨ(t) = I+∫^t_0 Ψ(s){-(b_x(s)+𝔼̃[b_μ(s)])+(σ_x(s)+𝔼̃[σ_μ(s)])^2}ds - ∫^t_0 Ψ(s){σ_x(s)+𝔼̃[σ_μ(s)]}dB_s.Applying Itô's formula to [Ψ_tΦ_t], we can easily get d[Ψ_tΦ_t]=0, which means Ψ_tΦ_t≡ I, i.e. Ψ_t=Φ_t^-1. Applying Itô's formula to d[Ψ_tX^v̂,1_t], by Lemma <ref>, we then get our desire result. Now we continue to derive the expansion. Applying Itô's formula to Y_t:=X^v̂,1_tX^*v̂,1_t, we havedY_t = { Y_tb_x^*(t)+X^v̂,1_t𝔼̃[X̃^*v̂,1_t b_μ^*(t)]+X^v̂,1_t△b^*(t;v̂)+b_x(t)Y_t+𝔼̃[b_μ(t)X̃^v̂,1_t]X^*v̂,1_t +△ b(t;v)X^*v̂,1_t+(σ_x(t)X^v̂,1_t+𝔼̃[σ_μ(t)X^v̂,1_t])(X^*v̂,1_tσ^*_x(t)+𝔼̃[X^*v̂,1_tσ^*_μ(t)])}dt+{ Y_tσ_x^*(t)+X^v̂,1_t𝔼̃[X̃^*v̂,1_t σ_μ^*(t)]+σ_x(t)Y_t+𝔼̃[σ_μ(t)X̃^v̂,1_t]X^*v̂,1_t}dB_t. Applying Itô's formula to P_tY_t, according to (<ref>), we obtain𝔼∫^T_0 Trace{H_xx(s)X^v̂,1_s X^*v̂,1_s} ds + 2𝔼𝔼̃∫^T_0 Trace{H_xμ(s)X^v̂,1_sX̃^*v̂,1_s} ds +𝔼𝔼̃∫^T_0 Trace{H_yμ(s)X̃^v̂,1_s X̃^*v̂,1_s }ds +𝔼𝔼̃𝔼̅∫^T_0 Trace {H_μμ(s)X̃^v̂,1_s X̅^*v̂,1_s} ds= 𝔼∫^T_0 Trace{P_sX^v̂,1_s△b^*(t;v̂) +P_s△ b(s;v̂)X^*v̂,1_s}ds+o(ρ^2)= 2𝔼∫^T_0[△b^*(s;v̂) P_s X^v̂,1_s]ds+o(ρ^2). Putting (<ref>) into (<ref>), we concludeJ(v̂_·)-J(u_·) = 𝔼∫^T_0 △ H(s;v̂)ds+𝔼∫^T_0 △ H_x(s;v̂)X^v̂,1_sds +𝔼𝔼̃∫^T_0△ H_μ(s;v̂)X̃^v̂,1_sds 𝔼∫^T_0[△b^*(s;v̂) P_sX^v̂,1_s]ds+o(ρ^2). From Lemma 5.1, (<ref>) and (<ref>), we obtainJ(v̂_·)-J(u_·)= 𝔼∫^T_0 △ H(s;v̂)ds+ρ𝔼∫^T_0 △ H_x(s;v̂)X^v,1_sds +ρ𝔼𝔼̃∫^T_0△ H_μ(s;v̂)X̃^v,1_sds+ρ𝔼∫^T_0[△b^*(s;v̂) P_sX^v,1_s]ds +o(ρ^2)= 𝔼∫_I_ρ△ H(s;v)ds+ρ𝔼∫_I_ρ△ H_x(s;v)X^v,1_sds +ρ𝔼𝔼̃∫_I_ρ△ H_μ(s;v)X̃^v,1_sds+ρ𝔼∫_I_ρ[△b^*(s;v) P_sX^v,1_s]ds +o(ρ^2). Finally, according to (<ref>), we haveJ(v̂_·)-J(u_·)= ρ𝔼∫_0^T △ H(s;v)ds+ρ^2𝔼∫_0^T △ H_x(s;v)X^v,1_sds +ρ^2𝔼𝔼̃∫_0^T△ H_μ(s;v)X̃^v,1_sds+ρ^2𝔼∫_0^T[△b^*(s;v) P_sX^v,1_s]ds +o(ρ^2).§ THE PROOFS OF THE STOCHASTIC MAXIMUM PRINCIPLE.Although the first-order SMP has been obtained by <cit.>, we give a proof here for completeness. In fact, after the preparation of the previous sections which will also be needed in the proof of the second-order SMP, this proof does not take too much extra effort.Proof of first-order SMP: Since (X^u_·, u_·) is an optimal pair of our system, it follows from (<ref>) thatJ(v̂_·)-J(u_·)=ρ𝔼∫_0^T △ H(s;v)ds+o(ρ)≥ 0. for any ρ∈ [0,T], ∀ v(·)∈𝒰.Setting ρ→ 0+, we obtain 𝔼∫_0^T △ H(s;v)ds≥ 0, ∀ v(·)∈𝒰.Then we can deduce that, for any fixed v∈𝒰, there exists a null subset S^v⊂ [0,T]×Ω, such that for each (t,ω)∈(S^v)^c,△ H(s;v)≥ 0.Otherwise, suppose thatA={(s,ω): △ H(s;v^*)<0}has positive measure in [0,T]×Ω, for a v^*∈𝒰. Letv̂^*=v^*1_A+u1_A^c.Then, 𝔼∫^T_0 △ H(s;v̂^*)ds =𝔼∫_0^t △ H(s;v^*)1_A ds<0.This contradicts from (<ref>).Select a countable dense subset {v_s^(i)}^∞_i=1⊂ U, set S_0=⋃^∞_i=1 S^v^(i). Then, S_0 is a null subset of [0,r]×Ω, and for (t,ω)∈ S:= (S_0)^c, we get △ H(s;v^(i))≥ 0.By Fubini's theorem, it is easy to see that there exists a null subset T_0 of [0,T], such that ∀ t∈ T_0^c, (<ref>) holds a.s..Finally, from the continuity of the function and the denseness of {v^(i)}^∞_i=1, we have for t∈(T_0)^c, △ H(s;v)≥ 0, ∀ v∈ U, a.s.. Now, we proceed to presenting the proof of the second-order stochastic maximum principle for singular generalized mean-field control problem. Proof of Theorem <ref>.The optimality and the singularity imply that△ H(t;v)≡ 0,∀ v∈ V.According to (<ref>), we have𝔼∫^t_2_t_1{△ H_x(s;v)X^v,1_s+𝔼̃[△ H_μ(s;v)X̃^v,1_s]+△b^*(s;v) P_sX^v,1_s}ds≥0, ∀ v∈𝒱(t_1,t_2), a.s., where 𝒱(t_1,t_2):={v(·)∈𝒰|v_t∈ V, a.s., a.e., t∈[t_1,t_2]; v(t)=u(t),t∈[0,T]∖ [t_1,t_2] }. As in <cit.><cit.>, denote by { t_i }^∞_i=1 the collection of all rational numbers in [0,T], and {v_k}^∞_k=1 a dense subset of V. Because of the fact that ℱ_t is countability generated for t∈[0,T], we can assume {A_i,j}^∞_j=1 generates ℱ_t_i,i=1,2,⋯. For any τ∈ [t_i,T) and θ∈ (0,T-τ), write E^i_θ=[τ,τ+θ), and definev^k_i,j(t,ω)={[v_k(t,ω), (t,ω)∈ E^i_θ× A_i,j,; u(t,ω), (t,ω)∈(E^i_θ× A_i,j)^c. ].Let X^1k_ij be the solution to the equation (<ref>) with respect to v^k_i,j(·). Notice that we can always choose suitable I_θ, such that I_θ∩ E^i_θ= E^i_θ. So Lemma <ref> holds for X^1k_ij.By Lemma 4.1 <cit.>, lemma <ref> and Lebesgue differential theorem, there is a null subset T^k_ij⊂ [0,T] such that for τ∈(T^k_ij)^c, we have0 ≤ lim_θ→ 0+1/θ^2∫^τ+θ_τ𝔼{△ H_x(s;v^k_i,j)X^1k_ij(s)+𝔼̃[△ H_μ(s;v^k_i,j)X̃^1k_ij(s)] +△b^*(s;v^k_i,j) P_s X^1k_ij(s)}ds= lim_θ→ 0+1/θ^2∫^τ+θ_τ𝔼{△ H_x(s;v^k_i,j)Φ(s)∫^s_τΦ^-1(r) △ b(r;v_k)1_A_ijdr +𝔼̃[△ H_μ(s;v^k_i,j)Φ(s)∫^s_τΦ^-1(r) △b̃(r;v_k)1_A_ijdr] +△b^*(r;v^k_i,j) P_s Φ(s)∫^s_τΦ^-1(r) △b(r;v_k)1_A_ijdr}ds= 𝔼{△ H_x(τ;v_k)△ b(τ;v_k)1_A_ij+𝔼̃[△ H_μ(τ;v_k)△b̃(τ;v_k)1_A_ij] +△b^*(τ;v_k) P_τ△ b(τ;v_k)1_A_ij}. SetT_0=⋃_1≤ i,j,k≤∞T^k_i,j.Then, T_0 is a null subset of [0,T]. For s∈ [0,T]∖ T_0 and i, we deduce that𝔼{△ H_x(s;v_k)△ b(s;v_k)1_A_ij+𝔼̃[△ H_μ(s;v_k)△b̃(s;v_k)1_A_ij] +△b^*(s;v_k) P_s△ b(s;v_k)1_A_ij}≥ 0, ∀ j,k=1,2,⋯,which means𝔼{△ H_x(s;v)△ b(s;v)1_A+𝔼̃[△ H_μ(s;v)△b̃(s;v)1_A] +△b^*(s;v) P(s)△ b(s;v)1_A}≥ 0, ∀ v∈ V,A∈ℱ_t.By virtue of the continuity of the function and the denseness of {v_k}^∞_k=1, we finish the proof. We now come back to Example <ref>. It is not hard to check that the second order adjoint process (P_t,Q_t)≡(1,0). Then△ H_x(t;v)△ b(t;v)+𝔼̃[△ H_μ(t;v)△b̃(t;v)] +△b^*(t;v) P_t△ b(t;v)=v_t^2≥ 0, ∀ v∈ U, a.s..So we can say u_t≡ 0 is the only candidate for optimal controls. 99 AS1979 V. Arkin and I. Saksonov,Necessary optimality conditions of optimality in the problems of controlof stochastic differential-equations, Doklady Akademii Nauk SSSR., 244 (1979), 11–15.BJ1975D. J. Bell and D. H. Jacobson,Singular Optimal Control Problems,Vol. 117. Elsevier, 1975.B1982 [10.1007/BFb0064859] A. Bensoussan,Lectures on stochastic control.Nonlinear filtering and stochastic control,(1982), 1–62. B1978 [10.1137/1020004] J. M. Bismut,An introductory approach to duality in optimal stochastic control, SIAM Review,20.1 (1978), 62–78.BLM2016 [10.1007/s00245-016-9394-9] R. Buckdahn, J. Li and J. Ma, A Stochastic Maximum Principle for General Mean-Field Systems.Applied Mathematics and Optimization, 74.3 (2016), 507–534.BLP2009[10.1016/j.spa.2009.05.002] R. Buckdahn, J. Li and S. Peng,Mean-field backward stochastic differential equations and related partial differential equations, Stoch. Proc. App., 119.10 (2009), 3133-3154.BLP2014 R. Buckdahn, J. Li, S. Peng and C. Rainer, Mean-field stochastic differential equations and associated PDEs, preprint 1407.1215.C 2013[10.1007/978-3-319-06917-3_5] P. Cardaliaguet,Weak solutions for first order mean field games with local coupling,Analysis and Geometry in Control Theory and its Applications,(2015), 111–158.GK1972 [10.1137/0310012] R. Gabasov , F. M. Kirillova,High order necessary conditions for optimality,SIAM J. Control, 10 (1972), 127–168.H1986 [10.1007/BF00047571] U. G. Haussmann,A Stochastic Maximum Principle for Optimal Control of Diffusions, Essex, UK: Longman Scientific and Technical, 1986.H1976 [10.1007/BFb0120743] U. G. Haussmann,General necessary conditions for optimal control of stochastic systems,Math. Program. Study, 6 (1976), 30–48.KD1984 [10.1007/BF00935010] M. A. Kazemi-Dehkordi, Necessary conditions for optimality of singular controls, J. Optim. Theor. Appl., 43 (1984), 629–637.K1977 [10.1137/0315019] A. J. Krener, The high-order maximum principle and its application to singular extremals,SIAM J. Control, 15 (1977), 256–293. K1965 [10.1016/0022-247X(65)90070-3] H. J. Kushner, On the stochastic maximum principle: fixed time of control, Journal of Mathematical Analysis and Applications, 11 (1965), 78–92. K1972 [10.1137/0310041] H. J. Kushner,Necessary conditions for continuous parameter stochastic optimization problems, SIAM Journal of Control, 10 (1972), 550–565.L2012 [10.1016/j.automatica.2011.11.006] J. Li,Stochastic maximum principle in the mean-field controls, Automatica, 48 (2012), 366–373.L2016 [ 10.1109/ChiCC.2016.7553759] Q. Lu,Second order necessary conditions for optimal control problems of stochastic evolution equations, Control Conference (CCC), 2016 35th Chinese. IEEE, (2016), 2620-2625. MY2007 L. Mou and J. Yong, A variational formula for stochastic controls and some applications, Pure Appl. Math. Q, 3.2 (2007), 539–567. MW1992 [10.1080/00207729208949387] K. Mizukami and H. Wu, New necessary conditions for optimality of singular controls in optimal control problems, Int. J. Systems Sci., 23 (1992), 1335–1345.PP1990 [10.1016/0167-6911(90)90082-6] E. Pardoux and S. Peng,Adapted solution of a backward stochastic differential equation, System Control Lett., 14.1 (1990), 55–61.P1990 [10.1137/0328054] S. Peng, A general stochastic maximum principle for optimal control problem, SIAM J. Control and Optimization, 28.4 (1990), 966–979.P1962 L. S. Pontrvagin, V. G. Boltyanskii, R. V. Gamkerlidze and E. F. Mischenko. The Mathematical Theory of Optimal Control Processes,John Wiley, New York, 1962.T2010 [10.3934/dcdsb.2010.14.1581 ] S. J. Tang, A second-order maximum principle for singular optimal stochastic controls, Discrete and continuous dynamical system series B, 14.4 (2010), 1581–1599.YZ2000 J. Yong and X.Y. Zhou,Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer-Verlag, New York, Berlin, 2000.ZZ2015 [10.1137/14098627X] H. Zhang and X. Zhang, Pointwise second-order necessary conditions for stochastic optimal controls, Part I: The case of convex control constraint. SIAM Journal on Control and Optimization, 53.4 (2015), 2267–2296.Received xxxx 20xx; revised xxxx 20xx. | http://arxiv.org/abs/1704.08002v1 | {
"authors": [
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"Jie Xiong"
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"published": "20170426080606",
"title": "A second-order stochastic maximum principle for generalized mean-field control problem"
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𝐱 ŁL Li Li^∗, Zhu Li^∗, Xiang Ma^⋆, Haitao Yang^⋆ and Houqiang Li^†This paper is partially supported by the UMKC strategic funding on big imaging and smart city center ^∗ University of Missouri Kansas City ^⋆ Huawei Technologies Co., Ltd. ^† University of Science and Technology of China{lil1, lizhu}@umkc.edu, {maxiang6, haitao.yang}@huawei.com, [email protected] based 3-D padding for polyhedron projection for 360-degree video [NO \author GIVEN] December 30, 2023 ====================================================================================The polyhedron projection for 360-degree video is becoming more and more popular since it can lead to much less geometry distortion compared with the equirectangular projection.However, in the polyhedron projection, we can observe very obvious texture discontinuity in the area near the face boundary.Such a texture discontinuity may lead to serious quality degradation when motion compensation crosses the discontinuous face boundary.To solve this problem, in this paper, we first propose to fill the corresponding neighboring faces in the suitable positions as the extension of the current face to keep approximated texture continuity.Then a co-projection-plane based 3-D padding method is proposed to project the reference pixels in the neighboring face to the current face to guarantee exact texture continuity.Under the proposed scheme, the reference pixel is always projected to the same plane with the current pixel when performing motion compensation so that the texture discontinuity problem can be solved.The proposed scheme is implemented in the reference software of High Efficiency Video Coding.Compared with the existing method, the proposed algorithm can significantly improve the rate-distortion performance.The experimental results obviously demonstrate that the texture discontinuity in the face boundary can be well handled by the proposed algorithm.360-degree video compression, polyhedron projection, inter prediction, padding, high efficiency video coding § INTRODUCTIONAlong with the emergence and popularity of one virtual reality (VR) product after another, such as Oculus Rift, Gear VR, and HTC Vive, video contents are becoming one of the most important applications for the VR product. To support the content representation from all directions and create a fully immersed experience, the VR video needs to contain the information from all 360 degrees. Therefore, the VR video, also named as 360-degree video, should be with very high spatial resolution even higher than 8K to maintain relatively good visual quality. Such high resolution videos can bring many challenges to the video compression technologies, and the need to develop specified compression method for these video becomes quite urgent. Since the original 360-degree video is a sphere, to adapt to the modern video coding standards such as H.264/Advanced Video Coding (AVC) <cit.>, and H.265/High Efficiency Video Coding (HEVC) <cit.>, the 360-degree video is always projected to a 2-D format for compression. According to the investigation in <cit.>, there are actually lots of projection methods such as equirectangular and polyhedron including cube map, octahedron, icosahedron. Comparing the equirectangular and polyhedron formats, the polyhedron formats present less geometry distortion so that they can lead to better coding efficiency <cit.> <cit.>. However, the polyhedron formats also have their disadvantages that very obvious texture discontinuities exist in the area near the face boundary. The texture discontinuities can be divided into two kinds, which are obviously shown in Fig. <ref> for the typical 4×3 cubic format. One kind of the discontinuities is caused by the face unfold from 3-D cubic to 2-D image, which is represented by the green rectangles. The other kind of discontinuities is brought by the projection to different planes (or faces) from sphere to cubic format, which is shown by the red rectangles. When the motion vector (MV) happens to cross the face boundary, the current motion compensation (MC) scheme will obtain an unreasonable prediction block with quite obvious texture discontinuity, which will lead to serious coding efficiency decrease.In the current standard-based video coding scheme, a simple padding scheme, which extends the picture boundary pixel to the outside of the picture, is implemented in the HEVC reference software <cit.> to both guarantee the picture size as the multiple of the coding unit size and prevent the MC operation from crossing the picture boundary. Li et al. <cit.> have also tried to optimize the padding scheme for arbitrary size picture using the fundamental rate distortion optimization (RDO) theory.However, since these schemes only consider the picture itself and have not considered the specific 360-degree information of the 360-degree video, they are not the best ways to solve the problems of texture discontinuity in the face boundary for the 360-degree video. Therefore, in this paper, to better solve the problem of texture discontinuity in the face boundary, we try to make full use of all the information from the 360-degree video. To be more specific, we first fill the neighboring faces in the suitable positions for the current face to keep approximate texture continuity. Then we propose a co-projection-plane based 3-D padding method to project the reference pixels in the neighboring face to the current face to guarantee exact texture continuity. Under the proposed scheme, the reference pixel is always projected to the same plane with the current pixel when performing MC so that the texture discontinuity problem in the face boundary can be solved. This paper is organized as follows. In Section <ref>, we will give a brief introduction of the polyhedron projection. The proposed co-projection-plain based 3-D mapping method will be described in detail in Section <ref>. The detailed experimental results will be shown in Section <ref>. Section <ref> concludes the whole paper.§ A BRIEF INTRODUCTION OF THE POLYHEDRON PROJECTION As its name implies, polyhedron projection is to project the inscribed sphere (360-degree video) to each face of the polyhedron, such as cube, octahedron, and icosahedron. As a typical example, the detailed projection process from inscribed sphere to the cube map can be seen from Fig. <ref>. For each point N in the face of the cube, we will connect a line between the center point O and N. Then the line and the sphere will have an intersection point M, and the pixel value of point M will be used as the value of point N. Since the point M may not be in the integer sampling position of the sphere, the pixel value of point M will be interpolated through surrounding integer pixels. To be more specific, the Luma component is interpolated using the Lanczos3 (6×6) <cit.> interpolation filter, and the Chroma component is interpolated using the Lanczos2 (4×4) <cit.> interpolation filter.After the projection from a sphere to a polyhedron, the polyhedron will then be unfolded to obtain the 2-D image for compression. There are various kinds of unfolding methods for a polyhedron including non-compact and compact methods. Especially, for the cube map projection, as shown in Fig. <ref>, mainly two methods of unfolding by putting different faces in different positions are introduced, including 4×3 and 3×2 formates. And in the following sections, the 4×3 cube map projection will be used as an example to introduce the proposed co-projection-plain based 3-D mapping methods.§ THE PROPOSED CO-PROJECTION-PLAIN BASED 3-D PADDING The proposed co-projection-plain based 3-D padding method will be introduced in two aspects. We will first fill the corresponding neighboring faces in the suitable positions as the extension of the current face to keep approximated texture continuity in subsection <ref>. Then we will project the reference pixels in the neighboring face to the current face to guarantee exact texture continuity in subsection <ref>. Finally, in subsection <ref>, we will introduce some implementation details. §.§ Approximated texture continuity As each face of a cube has four edges, to achieve approximated texture continuity, we should first try to make all the four neighboring faces of the current face available. As shown in Fig. <ref> (a), the front face has three neighboring faces, the right and rear faces have two neighboring faces, and the top, bottom, and left faces have only one neighboring face. We will complement the neighboring faces of all the faces to four neighboring faces. Using the right face as an example, besides the existing front and rear faces, we will complement the top and bottom faces for the current face. The complementation result is shown in Fig. <ref> (a), and the actual result of a typical sequence is presented in Fig. <ref> (b).As can be obviously seen from Fig. <ref> (b), the complementation result still presents very obvious texture discontinuity in the common edges between the center face and top/bottom faces. The main reason is that the common edges of the neighboring faces are not aligned together. To guarantee the alignment of the common edges, the top face should be rotated by 90 degrees clockwise, and the bottom face should be rotated by 90 degrees anticlockwise. The final approximated texture continuity results are shown in Fig. <ref>. The above process is just a typical example for the right face, and the other faces can be done in a similar way to achieve approximated texture continuity. §.§ Exact texture continuity After the approximated texture continuity is achieved, if we take a look at Fig. <ref> (b) carefully, we can still see that straight lines on the car become broken lines when crossing the face boundary. This is mainly caused by the cube map projection from inscribed sphere to difference faces. Therefore, in this subsection, we will propose a co-projection-plain based 3-D padding to achieve exact texture continuity.As shown in Fig. <ref>, under the co-projection-plain based 3-D padding method, we will try to extend the current face ABCD into a larger one A'B'C'D', and the values of the extended pixels will be determined by the projection of the neighboring faces, which are generated in the approximated-texture-continuity step, to the current face. Using the bottom face as an example, for a point T in the extended zone of the bottom face, assume that the top left position A' is (0,0), the position T in the extension face is (x,y), the face extension range is S, and the edge length of the cube is a. Then the lengths of TK and JK can be calculated asTK = a/2+S-y JK = x-a-STherefore, according to the principle of similar triangles, we can obtain the length of HS asHS = ST/O'T× OO' = JK/O'K× OO'Similarly, we can also obtain the length of SJ asSJ = O'J/O'K× TKIn this way, the coordinate of the corresponding position in the right face can be derived. The other projection positions of the neighboring faces can be derived in a similar way.It should be noted that the calculated coordinate may not be always in the integer position. In the current implementation, the bilinear interpolation is used to interpolate the pixels in the fractional positions. It should also be mentioned that the pixels belonging to lines AA', BB', CC', and DD' will be projected to the common edges of two neighboring faces. If the bilinear interpolation is still used, the final pixel values will be interpolated from the neighboring pixels coming from two different faces, which is obviously unreasonable. In our implementation, the pixels belonging to lines AA', BB', CC', and DD' are derived through the average of the neighboring pixels in the extended zones. After these operations, the interpolation results are shown in Fig. <ref> (b). Compared with the results generated by the HEVC reference software as shown in Fig. <ref> (a), it can be obviously seen that the proposed algorithm can achieve exact texture continuity. Not only the gray zones but also the discontinuous face boundaries are filled with suitable values to guarantee exact texture continuity. §.§ Implementation details The proposed algorithm is implemented in the HEVC reference software. Our current implementation can be roughly divided into two parts and will not lead to any modification of the coding tools in the coding unit (CU) level. The first part is to get the extension for all the 6 faces for the reference frames. To be more specific, after the encoding of the current frame is finished, if the current frame is a reference frame, the neighbor faces of all the 6 faces will be first complemented using the method introduced in subsection <ref> to generate the image similar to Fig. <ref> (b). Then the method introduced in subsection <ref> will be used to generate the extended faces similar to Fig. <ref> (b) to achieve exact texture continuity.Then the second part is to fill the reference frame with the face extension when encoding each CU. For example, when we are encoding a CU in the right face, we will fill in the right face extension to the each reference frame for the current CU. The results can be seen from Fig. <ref>. It seems discontinuous for the whole frame but for the right face in a predefined search range S, the texture is continuous. And after the coding of CUs belonging to the current face, the reference frame will be refilled with the original values and prepare to be filled with the extension of other faces in the future encoding process. It should be noted that in the decoding process before the reference frame will be used for each CU, we will already know the MV of the current CU. Therefore, we can determine whether the current CU needs to fill in the extension of a current face or not according to the value of MV so as to avoiding the unnecessary extension operations and reducing decoding complexity.§ EXPERIMENTAL RESULTS The proposed co-projection-plain based 3-D padding method is implemented in the HEVC reference software HM-16.6 to compare with HEVC without the proposed algorithm. All the test conditions specified for inter frames including random access (RA) main 10, low delay (LD) main 10, low delay P (LDP) main 10 are used as the test conditions. The quantization parameters (QP) tested in our experiments are 22, 27, 32, 37 following the HEVC common test conditions. The face extension range S is set as 64 in our experiments. Besides, the BD-rate (Bjontegaard Delta rate) <cit.> is used to measure the difference between the anchor and the proposed algorithm. In the current implementation, the Peak Signal to Noise Ratio (PSNR) is used to measure the quality of between the reconstructed and original sequences. We will use the quality metrics, which are more suitable for 360-degree videos such as WS-PSNR <cit.> and S-PSNR <cit.>, as the quality measurements in our future work.For the test sequences, we use the test sequences specified in <cit.> to measure the performance of the proposed algorithm. To be more specific, we used the conversion tool specified in <cit.> to convert the high fidelity input test sequences in equirectangular format to the 10 bit 4×3 cubic formate test sequences. The detailed information and characteristics of the test sequences can be seen in Fig. <ref>. The frame count tested is approximated as 1 second as shown in Fig. <ref>.The test results of the proposed algorithm in RA main10, LD main10, and LDP main10 are shown in Table <ref>, Table <ref>, and Table <ref>, respectively. From the test results, we can see that about for the Y component, compared with the HEVC anchor, about averagely 1.1%, 1.2% and 1.2% R-D performance improvement can be achieved in RA, LD, and LDP cases, respectively. For U and V components, about averagely 1.3%, 1.5%, and 1.3% bitrate reduction are observed accordingly. Besides, we can also see from these tables that for the sequence with relatively larger motion, the maximum bitrate saving for the Y component can be as high as 3.3%, 3.4%, and 3.3% in RA, LD, and LDP cases, respectively. Except for the average and maximum bitrate reduction, we can also see that the proposed algorithm can lead to consistently better R-D performance for all the test sequences even if the RDO based selection between the proposed reference frame and the original reference frame is not used in the proposed framework. This can obviously demonstrate that the reference frame in the proposed framework can always lead to better or equivalent compression results compared with that in the original framework. However, we can also see that the performance improvement may vary due to the differences of the characteristics of various sequences. For the sequences with large motion in the face boundary such as the sequence DrivingInCountry, the situation where the MC cross the face boundary will be quite a lot, thus the proposed algorithm can lead to significant bitrate reduction. On the contrary, for the sequences with almost zero motion in the face boundary such as the sequence Harbor, the situation where the MC cross the face boundary will be very rare, thus the proposed algorithm cannot provide an obvious performance improvement.Some typical R-D curves in various test conditions with different test sequences are shown in Fig. <ref>. The R-D curves also demonstrate that the proposed algorithm can lead to some performance improvement compared with HEVC anchor. Besides, from these typical R-D curves, we can also see that the proposed algorithm can lead to similar performance improvement for both high bitrate and low bitrate.§ CONCLUSION In this paper, we first point out the existence and influences of the very serious texture discontinuities in the face boundary in the polyhedron projection. Then we propose to fill the corresponding neighboring faces in the suitable positions as the extension of the current face to keep approximated texture continuity. After that, a co-projection-plane based 3-D padding method is proposed to project the reference pixels in the neighboring face to the current face to guarantee exact texture continuity. The proposed scheme is implemented in the reference software of High Efficiency Video Coding.Compared with the existing method in the High Efficiency Video Coding reference software, the proposed algorithm can bring averagely 1.1% and maximum 3.4% bitrate savings in different test conditions. The experimental results obviously demonstrate that the texture discontinuity in the face boundary can be well handled by the proposed algorithm. IEEEbib | http://arxiv.org/abs/1704.08768v1 | {
"authors": [
"Li Li",
"Zhu Li",
"Xiang Ma",
"Haitao Yang",
"Houqiang Li"
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Schottky AnomalyThe Institute of Mathematical Sciences, Chennai 600113, India (affiliated to HBNI),[email protected],WWW home page:Analyzing the mesonic spectrum using the method of Schottky anomalyAritra Biswas1 December 30, 2023 ==================================================================== A possible diagnostic is proposed which may be used to infer the different scales underlying the dynamical structure of hadronic resonances using the phenomenon of Schottky anomaly.§ INTRODUCTION The search for exotic hadrons has been rejuvenated in recent years with the discovery of the so called “pentaquark states” <cit.>. Along with the X, Y, Z states <cit.> discovered in the last decade by the different experimental groups <cit.> and the recent confirmation of the Λ(1405) as a molecular state <cit.> (which had been proposed earlier <cit.>), this has rekindled the quest for a better theoretical understanding of these states. Various models have been proposed to this end over decades <cit.>. We propose a model independent analysis in order to identify states which differ in their underlying dynamics due to different interaction scales responsible for forming composite hadronic states. The tool used is the method of Schottky anomaly. We restrict ourselves to the analysis of the mesonic spectra only.§ C_V FOR AN IDEAL SYSTEMConsider a two-level system with energy-gap Δ. The canonical partition function is given byZ=1+e^-βΔ.where β=1/k_BT is the inverse temperature.[The temperature is introduced here simply as a mathematical parameter to define the partition function of the system and no assumption is made regarding the system being in a heat bath in equilibrium.] The specific heat of the system at constant volume may be defined asC_V=β^2[1/Z∂^2 Z/∂β^2 -(1/Z∂ Z/∂β)^2] =β^2[⟨ E^2⟩-⟨ E⟩^2]. Substituting for the partition function of the two level system given in eq.(<ref>) we haveC_V= β^2 Δ^2 e^-βΔ/(1+e^-βΔ)^2.When C_V is plotted against βΔ the Schottky peak appears at a value βΔ≈ 2.4 . The location of the peak is a function of the energy gap in the system. A realistic spectra in general might have more than one scale in operation. In order to simulate the effect such a system we first consider the spectrum of a three-dimensional Harmonic oscillator. The partition function of the system is given by Z_1=∑_n=0^∞D(n)e^-βħω(n+3/2) where the oscillator parameter ω defines the scale in the problem and D(n) is the degeneracy of the level. We combine the spectra of two such systems with different values for the scale parameters (Fig. <ref>). The effect of combining is to normalize the specific heat with a single partition function given by,Z(β)= Z_1(β,ħω_1)+Z_2(β,ħω_2). The peaks appear when there spectrum has a cut offotherwise, it will saturate as a function of temperature. In the next section we give our results for the analysis of the current heavy mesonic spectra taken from <cit.>. For a detailed discussion the interested reader is referred to <cit.>. § ANALYZING THE EXPERIMENTAL SPECTRA* The charmonium spectra: When all the cc̅ states are plotted together, the existence of two peaks at T≈40 MeV and T≈190 MeV are revealed, indicating the existence of two well defined scales, which might be identified with the “hyperfine” and the “confinement” scales respectively. When the states are grouped according to their individual J values J=0,1,2, the hyperfine peak vanishes, and only the confinement peak is retained. However, when only the “exotic” states in the charmonium mass range are plotted, we again find a double-peaked structure, where the confinement peak shifts to 100 MeV along with a sharp peak below 10 MeV. * The bottomonium spectra The two peaked structure with distinct hyperfine (22 MeV) and confinement (185 MeV) peaks are revealed for the case of the bottomonium spectrum also. The hyperfine peak disappears as expected when the states are grouped according to their J values and plotted again.The bottomonium exotics are however, scarce in number (three to be exact), with ill defined quantum numbers. It is hence premature to comment on these states. However, on plotting, they do retain the peak at 100 MeV, which is encouraging and may mean that the underlying mechanism for the charmonium and the bottomonium exotic states are the same.* The open charm spectra Compared to those discussed previously, this case is more complicated due to the presence of the isospin (I) symmetry along with the J symmetry. However, on plotting all the open charm mesons together, the corresponding C_V vs T plot shows the two-peaked structure with the confinement peak at T=160 MeV and the hyperfine peak at T=50 MeV approximately. On plotting states with the same J and I together, the hyperfine peak vanishes and only the confinement peak is retained. * The open bottom spectra The open bottom spectra is rather sparse as compared to the open charm scenario. Nevertheless, some features are already visible. The two peaked structure prevails on plotting all the states together. On grouping the states according to their J and I values, a well defined confinement peak is seen. However, due to the limited number of states, one should wait for the advent of more data in the future before drawing any conclusions for this sector. 6pentaquarkR. Aaij et al. [LHCb Collaboration],Phys. Rev. Lett.115, 072001 (2015) doi:10.1103/PhysRevLett.115.072001 [arXiv:1507.03414 [hep-ex]]. Olsen:2014S. L. Olsen,Front. Phys. (Beijing) 10, no. 2, 121 (2015) doi:10.1007/S11467-014-0449-6 [arXiv:1411.7738 [hep-ex]].olsen_ref For more detailed experimental references for all the exotic states, see <cit.> and the references therein. latticeJ. M. M. Hall, W. Kamleh, D. B. Leinweber, B. J. Menadue, B. J. Owen, A. W. Thomas and R. D. Young,Phys. Rev. Lett.114, no. 13, 132002 (2015) doi:10.1103/PhysRevLett.114.132002 [arXiv:1411.3402 [hep-lat]]. rajaji1R. H. Dalitz, T. C. Wong and G. Rajasekaran,Phys. Rev.153, 1617 (1967). doi:10.1103/PhysRev.153.1617Godfrey:olsenS. Godfrey and S. L. Olsen,Ann. Rev. Nucl. Part. Sci.58, 51 (2008) doi:10.1146/annurev.nucl.58.110707.171145 [arXiv:0801.3867 [hep-ph]]. pdgC. Patrignani et al. [Particle Data Group],Chin. Phys. C 40, no. 10, 100001 (2016). doi:10.1088/1674-1137/40/10/100001 SchottkyA. Biswas, M. V. N. Murthy and N. Sinha,Phys. Rev. D 92, no. 11, 114012 (2015) doi:10.1103/PhysRevD.92.114012 [arXiv:1509.06201 [hep-ph]]. | http://arxiv.org/abs/1704.08527v1 | {
"authors": [
"Aritra Biswas"
],
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"hep-ph"
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"primary_category": "hep-ph",
"published": "20170427120814",
"title": "Analyzing the mesonic spectrum using the method of Schottky anomaly"
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2017 Vol. X No. XX, 000–000 Department of Astronomy, Beijing Normal University,Beijing 100875,China; [email protected],[email protected] Laboratory of Earth and Planetary Physics, Instituteof Geology and Geophysics, Chinese Academy of Sciences, Beijing100029, ChinaCollege of Earth Sciences, University of Chinese Academyof Sciences, Beijing,China Received ——-; accepted ——We analyze sunspots rotation and magnetic transients in NOAA AR 11429 during two X-class (X5.4 and X1.3) flares using the data from the Helioseismic and Magnetic Imager on board the Solar Dynamics Observatory. A large leading sunspot with positive magnetic polarity rotated counterclockwise. As expected, the rotation was significantly affected by the two flares. The magnetic transients induced by the flares were clearly evident in the sunspots with negative polarity. They were moving across the sunspots with speed of order 3-7kms^-1. Furthermore, the trend of magnetic flux evolution of these sunspots exhibited changes associated with the flares. These results may shed light on the understanding of the evolution of sunspots. J.-C. Zheng et al. Sunspots rotation and magnetic transients associated with flares Sunspots rotation and magnetic transients associated with flares in NOAA AR 11429 ^* * Supported by the National Natural Science Foundation of China.Jianchuan Zheng1 Zhiliang Yang1 Jianpeng Guo1, 2 Kaiming Guo1 Hui Huang1 Xuan Song1 Weixing Wan2, 3 December 30, 2023 ======================================================================================================================================================§ INTRODUCTION Sunspots are concentrations of strong magnetic field on the solar surface, consisting of a dark umbra and a fibrous penumbra <cit.>. It is generally accepted that eruption events such as flares occurring in upper atmosphere are associated with magnetic field generated from the base of convection zone, emerging as bipolar magnetic regions at solar surface, then extending to corona <cit.>. So the visible sunspots play an important role in studying a variety of phenomena on the Sun.The evolution of sunspots includes formation, movement, deformation and disappearance. Rotation is an important feature during evolution. In general, there are two kinds of rotation: one sunspot rotates around its umbral centre and one sunspot rotates around the other <cit.>. The rotation of sunspots and spot-groups have been studied for many decades (see, e.g., ). <cit.> found 182 significantly rotating sunspots among 2959 active regions. Some authors believed the rotational motion of sunspots may be involved with energy build-up and later release by flares (e.g., ). Furthermore, the relationships between the sunspot rotation and coronal consequence <cit.>, flare productivity <cit.>, direction of the rotation <cit.>, and magnetic helicities <cit.> have been investigated. The association of flares with abnormal rotation rates <cit.> has been found. Specifically, <cit.> reported two sunspots in AR 11158 rotating along with a X2.2 flare. <cit.> reported a non-uniform rotation induced by a M6.5 flare in NOAA AR 12371, and <cit.> found an abrupt reverse rotation of sunspot in NOAA AR 12158 caused by back reaction of X1.6 flare.Besides, the photospheric magnetic field changes during the flares. The magnetic field is distorted enough to store energy powering flare. After a flare taking place, the distorted field relax and restructure <cit.>. There are two kinds of changes of magnetic field based on observations: the first is rapid short-term changes <cit.>, which is due to flare-induced spectral line changes <cit.>. It is generally believed that the rapid short-term changes are not the real changes of magnetic field (see, ). However, <cit.> also analyzed the magnetic transients during the M7.9 flare in NOAA AR 11429 on 13 March 2012 and suggested that the magnetic transients represented a real change in the photospheric magnetic filed. Another isirreversible changes from pre-flare to post-flare (e.g., ). The irreversible changes were first noticed by <cit.>. They showed that magnetic shear increased after flares with vector magnetic field data. <cit.> reported a change in the line-of-sight (LOS) field during the X9.3 flare on 24 May 1990 with the videomagnetograph data from the Big Bear Solar Observatory (BBSO). <cit.> and <cit.> analyzed the changes of the LOS magnetic field during some X- and M- class flares with Global Oscillation Network Group (GONG) magnetograms. The median duration of changes was about 15 minutes and the median absolute value of changes was about 69 G. <cit.> demonstrated that the change of LOS magnetic field is due to increase of horizontal magnetic field near the polarity inversion line. With the seeing-free data of vector magnetograms, <cit.> found that the photospheric transverse magnetic field enhancement was associated with the M6.6 flare in NOAA AR 11158. <cit.> observed the same phenomenon associated with the X2.2 flare in this active region. These results suggested that the magnetic field near the polarity inversion line could become more horizontal after a flare.In this study, we analyze the evolution of sunspots in NOAA AR 11429. We will explore the rotation of sunspots along with the flares, i.e., how do the flares and the rotation of sunspots affect each other. The magnetic transients induced by flares are studied in prior works. We further study the motion of the magnetic transients to understand how the flares influence sunspots. The paper is organized as follows. In Section <ref> we describe the data used and the entire structure of the active region. Section <ref> shows data processing. We summarize and discuss the results in Section <ref>.§ OBSERVATIONAL DATAThe 45 s cadence and spatial sampling of 0”.5 pixel^-1 full-disk continuum intensity images and line-of-sight magnetograms observed by the Helioseismic and Magnetic Imager (HMI; ) on board the Solar Dynamics Observatory ( SDO) are used to analyze the variation of sunspots in NOAA AR 11429. The images from the Atmospheric Imaging Assembly (AIA; ) on board SDO are used to investigate the chromospheric and coronal context: AIA 304 Å formed in transition region and chromosphere, and AIA 1600 Å formed in upper photosphere and transition region.AR 11429 appeared on the eastern limb on 3 March 2012, and then rapidly became complicated reverse-polarity βγδ active region. The evolution and development of AR 11429 from 5 March to 12 March are displayed in Figure <ref>. It is interesting to note that the sunspots in this active region were moving away from each other, i.e., the leading sunspots moving westward and the following sunspots moving eastward. Several major eruptions were clearly visible in this active region. Typically, three X-class flares occurred: one (X1.1) on 5 March and two of interest (X5.4 and X1.3) on 7 March. The GOES soft X-ray flux (Figure <ref>a) indicates that the X5.4 flare started at 00:02 UT, peaked at 00:27 UT, and ended at 00:40 UT, and the X1.3 flare started at 01:05 UT, peaked at 01:14 UT, and ended at 01:23 UT. The evolution of five prominent sunspots, two positive polarity sunspots (P1 and P2) and three negative polarity sunspots (N1, N2 and N3), were associated with these two flares, as be discussed in the next section. § ANALYSIS AND RESULTS §.§ Sunspots rotating associated with flaresTo study the rotation, we take the rotation of individual sunspots as a simplified solid-body rotation (i.e., a sunspot rotates around its umbral center), and track the variation of the ellipses that best fit individual sunspots on time-series intensity images (see ). Take sunspot P1 as an example. The best-fit ellipses on the intensity images at 23:29:23 UT of 6 March and 01:28:38 UT of 7 March are shown in Figure <ref>(a) and Figure <ref>(b), respectively. Obviously, the major axis rotates counterclockwise about 7^∘. In the present study, the changes of the angle between the major axis and the horizontal direction of the best-fit ellipses are used to describe the rotation of the sunspot. The results together with the GOES soft X-ray flux are shown in Figure <ref>.Figure <ref>(b) shows that P1 rotated mainly counterclockwise from ∼23:30 UT of 6 March to ∼01:30 UT of 7 March. Before the start of the X5.4 flare, the rotation was very fast with speed up to 7.92^∘hr^-1. From start to peak of the flare, the rotation speed decreased slowly to about 4.89^∘hr^-1 and the rotation trend kept counterclockwise. Just after the peak of the flare, the rotation trend reversed from counterclockwise to clockwise. Its speed was about 4.07^∘hr^-1. After the flare, the rotation direction returned to counterclockwise with the rotation speed of about 6.56^∘hr^-1, and the rotation speed increased gradually until the beginning of the X1.3 flare. Later, the rotation trend reversed to clockwise again with the rotation speed of about 6.44^∘hr^-1 and slowed down. These results indicate that the rotation of sunspot P1 was significantly affected by both flares. The relative mean intensity profiles in different AIA wavelengths, obtained from the co-aligned with HMI images, are also plotted in Figure <ref>. The intensity in the AIA wavelengths of 304 Å (green curve) and AIA 1600 Å (blue curve) started rising around 00:05 UT, peaked at about 00:07 UT, and then decreased during the impulsive phase of the X5.4 flare. At about 01:05 UT, the intensities started rising again and peaked at about 01:14 UT. It provides further support for the view that solar flares are often related to the rotation of sunspots.The rotation of the sunspot N1 was complicated (Figure <ref>c). Before the peak of the X5.4 flare, the rotation trend was mainly counterclockwise, and then reversed to clockwise. The situation around the X1.3 flare was similar to that around the former flare. The intensity of AIA wavelengths increased and the rotation changed significantly during the impulsive phases of the X5.4 and X1.3 flares. As shown in Figure <ref>(d), sunspot N2 rotated clockwise from ∼00:15 UT to ∼00:40 UT. The relative mean intensity of 1600 Å in this sunspot started rising at ∼00:05 UT and reached its maximum at ∼00:20 UT. The relative mean intensity of 304 Å started rising at ∼00:05 UT and reached its maximum at ∼00:30 UT. So the X5.4 flare affected the rotation of sunspot N2. The sunspot N3 remained stable during the two flares (Figure <ref>e).The rotation speed was poorly correlated with the intensity of AIA wavelengths, and therefore the X5.4 and X1.3 flares. §.§ The motion of Magnetic TransientMagnetic transients induced by flares have been well investigated by <cit.> and <cit.>. They analyzed the cause, location and level of the magnetic transients. Here, we find that there are some magnetic transients moving through sunspots N1, N2 and N3, and there is no apparent magnetic transient motion in the sunspots P1 and P2. Because the polarity of the sunspots N1, N2 and N3 is negative (black regions in the magnetograms), the magnetic transients means that the polarity is reversed.Figure <ref> shows the changes of the magnetic field in sunspot N1. At 01:09 UT of 7 March, the magnetic transients marked by red arrows in the umbra of the sunspot can be clearly seen. These magnetic transients (the positive magnetic field) moved toward the southeast direction. At 01:14 UT, the areawith positive magnetic field became larger while the strength became weaker, as shown in Figure <ref>(a2). The green line in Figure <ref>(a3) indicates the moving trajectory of the positive polarity points. To investigate the kinematic evolution of the reversed polarity, the time-distance plots obtained along the green trajectory line are shown in Figure <ref>(d). A white streak is clearly visible. It started at ∼01:07 UT and ended at ∼01:16 UT,consistent with the time of the X1.3 flare. It might imply that the reversed magnetic polarity was caused by this flare. The distance corresponding to the streakwas about 5 arcsec. Thus, the moving speed of the magnetic transients can be estimated to be about 6.59kms^-1. Similarly, some magnetic transients (i.e., the reversed magnetic field) were presented in sunspots N2 and N3, and probably associated with the X5.4 flare, as shown in Figures <ref> and <ref>. They were moving toward southeast with speed about 3.69kms^-1 and 3.22kms^-1 respectively.In order to check whether the magnetic transients were co-spatial with flare ribbons, we further examined the co-aligned AIA 304 Å and AIA 1600 Å images corresponding to HMI magnetograms, which are displayed in panels (b1)–(b3) and panels (c1)–c(3) in Figures <ref>, <ref> and <ref> respectively. The flare ribbons can be seen in these panels. They were separating from each other. The locations of sunspots N1, N2 and N3 in AIA images, marked by magnetogram contours at -800 G, were moving with flare ribbons. This implies the magnetic transients were co-spatial with flare ribbons. We calculated magnetic flux of sunspots N1, N2 and N3 for the regions marked by the red rectangular boxes in Figure <ref>. The magnetic flux of sunspots with negative polarity together with GOES soft X-ray flux are shown in Figure <ref>. Before the start of the X5.4 flare, the absolute value of magnetic flux in sunspot N1 decreased slowly. From the start to peak of the flare, it decreased gradually. Then, the value kept almost unchanged. At the end of the flare, it increased slightly. During the period of the X1.3 flare, the absolute value of magnetic flux decreased first and then increased abruptly. These results suggest that the changes of trend of magnetic flux in sunspot N1 were caused by both the X5.4 flare and the X1.3 flare.The magnetic flux of sunspot N2 mainly increased from 23:30 UT of 6 March to 01:30 UT of 7 March. There was an obvious change near the peak of X5.4 flare. Specifically, the absolute value decreased first and then increased, as shown in Figure <ref>(c). The absolute magnetic flux value of sunspot N3 (Figure <ref>d) almost kept unchanged before the start of the X5.4 flare. Similarly, during the X5.4 flare , the magnetic flux decreased rapidly, and then increased monotonically. These results suggest that the changes of trend of magnetic flux in sunspots N2 and N3 were mainly due to the X5.4 flare. § SUMMARY AND DISCUSSIONAR 11429 is a complicated active region with a reversed polarity structure, in contrast with the nature of solar cycle 24. It has a large leading sunspot with positive magnetic polarity (P1) exhibiting significant counterclockwise rotation. The rotation underwent some changes during two X-class (X5.4 and X1.3) flares. It is possible that the flares were triggered by the sunspot rotation (see ), and in turn they reacted back and affected the sunspot P1 rotation.The magnetic transients induced by the two X-class flares can be seen in sunspots with negative magnetic polarity (N1, N2 and N3). They moved outward with respect to the center of the active region and moved through the sunspots quickly with speed up to about 6.59kms^-1. The magnetic transients lasted for about ten minutes within the period of the X5.4 and X1.3 flare. As expected, the magnetic transients, which are clearly illustrated in sunspots N1, N2 and N3 were moving away from the polarity inversion line shown in Figures <ref>, <ref> and <ref>. The magnetic transients occurred during the X5.4 and X1.3 flares were co-spatial with flare ribbons. This is consistent with <cit.>, who found that magnetic transients in AR 11158 during a X2.2 flare persisted for a few minutes and showed co-spatial with flare ribbons, which were separating out with a mean velocity of 8 km s^-1.During the evolution of sunspots (N1, N2 and N3), changes in the trend of magnetic field evolution were presented, and are associated with the two X-class flares. Similar results in five δ sunspots were found by <cit.>. The increasing and decreasing tendency of the magnetic flux in active region might be attributed to magnetic flux emergence and magnetic cancellation <cit.>.The evolution of sunspots with complicated structure can be affected by many factors such as interaction among sunspots within same active region, physical change induced by flare, and interaction between magnetic field and plasma. In this study, we have found that the motion and the trend of magnetic field evolution of sunspots are significantly affected by flares. Other influences on the evolution of sunspots will be examined in future studies. The authors thank the referees for their corrections and valuable suggestions to improve the paper This work is supported by the National Natural Science Foundations of China (41231068, 41374187, 41531073, and 41674147). We acknowledge the SDO/HMI and SDO/AIA teams for providing the data. The GOES soft X-ray flux data is available at NASA/GSFC Solar Data Analysis Center (SDAC). raa | http://arxiv.org/abs/1704.08018v2 | {
"authors": [
"Jianchuan Zheng",
"Zhiliang Yang",
"Jianpeng Guo",
"Kaiming Guo",
"Hui Huang",
"Xuan Song",
"Weixing Wan"
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"published": "20170426090553",
"title": "Sunspots rotation and magnetic transients associated with flares in NOAA AR 11429"
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0 cm 0 cm-1.0 cm 22 cm 16.5 cm-2.0 cm 23.6 cm 16.5 cm #1 | http://arxiv.org/abs/1704.08725v2 | {
"authors": [
"Robert B. Griffiths"
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"quant-ph"
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"primary_category": "quant-ph",
"published": "20170427192430",
"title": "What Quantum Measurements Measure"
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arabic Hjalmar Rosengren and S. Ole Warnaar CHAPTER: ELLIPTIC HYPERGEOMETRIC FUNCTIONS ASSOCIATED WITH ROOT SYSTEMS § INTRODUCTION Let f=∑_n≥ 0 c_n. The series f is called hypergeometric if the ratio c_n+1/c_n, viewed as a function of n, is rational. A simple example is the Taylor series exp(z)=∑_n=0^∞ z^n/n!. Similarly, if the ratio of consecutive terms of f is a rational function of q^n for some fixed q — known as the base — then f is called a basic hypergeometric series. An early example of a basic hypergeometric series is Euler's q-exponentialfunction e_q(z)=∑_n≥ 0 z^n/((1-q)⋯(1-q^n)). If we express the base as q=exp(2π/ω) then c_n+1/c_n becomes a trigonometric function in n, with period ω.This motivates the more general definition of an elliptic hypergeometric series as a series f for which c_n+1/c_n is a doubly-periodic meromorphic function of n.Elliptic hypergeometric series first appeared in 1988 in the work of Dateet al. on exactly solvable lattice models in statisticalmechanics <cit.>. They were formally defined and identified as mathematical objects ofinterest in their own right by Frenkel and Turaev in 1997 <cit.>. Subsequently, Spiridonov introduced the elliptic beta integral, initiating a parallel theory of elliptic hypergeometric integrals <cit.>. Together with Zhedanov <cit.> he also showed that Rahman's<cit.> and Wilson's <cit.> theory of biorthogonal rationalfunctions — itself a generalization of the Askey scheme <cit.>of classical orthogonal polynomials — can be lifted to the elliptic level.All three aspects of the theory of elliptic hypergeometric functions(series, integrals and biorthogonal functions) have been generalized to higher dimensions, connecting them to root systems andMacdonald–Koornwinder theory. In <cit.> Warnaar introduced elliptic hypergeometric series associated to root systems, including a conjectural series evaluation of type C_n.This was recognized by van Diejen and Spiridonov <cit.> as a discrete analogue of a multiple elliptic beta integral (or elliptic Selberg integral). They formulated the corresponding integral evaluation, again as a conjecture. This in turn led Rains <cit.> to develop an elliptic analogue of Macdonald–Koornwinder theory, resulting in continuous as well as discrete biorthogonal elliptic functions attached to the non-reduced rootsystem BC_n.In this theory, the elliptic multiple beta integral and its discreteanalogue give the total mass of the biorthogonality measure.Although a relatively young field, the theory of elliptic hypergeometricfunctions has already seen some remarkable applications. Many of these involve the multivariable theory. In 2009, Dolan and Osborn showed that supersymmetric indices offour-dimensional supersymmetric quantum field theories are expressiblein terms of elliptic hypergeometric integrals <cit.>.Conjecturally, such field theories admit electric–magnetic dualities known as Seiberg dualities, such that dual theories have the same index. This leads to non-trivial identities between elliptic hypergeometricintegrals (or, for so called confining theories, to integral evaluations).In some cases these are known identities, which thus gives a partialconfirmation of the underlying Seiberg duality. However, in many cases it leads to new identities that are yet to be rigorously proved, see e.g.<cit.> and the recent survey <cit.>. Another application of elliptic hypergeometric functions is to exactly solvable lattice models in statistical mechanics.We already mentioned the occurrence of elliptic hypergeometric series in the work of Date et al., but more recently it was shown that elliptic hypergeometric integrals are related to solvable lattice models with continuous spin parameters <cit.>. In the one-variable case, this leads to a generalization of many well-known discrete models such as the two-dimensional Ising model and the chiral Potts model. This relation to solvable lattice models has been extended to multivariable elliptic hypergeometric integrals in <cit.>. Further applications of multivariable elliptic hypergeometric functions pertain to elliptic Calogero–Sutherland-type systems <cit.> and the representation theory of elliptic quantum groups <cit.>.In the current chapter we give a survey of elliptic hypergeometric functions associated with root systems, comprised of three main parts. The first two form in essence an annotated table of the main evaluation and transformation formulas for elliptic hypergeometric integrals and series on root systems.The third and final part gives an introduction to Rains' elliptic Macdonald–Koornwinder theory (in part also developed by Coskun and Gustafson <cit.>). Due to space limitations, applications will not be covered here and we refer the interested reader to the above-mentioned papers and references therein.Rather than throughout the text, references for the main resultsare given in the form of separate notes at the end of each section. These notes also contain some brief historical comments and further pointersto the literature.Acknowledgements: We thank Ilmar Gahramanov,Eric Rains, Michael Schlosser and Vyacheslav Spiridonov forvaluable comments. §.§ Preliminaries Elliptic functions are doubly-periodic meromorphic functions on ℂ. That is, a meromorphic function g:ℂ→ℂ is elliptic if there exist ω_1,ω_2 with Im(ω_1/ω_2)>0such that g(z+ω_1)=g(z+ω_2)=g(z) for all z∈ℂ. If we define the elliptic nome p by p=^2πω_1/ω_2 (so that p<1) then z↦^2π z/ω_2 maps the period parallelogram spanned by ω_1,ω_2 to an annulus with radii p and 1. Given an elliptic function g with periods ω_1 and ω_2, the function f:ℂ^∗→ℂ defined byg(z)=f(^2π z/ω_2)is thus periodic in an annulus:f(pz)=f(z).By mild abuse of terminology we will also refer to such f as an elliptic function. A more precise description would be elliptic function in multiplicative form.The basic building blocks for elliptic hypergeometric functions areθ(z)=θ(z;p) =∏_i=0^∞(1-zp^i)(1-p^i+1/z), (z)_k=(z;q,p)_k =∏_i=0^k-1θ(zq^i;p),Γ(z)=Γ(z;p,q) = ∏_i,j=0^∞1-p^i+1q^j+1/z/1-zp^iq^j,known as the modified theta function, elliptic shifted factorial andelliptic gamma function, respectively. Note that the dependence on the elliptic nome p and base q will mostly be suppressed from our notation. One exception is the q-shifted factorial(z;q)_∞=∏_i≥ 0(1-zq^i) which, to avoid possible confusion, will never be shortened to (z)_∞.For simple relations satisfied by the above three functions we referthe reader to <cit.>. Here we only note that the elliptic gamma function is symmetric in p and q and satisfiesΓ(pq/z)Γ(z)=1 andΓ(qz)=θ(z)Γ(z). For each of the functions θ(z), (z)_k and Γ(z), weemploy condensed notation as exemplified byθ(z_1,…,z_m) =θ(z_1)…θ(z_m),(az^±)_k =(az)_k(a/z)_k, Γ(tz^±w^±) =Γ(tzw)Γ(tz/w) Γ(tw/z)Γ(t/zw). In the trigonometric case p=0 we have θ(z)=1-z, so that(z)_k becomes a standard q-shifted factorial and Γ(z) arescaled version of the q-gamma function.We also need elliptic shifted factorials indexed by partitions. A partition λ=(λ_1,λ_2,…) is a weakly decreasingsequence of non-negative integers such that only finitely many λ_i are non-zero. The number of positive λ_i is called the length of λ and denoted by l(λ). The sum of the λ_i will be denoted by λ.Thediagram of λ consists of the points (i,j)∈ℤ^2 such that 1≤ i≤ l(λ) and 1≤ j≤λ_i.If these inequalities hold for (i,j)∈ℤ^2 we write(i,j)∈λ. Reflecting thediagram in the main diagonal yields the conjugate partition λ'. In other words, the rows of λ are the columns of λ' and vice versa. A standard statistic on partitions isn(λ)=∑_i≥ 1 (i-1)λ_i =∑_i≥ 1λ'_i2.For a pair of partitions λ,μ we write μ⊂λ if μ_i≤λ_i for all i≥ 1. In particular, when l(λ)≤ n and λ_i≤ N for all 1≤ i≤ N we write λ⊂ (N^n). Similarly, we write μ≺λ if the interlacing conditions λ_1≥μ_1≥λ_2≥μ_2≥⋯ hold.With t an additional fixed parameter, we will need the following threetypes of elliptic shifted factorials index by partitions:(z)_λ=(z;q,t;p)_λ =∏_(i,j)∈λθ(zq^j-1t^1-i) =∏_i≥ 1(zt^1-i)_λ_i, C^-_λ(z)=C^-_λ(z;q,t;p) =∏_(i,j)∈λθ(zq^λ_i-jt^λ'_j-i), C^+_λ(z)=C^+_λ(z;q,t;p) =∏_(i,j)∈λθ(zq^λ_i+j-1t^2-λ'_j-i).By θ(pz)=-z^-1θ(z) it follows that (a)_λ is quasi-periodic:(p^k z)_λ=[(-z)^-λq^-n(λ') t^n(λ)]^kp^-k2λ(z)_λ,k∈ℤ.Again we use condensed notation so that, for example, (a_1,…,a_k)_λ=(a_1)_λ⋯(a_k)_λ.§.§ Elliptic Weyl denominators Suppressing their p-dependence we defineΔ^A(x_1,…,x_n+1) =∏_1≤ i<j≤ n+1x_j θ(x_i/x_j), Δ^C(x_1,…,x_n) =∏_j=1^nθ(x_j^2) ∏_1≤ i<j≤ nx_j θ(x_ix_j^±),which are essentially the Weyl denominators of the affine root systems A_n^(1) and C_n^(1) <cit.>. Although we have no need for the theory of affine root systems here, it may be instructive to explain the connection to the root systemC_n^(1) (the case of A_n^(1) is similar). TheWeyl denominator of an affine root system R is the formal product ∏_α∈ R_+(1-^-α)^m(α), where R_+ denotes the set of positive roots and m is a multiplicity function. For C_n^(1), the positive roots arem δ,m≥ 1, m δ+2ε_i,m≥ 0, 1≤ i≤ n, m δ-2ε_i,m≥ 1, 1≤ i≤ n, m δ+ε_i±ε_j,m≥ 0,1≤ i<j≤ n, m δ-ε_i±ε_j, m≥ 1,1≤ i<j≤ n,where ε_1,…,ε_n are the coordinate functionson ^n and δ is the constant function 1. The roots m δ have multiplicity n, while all other roots havemultiplicity 1. Thus, the Weyl denominator for C_n^(1) is∏_m=0^∞((1-^-(m+1)δ)^n ∏_i=1^n (1-^-m δ-2ε_i) (1-^-(m+1)δ+2ε_i) ×∏_1≤ i<j≤ n(1-^-m δ-ε_i-ε_j) (1-^-(m+1)δ+ε_i+ε_j) (1-^-m δ-ε_i+ε_j) (1-^-(m+1)δ+ε_i-ε_j)). It is easy to check that this equals(p;p)_∞^n x_1^0x_2^-1⋯ x_n^1-n Δ^C(x_1,…,x_n), wherep=^-δ and x_i=^-ε_i.We will consider elliptic hypergeometric series containing the factor Δ^A(xq^k) or Δ^C(xq^k), where xq^k=(x_1q^k_1,x_2q^k_2,…,x_rq^k_r),the k_i∈ℤ being summation indices, and r=n+1 in thecase of A_n and r=n in the case of C_n.We refer to these as A_n and C_n series,respectively. In the case of A_n,the summation variables typicallysatisfy a restriction of the form k_1+…+k_n+1=N. Eliminating k_n+1 gives series containing the A_n-1^(1)Weyl denominator times ∏_i=1^nθ(ax_iq^k_i+k), where a=q^-N/x_n+1; these will also be viewed as A_n series.Similarly, A_n integrals contain the factor 1/∏_1≤ i<j≤ n+1Γ(z_i/z_j,z_j/z_i),where z_1⋯ z_n+1=1, while C_n integrals contain1/∏_i=1^nΓ(z_i^± 2)∏_1≤ i<j≤ nΓ(z_i^±z_j^±). If we denote the expression (<ref>) by g(z) then it is easy toverify that, for k∈ℤ^n,g(zq^k)/g(z) =( ∏_i=1^nq^-nk_i-(n+1)k_i^2z_i^-2(n+1)k_i) Δ^C(zq^k)/Δ^C(z).A similar relation holds for the A-type factors. This shows that the series can be considered as discrete analogues ofthe integrals. In fact in many instances the series can be obtained from the integrals via residue calculus.It is customary to attach a “type” to hypergeometric integrals associated with root systems, although different authors have used slightly different definitions of type. As the terminology will be used here, in type I integrals the only factors containing more than one integration variable are (<ref>), while type II integrals contain twice the number of such factors.For example, C_n^(II) integrals contain the factor∏_i<jΓ(tz_i^±z_j^±)/Γ(z_i^±z_j^±).It may be noted that, under appropriate assumptions on the parameters,lim_q→ 1lim_p→ 0∏_i<jΓ(q^t± z_i± z_j)/Γ(q^± z_i± z_j) =lim_q→ 1∏_i<j(q^± z_i± z_j;q)_∞/(q^t± z_i± z_j;q)_∞=∏_i<j(1-z_i^±z_j^±)^t= ∏_i<j((z_i+z_i^-1)-(z_j+z_j^-1))^2t.For this reason C_n beta integrals of type II are sometimes referred to as elliptic Selberg integrals. There are also integrals containing an intermediate number of factors.We will refer to these as integrals of mixed type.§ INTEGRALS Throughout this section we assume that q<1.Whenever possible, we have restricted the parameters in such a way that theintegrals may be taken over the n-dimensional complex torus 𝕋^n. However, all results can be extended to more general parameter domains by appropriately deforming 𝕋^n.When n=1 all the stated A_n and C_n beta integral evaluations reduce to Spiridonov's elliptic beta integral. §.§ A_n beta integrals We will present four A_n beta integrals. In each of these the integrand contains a variable z_n+1 which is determined from the integration variables z_1,…,z_n by the relationz_1⋯ z_n+1=1.To shorten the expressions we define the constant κ_n^Abyκ_n^A=(p;p)_∞^n(q;q)_∞^n/(n+1)!(2π)^n. For 1≤ i≤ n+2, let s_i<1 and t_i<1, such that ST=pq, where S=s_1⋯ s_n+2 and T=t_1⋯ t_n+2.Then we have the type I integralκ_n^A∫_𝕋^n∏_i=1^n+2∏_j=1^n+1Γ(s_iz_j,t_i/z_j)/∏_1≤ i<j≤ n+1Γ(z_i/z_j,z_j/z_i)z_1/z_1⋯ z_n/z_n=∏_i=1^n+2Γ(S/s_i,T/t_i) ∏_i,j=1^n+2Γ(s_it_j). Next, let s<1, t<1, s_i<1 and t_i<1for 1≤ i≤ 3, such that s^n-1t^n-1s_1s_2s_3t_1t_2t_3=pq. Then we have the type II integralκ_n^A∫_𝕋^n∏_1≤ i<j≤ n+1Γ(sz_iz_j,t/z_iz_j)/Γ(z_i/z_j,z_j/z_i)∏_i=1^3 ∏_j=1^n+1Γ(s_iz_j,t_i/z_j)z_1/z_1⋯ z_n/z_n=∏_m=1^N(Γ(s^mt^m)∏_1≤ i<j≤ 3Γ(s^m-1t^ms_is_j,s^mt^m-1t_it_j) ∏_i,j=1^3Γ(s^m-1t^m-1s_it_j)) × Γ(s^N-1s_1s_2s_3,t^N-1t_1t_2t_3) ∏_i=1^3Γ(s^Ns_i,t^Nt_i),n=2N, ∏_m=1^N (Γ(s^mt^m)∏_1≤ i<j≤ 3Γ(s^m-1t^ms_is_j,s^mt^m-1t_it_j)) ∏_m=1^N+1∏_i,j=1^3Γ(s^m-1t^m-1s_it_j) × Γ(s^N+1,t^N+1) ∏_1≤ i<j≤ 3Γ(s^Ns_is_j,t^Nt_it_j),n=2N+1. Let t<1, t_i<1 for 1≤ i≤ n+3 andt<s_i<t^-1 for 1≤ i≤ n,where t^2t_1⋯ t_n+3=pq. Then,κ_n^A∫_𝕋^n∏_1≤ i<j≤ n+11/Γ(z_i/z_j,z_j/z_i,t^2z_iz_j)∏_j=1^n+1(∏_i=1^n Γ(ts_i^± z_j) ∏_i=1^n+3Γ(t_i/z_j) ) z_1/z_1⋯ z_n/z_n=∏_i=1^n∏_j=1^n+3Γ(ts_i^±t_j) ∏_1≤ i<j≤ n+31/Γ(t^2t_it_j),which is an integral of mixed type.Finally, let t<1, s_i<1 for 1≤ i≤ 4 and t_i<1 for 1≤ i≤ n+1 such that t^n-1s_1⋯ s_4T=pq,where T=t_1⋯ t_n+1. Then we have a second mixed-type integral:κ_n^A∫_𝕋^n∏_1≤ i<j≤ n+1Γ(tz_iz_j)/Γ(z_i/z_j,z_j/z_i)∏_j=1^n+1(∏_i=1^4Γ(s_iz_j)∏_i=1^n+1Γ(t_i/z_j))z_1/z_1⋯ z_n/z_n=Γ(T)∏_i=1^4Γ(t^Ns_i)/Γ(t^NTs_i)∏_1≤ i<j≤ n+1Γ(tt_it_j) ∏_i=1^4 ∏_j=1^n+1Γ(s_it_j), n=2N, Γ(t^N+1,T)/Γ(t^N+1T)∏_1≤ i<j≤ 4Γ(t^Ns_is_j) ∏_1≤ i<j≤ n+1Γ(tt_it_j) ∏_i=1^4 ∏_j=1^n+1Γ(s_it_j),n=2N+1.§.§ C_n beta integrals We will give three C_n beta integrals.They all involve the constantκ_n^C=(p;p)_∞^n(q;q)_∞^n/n!2^n(2π)^n. Let t_i<1 for 1≤ i≤ 2n+4 such that t_1⋯ t_2n+4=pq.We then have the following C_n beta integral of type Iκ_n^C∫_𝕋^n∏_1≤ i<j≤ n1/Γ(z_i^±z_j^±)∏_j=1^n∏_i=1^2n+4Γ(t_iz_j^±)/Γ(z_j^± 2)z_1/z_1⋯ z_n/z_n=∏_1≤ i<j≤ 2n+4Γ(t_it_j). Next, let t<1 and t_i<1 for 1≤ i≤ 6 such that t^2n-2t_1⋯ t_6=pq. We then have the type II C_n beta integral κ_n^C∫_𝕋^n∏_1≤ i<j≤ nΓ(tz_i^±z_j^±)/Γ(z_i^±z_j^±)∏_j=1^n∏_i=1^6Γ(t_iz_j^±)/Γ(z_j^± 2)z_1/z_1⋯ z_n/z_n=∏_m=1^n( Γ(t^m)/Γ(t)∏_1≤ i<j≤ 6Γ(t^m-1t_it_j)).This is the elliptic Selberg integral mentioned in the introduction.At this point it is convenient to introduce notation for more general C_n integrals of type II.For m a non-negative integer, let t<1 and t_i<1for 1≤ i≤ 2m+6 such thatt^2n-2t_1⋯ t_2m+6=(pq)^m+1. We then defineJ_C_n^(m)(t_1,…,t_2m+6;t)= κ_n^C∫_𝕋^n∏_1≤ i<j≤ nΓ(tz_i^±z_j^±)/Γ(z_i^±z_j^±)∏_j=1^n∏_i=1^2m+6Γ(t_iz_j^±)/Γ(z_j^± 2)z_1/z_1⋯ z_n/z_n.Note that (<ref>) gives a closed-form evaluation for the integral J_C_n^(0).As outlined in <cit.>, J_C_n^(m) can becontinued to a single-valued meromorphic function in the parameters t_i and t subject to the constraint (<ref>).For generic values of the parameters this continuation is obtainedby replacing the integration domain with an appropriate deformation of 𝕋^n. We can now state the second C_n beta integral of type II asJ_C_n^(n-1)(t_1,…,t_4,s_1,…,s_n,pq/ts_1,…,pq/ts_n;t) =Γ(t)^n ∏_l=1^n∏_1≤ i<j≤ 4Γ(t^l-1t_it_j) ∏_i=1^n∏_j=1^4 Γ(s_it_j)/Γ(ts_i/t_j),wheret^n-2t_1t_2t_3t_4=1.In this identity it is necessary to work with an analytic continuation of (<ref>) since the inequalities t_i, t<1 areincompatible with t^n-2t_1t_2t_3t_4=1 for n≥ 2. §.§ Integral transformations Wenow turn to integral transformations, starting with integrals of type I.For m a non-negative integer we introduce the notationI_A_n^(m)(s_1,…,s_m+n+2;t_1,…,t_m+n+2)= κ_n^A∫_𝕋^n∏_i=1^m+n+2∏_j=1^n+1Γ(s_iz_j,t_i/z_j)/∏_1≤ i<j≤ n+1Γ(z_i/z_j,z_j/z_i)z_1/z_1⋯ z_n/z_n,where s_i<1 and t_i<1 for all i,∏_i=1^m+n+2s_it_i=(pq)^m+1 and z_1⋯ z_n+1=1.We also defineI_C_n^(m)(t_1,…,t_2m+2n+4) =κ_n^C∫_𝕋^n∏_1≤ i<j≤ n1/Γ(z_i^±z_j^±)∏_j=1^n∏_i=1^2m+2n+4Γ(t_iz_j^±)/Γ(z_j^± 2)z_1/z_1⋯ z_n/z_n,wheret_i<1 for all i and t_1⋯ t_2m+2n+4=(pq)^m+1.The A_n integral satisfiesI_A_n^(m)(s_1,…,s_m+n+2;t_1,…,t_m+n+2)= I_A_n^(m)(s_1ζ,…,s_m+n+2ζ; t_1/ζ,…,t_m+n+2/ζ)for ζ any (n+1)-th root of unity, whereas the C_n integral is invariant under simultaneous negation of all of the t_i. We further note that (<ref>) and (<ref>) provide closed-formevaluations of I_A_n^(0) and I_C_n^(0), respectively.For the integral I_A_n^(m), the following transformation reverses the roles of m and n:I_A_n^(m)(s_1,…,s_m+n+2;t_1,…,t_m+n+2)=∏_i,j=1^m+n+2Γ(s_it_j)· I_A_m^(n)(λ/s_1,…,λ/s_m+n+2;pq/λ t_1,…, pq/λ t_m+n+2),where λ^m+1=s_1⋯ s_m+n+2,(pq/λ)^m+1=t_1⋯ t_m+n+2. Moreover, for t_1⋯ t_2m+2n+4=(pq)^m+1, there is an analogoustransformation of type C:I_C_n^(m)(t_1,…,t_2m+2n+4) =∏_1≤ i<j≤ 2m+2n+4Γ(t_it_j)· I_C_m^(n)(√(pq)/t_1,…,√(pq)/t_2m+2n+4). It is easy to check thatI_A_1^(m)(t_1,…,t_m+3;t_m+4,…,t_2m+6)= I_C_1^(m)(t_1,…,t_2m+6). Thus, combining (<ref>) and (<ref>) leads toI_A_n^(1)(s_1,…,s_n+3;t_1,…,t_n+3) =∏_1≤ i<j≤ n+3Γ(S/s_is_j,T/t_it_j)· I_C_n^(1) (s_1/v,…,s_n+3/v,t_1v,…,t_n+3v),where S=s_1⋯ s_n+3, T=t_1⋯ t_n+3 and ν^2=S/pq=pq/T. Since I_C_n^(1) is symmetric,(<ref>) implies non-trivial symmetries of I_A_n^(1), such asI_A_n^(1)(s_1,…,s_n+3;t_1,…,t_n+3)= ∏_i=1^n+2Γ(s_it_n+3,t_is_n+3,S/s_is_n+3,T/t_it_n+3) × I_A_n^(1)(s_1/v,…,s_n+2/v,s_n+3v^n; t_1v,…,t_n+2v,t_n+3/v^n),where, with the same definitions of S and T as above, ST=(pq)^2 and ν^n+1=St_n+3/pqs_n+3. J_C_n^(1)(t_1,…,t_8;t)= ∏_m=1^n (∏_1≤ i<j≤ 4Γ(t^m-1t_it_j) ∏_5≤ i<j≤ 8Γ(t^m-1t_it_j))× J_C_n^(1)(t_1v,…,t_4v,t_5/v,…,t_8/v;t),wheret^2n-2t_1⋯ t_8=(pq)^2 andv^2=pqt^1-n/t_1t_2t_3t_4=t^n-1t_5t_6t_7t_8/pq. Iterating this transformation yields a symmetry of J_C_n^(1) under the Weyl group of type E_7<cit.>.We conclude with a transformation between C_n and C_m integrals of type II:J_C_n^(m+n-1) (t_1,…,t_4,s_1,…, s_m+n,pq/ts_1,…,pq/ts_m+n;t) =Γ(t)^n-m∏_1≤ i<j≤ 4∏_l=1^n Γ(t^l-1t_it_j)/∏_l=1^m Γ(t^l+n-m-1t_it_j)∏_i=1^m+n∏_j=1^4 Γ(s_it_j)/Γ(ts_i/t_j)× J_C_m^(m+n-1)(t/t_1,…,t/t_4,s_1,…, s_m+n,pq/ts_1,…,pq/ts_m+n;t),where t_1t_2t_3t_4=t^m-n+2. §.§ Notes For p=0 the integrals (<ref>), (<ref>), (<ref>) and(<ref>) are due to Gustafson <cit.>, the integral (<ref>) to Gustafson and Rakha <cit.> and the transformation (<ref>) to Denis and Gustafson <cit.>.None of the p=0 instances of (<ref>), (<ref>)–(<ref>),(<ref>) and (<ref>) were known prior to the elliptic case.For general p, van Diejen and Spiridonov conjectured the type IC_n beta integral (<ref>) and showed that it implies the elliptic Selberg integral (<ref>) <cit.>. A rigorous derivation of the classical Selberg integral as a special limit of (<ref>) is due to Rains <cit.>. Spiridonov <cit.> conjectured the type I A_n beta integral (<ref>) and showed that, combined with (<ref>), it implies the type II A_n beta integral (<ref>), as well as the integral (<ref>) of mixed type. He also showed that (<ref>) implies (<ref>).The first proofs of the fundamental type I integrals (<ref>) and(<ref>) were obtained by Rains <cit.>. For subsequent proofs of (<ref>), (<ref>) and (<ref>), see <cit.>, <cit.> and <cit.>,respectively. In <cit.> Rains also proved the integral transformations(<ref>), (<ref>) and (<ref>), and gave furthertransformations analogous to (<ref>). The integral (<ref>) of mixed type is due to Spiridonov and Warnaar <cit.>. The transformation (<ref>), which includes (<ref>) as its m=0 case, was conjectured by Rains <cit.> and also appearsin <cit.>. It was first proved by van der Bult in <cit.> and subsequently proved and generalized to an identity for the “interpolation kernel” (an analytic continuation of the elliptic interpolation functions R^∗_λ of Section <ref>) in <cit.>.Several of the integral identities surveyed here have analogues for q=1. In the case of(<ref>), (<ref>) and (<ref>) these were found in <cit.>, and the unit-circle analogue of (<ref>) is given in <cit.>.In <cit.> Spiridonov gives one more C_n beta integral,which lacks the p↔ q symmetry present in all the integralsconsidered here, and is more elementary in that it follows as a determinant of one-variable beta integrals.In <cit.> Rains conjectured several quadratic integral transformations involving the interpolation functions R^*_λ.These conjectures were proved in <cit.>.In special cases, they simplify to transformations for the functionJ_C_n^(2).Motivated by quantum field theories on lens spaces, Spiridonov<cit.> evaluated certain finite sums of C_n integrals, both for type I and type II. In closely related work, Kels and Yamazaki <cit.> obtainedtransformation formulas for finite sums of A_n and C_n integrals of type I.As mentioned in the introduction, the recent identification of elliptichypergeometric integrals as indices in supersymmetric quantum field theory by Dolan and Osborn <cit.> has led to a large number of conjectured integral evaluations and transformations <cit.>.It is too early to give a survey of the emerging picture, but it is clear that the identities stated in this section are a small samplefrom a much larger collection of identities.§ SERIES In this section we give the most important summation and transformation formulas for elliptic hypergeometric series associated to A_n and C_n.In the n=1 case all summations except for (<ref>) simplify to theelliptic Jackson summation of Frenkel and Turaev. Similarly, most transformations may be viewed as generalizations of the elliptic Bailey transformation. §.§ A_n summations The following A_n elliptic Jackson summation is a discreteanalogue of the multiple beta integral (<ref>):∑_k_1,…,k_n+1≥ 0k_1+⋯+k_n+1=NΔ^A(xq^k)/Δ^A(x)∏_i=1^n+1∏_j=1^n+2(x_ia_j)_k_i/(bx_i)_k_i∏_j=1^n+1(qx_i/x_j)_k_i =(b/a_1,…,b/a_n+2)_N/(q,bx_1,…,bx_n+1)_N,where b=a_1… a_n+2x_1⋯ x_n+1. Using the constraint on the summation indices to eliminate k_n+1,this identity can be written less symmetrically as∑_k_1,…, k_n≥ 0 k≤ NΔ^A(xq^k)/Δ^A(x)∏_i=1^n(θ(ax_iq^k_i+k)/θ(ax_i) (ax_i)_k∏_j=1^n+2(x_ib_j)_k_i/(aq^N+1x_i,aqx_i/c)_k_i∏_j=1^n(qx_i/x_j)_k_i)(q^-N,c)_k/∏_i=1^n+2(aq/b_i)_k q^k=c^N∏_i=1^n(aqx_i)_N/(aqx_i/c)_N∏_i=1^n+2(aq/cb_i)_N/(aq/b_i)_N,where b_1… b_n+2cx_1… x_n=a^2q^N+1. By analytic continuation one can then deduce the companion identity∑_k_1,…,k_n=0^N_1,…,N_n ( Δ^A(xq^k)/Δ^A(x)∏_i=1^n(θ(ax_iq^k_i+k)/θ(ax_i) (ax_i)_k(dx_i,ex_i)_k_i/(aq^N_i+1x_i)_k(aqx_i/b,aqx_i/c)_k_i) ×(b,c)_kq^k/(aq/d,aq/e)_k ∏_i,j=1^n(q^-N_jx_i/x_j)_k_i/(qx_i/x_j)_k_i) =(aq/cd,aq/bd)_N/(aq/d,aq/bcd)_N∏_i=1^n(aqx_i,aqx_i/bc)_N_i/(aqx_i/b,aqx_i/c)_N_i,where bcde=a^2q^N+1. For the discrete analogue of the type II integral (<ref>) we refer the reader to the Notes at the end of this section.Our next result corresponds to a discretization of (<ref>):∑_k_1,…,k_n+1≥ 0 k_1+…+k_n+1=NΔ^A(xq^k)/Δ^A(x)∏_1≤ i<j≤ n+11/(x_ix_j)_k_i+k_j∏_i=1^n+1q^k_i2x_i^k_i∏_j=1^n (x_ia_j^±)_k_i/(bx_i,q^1-Nx_i/b)_k_i∏_j=1^n+1(qx_i/x_j)_k_i=(-bq^N-1)^N∏_i=1^n(ba_i^±)_N/(q)_N∏_i=1^n+1(bx_i^±)_N.Mimicking the steps that led from (<ref>) to (<ref>), the identity (<ref>) can be rewritten as a sum over an n-dimensional rectangle, see <cit.>. Some authors have associated (<ref>) and related results with the root system D_n rather than A_n.Finally, the following summation is a discrete analogue of (<ref>):∑_k_1,…,k_n+1≥ 0, k_1+…+k_n+1=NΔ^A(xq^k)/Δ^A(x)∏_1≤ i<j≤ n+1q^k_ik_j(x_ix_j)_k_i+k_j∏_i=1^n+1∏_j=1^4(x_ib_j)_k_i/x_i^k_i∏_j=1^n+1(qx_i/x_j)_k_i=(Xb_1,Xb_2,Xb_3,Xb_4)_N/X^N(q)_N, n odd, (X,Xb_1b_2,Xb_1b_3,Xb_1b_4)_N/(Xb_1)^N(q)_N, n even,where X=x_1… x_n+1 and q^N-1b_1… b_4X^2=1. §.§ C_n summations The following C_n elliptic Jackson summation is a discreteanalogue of (<ref>):∑_k_1,…,k_n=0^N_1,…,N_nΔ^C(xq^k)/Δ^C(x)∏_i=1^n(bx_i,cx_i,dx_i,ex_i)_k_i q^k_i/(qx_i/b,qx_i/c,qx_i/d,qx_i/e)_k_i∏_i,j=1^n(q^-N_jx_i/x_j,x_ix_j)_k_i/(qx_i/x_j,q^N_j+1x_ix_j)_k_i=∏_i,j=1^n(qx_ix_j)_N_i/∏_1≤ i<j≤ n(qx_ix_j)_N_i+N_j (q/bc,q/bd,q/cd)_N/∏_i=1^n(qx_i/b,qx_i/c,qx_i/d,q^-N_ie/x_i)_N_i,where bcde=q^N+1.The discrete analogues of the type II integrals (<ref>) and (<ref>)are most conveniently expressed in terms of the seriesr+1rn(a;b_1,…,b_r-4) =∑_λ(∏_i=1^nθ(at^2-2iq^2λ_i)/θ(at^2-2i) (at^1-n,b_1,…,b_r-4)_λ/(qt^n-1,aq/b_1,…,aq/b_r-4)_λ ×∏_1≤ i<j≤ n( θ(t^j-iq^λ_i-λ_j,at^2-i-jq^λ_i+λ_j)/θ(t^j-i,at^2-i-j) (t^j-i+1)_λ_i-λ_j(at^3-i-j)_λ_i+λ_j/(qt^j-i-1)_λ_i-λ_j(aqt^1-i-j)_λ_i+λ_j)q^λt^2n(λ)),where the summation is over partitions λ=(λ_1,…,λ_n)of length at most n. Note that this implicitly depends on t as well asp and q. When b_r-4=q^-N with N∈ℤ_≥ 0, this becomes a terminating series, with sum ranging over partitions λ⊂(N^n). The series (<ref>) is associated with C_n since∏_i=1^n θ(at^2-2iq^2λ_i)/θ(at^2-2i)∏_1≤ i<j≤ nθ(t^j-i q^λ_i-λ_j, at^2-i-jq^λ_i+λ_j)/θ(t^j-i,at^2-i-j) =q^-n(λ)Δ^C(xq^λ)/Δ^C(x),with x_i=√(a)t^1-i.Using the above notation, the discrete analogue of (<ref>) is109n(a;b,c,d,e,q^-N)= (aq,aq/bc,aq/bd,aq/cd)_(N^n)/(aq/b,aq/c,aq/d,aq/bcd)_(N^n),where bcdet^n-1=a^2q^N+1.This is the C_n summation mentioned in the introduction.Next, we give a discrete analogue of (<ref>):2r+82r+7n(a;t^1-nb,a/b,c_1q^k_1,…,c_rq^k_r, aq/c_1,…,aq/c_r ,q^-N)=(aq,qt^n-1)_(N^n)/(bq,aqt^n-1/b)_(N^n)∏_i=1^r(c_ib/a,c_it^n-1/b)_(k_i^n)/(c_i,c_it^n-1/a)_(k_i^n),where the k_i are non-negative integers such that k_1+…+k_r=N. As the summand contains the factors(c_iq^k_i)_λ/(c_i)_λ,this is a so-called Karlsson–Minton-type summation.Finally, we have the C_n summation ∑_k_1,…,k_n=0^NΔ^C(xq^k)/Δ^C(x)∏_i=1^n (x_i^2,bx_i,cx_i,dx_i,ex_i,q^-N)_k_i/(q,qx_i/b,qx_i/c,qx_i/d,qx_i/e,q^N+1x_i^2)_k_iq^k_i=∏_1≤ i<j≤ nθ(q^Nx_ix_j)/θ(x_ix_j)∏_i=1^n(qx_i^2,q^2-i/bc,q^2-i/bd,q^2-i/cd)_N/(qx_i/b,qx_i/c,qx_i/d,q^-Ne/x_i)_N,where bcde=q^N-n+2.There is acorresponding integral evaluation <cit.>, which was mentioned in <ref>.§.§ Series transformations Several of the transformations stated below have companion identities(similar to the different versions of (<ref>)) which will not bestated explicitly.The following A_n Bailey transformation is a discrete analogue of (<ref>):∑_k_1,…,k_n=0^N_1,…,N_n(Δ^A(xq^k)/Δ^A(x)∏_i=1^n (θ(ax_iq^k_i+k)/θ(ax_i) (ax_i)_k(ex_i,fx_i,gx_i)_k_i/(aq^N_i+1x_i)_k(aqx_i/b,aqx_i/c,aqx_i/d)_k_i) ×(b,c,d)_kq^k/(aq/e,aq/f,aq/g)_k ∏_i,j=1^n (q^-N_jx_i/x_j)_k_i/(qx_i/x_j)_k_i) =(a/λ)^N(λ q/f,λ q/g)_N/(aq/f,aq/g)_N∏_i=1^n(aqx_i,λ qx_i/d)_N_i/(λ qx_i,aqx_i/d)_N_i×∑_k_1,…,k_n=0^N_1,…,N_n(Δ^A(xq^k)/Δ^A(x)∏_i=1^n(θ(λ x_i q^k_i+k)/θ(λ x_i) (λ x_i)_k(λ ex_i/a,fx_i,gx_i)_k_i/(λ q^N_i+1x_i)_k(aqx_i/b,aqx_i/c,λ qx_i/d)_k_i) ×(λ b/a,λ c/a,d)_kq^k/(aq/e,λ q/f,λ q/g)_k ∏_i,j=1^n (q^-N_jx_i/x_j)_k_i/(qx_i/x_j)_k_i),where bcdefg=a^3q^N+2 and λ=a^2q/bce. For be=aq the sum on the right trivializes and the transformation simplifies to (<ref>).The next transformation, which relates an A_n anda C_n series, is a discrete analogue of (<ref>):∑_k_1,…,k_n=0^N_1,…,N_nΔ^C(xq^k)/Δ^C(x)∏_i=1^n(bx_i,cx_i,dx_i,ex_i,fx_i,gx_i)_k_iq^k_i/(qx_i/b,qx_i/c,qx_i/d,qx_i/e,qx_i/f,qx_i/g)_k_i∏_i,j=1^n(q^-N_jx_i/x_j,x_ix_j)_k_i/(qx_i/x_j,q^N_j+1x_ix_j)_k_i=∏_i,j=1^n(qx_ix_j)_N_i/∏_1≤ i<j≤ n(qx_ix_j)_N_i+N_j (λ q/e,λ q/f,q/ef)_N/∏_i=1^n(λ qx_i,qx_i/e,qx_i/f,q^-N_ig/x_i)_N_i×∑_k_1,…,k_n=0^N_1,…,N_n(Δ^A(xq^k)/Δ^A(x)∏_i=1^n(θ(λ x_i q^k_i+k)/θ(λ x_i) (λ x_i)_k(ex_i,fx_i,gx_i)_k_i/(λ q^N_i+1x_i)_k(qx_i/b,qx_i/c,qx_i/d)_k_i)×(λ b,λ c,λ d)_kq^k/(λ q/e,λ q/f,λ q/g)_k∏_i,j=1^n(q^-N_jx_i/x_j)_k_i/(qx_i/x_j)_k_i),where bcdefg=q^N+2 and λ=q/bcd. For bc=q this reduces to (<ref>).The discrete analogue of (<ref>) provides a duality betweenA_n and A_m elliptic hypergeometric series:∑_k_1,…,k_n+1≥ 0k_1+…+k_n+1=NΔ^A(xq^k)/Δ^A(x)∏_i=1^n+1∏_j=1^m+n+2(x_ia_j)_k_i/∏_j=1^m+1(x_iy_j)_k_i∏_j=1^n+1(qx_i/x_j)_k_i=∑_k_1,…,k_m+1≥ 0k_1+…+k_m+1=NΔ^A(yq^k)/Δ^A(y)∏_i=1^m+1∏_j=1^m+n+2(y_i/a_j)_k_i/∏_j=1^n+1(y_ix_j)_k_i∏_j=1^m+1(qy_i/y_j)_k_i,where w_1⋯ w_m+1=x_1⋯ x_n+1a_1⋯ a_m+n+2. For m=0 this reduces to (<ref>).We next give a discrete analogue of (<ref>).When M_i and N_i for i=1,…,n are non-negative integers andbcde=q^N-M+1, then∑_k_1,…,k_n=0^N_1,…,N_n(Δ^C(xq^k)/Δ^C(x)∏_i=1^n(bx_i,cx_i,dx_i,ex_i)_k_i q^k_i/(qx_i/b,qx_i/c,qx_i/d,qx_i/e)_k_i×∏_i=1^n∏_j=1^m(q^M_jx_iy_j,qx_i/y_j)_k_i/(x_iy_j,q^1-M_jx_i/y_j)_k_i∏_i,j=1^n(q^-N_jx_i/x_j,x_ix_j)_k_i/(qx_i/x_j,q^N_j+1x_ix_j)_k_i) =q^-NM(q/bc,q/bd,q/cd)_N/(q^-Nbc,q^-Nbd,q^-Ncd)_M∏_i=1^m∏_j=1^n(q^-N_jy_i/x_j)_M_i/(y_i/x_j)_M_i×∏_i,j=1^n(qx_ix_j)_N_i∏_1≤ i<j≤ m(y_iy_j)_M_i+M_j/∏_i,j=1^m(y_iy_j)_M_i∏_1≤ i<j≤ n(qx_ix_j)_N_i+N_j∏_i=1^m(by_i,cy_i,dy_i,q^1-M_i/y_ie)_M_i/∏_i=1^n(qx_i/b,qx_i/c,qx_i/d,q^-N_ie/x_i)_N_i×∑_k_1,…,k_m=0^M_1,…,M_m(Δ^C(q^-1/2yq^k)/Δ^C(q^-1/2y)∏_i=1^m(y_i/b,y_i/c,y_i/d,y_i/e)_k_iq^k_i/(by_i,cy_i,dy_i,ey_i)_k_i×∏_i=1^m ∏_j=1^n (q^N_jy_ix_j,y_i/x_j)_k_i/(y_ix_j,q^-N_jy_i/x_j)_k_i∏_i,j=1^m(q^-M_jy_i/y_j,q^-1y_iy_j)_k_i/(qy_i/y_j,q^M_jy_iy_j)_k_i). Recalling the notation (<ref>), we have the following discreteanalogue of (<ref>):1211n(a;b,c,d,e,f,g,q^-N)=(aq,aq/ef,λ q/e,λ q/f)_(N^n)/(λ q,λ q/ef,a q/e,aq/f)_(N^n) 1211n(λ;λ b/a,λ c/a, λ d/a,e,f,g,q^-N),where bcdefgt^n-1=a^3q^N+2 and λ=a^2q/bcd.Finally, the following Karlsson–Minton-type transformation is an analogue of (<ref>): 2r+82r+7n (a;bt^1-n,aq^-M/b,c_1q^k_1, …,c_rq^k_r,aq/c_1,…,aq/c_r,q^-N) =(aq,t^n-1q)_(N^n)/(bq,t^n-1aq/b)_(N^n) (bq,t^n-1bq/a)_(M^n)/(b^2q/a,t^n-1q)_(M^n)∏_i=1^r(bc_i/a,t^n-1c_i/b)_(k_i^n)/(c_i,t^n-1c_i/a)_(k_i^n)× 2r+82r+7n(b^2/a;bt^1-n,bq^-N/a, bc_1q^k_1/a,…,bc_rq^k_r/a, bq/c_1,…,bq/c_r,q^-M),where the k_i are non-negative integers such that k_1+…+k_r=M+N. §.§ Notes For p=0 the A_n summations (<ref>), (<ref>) and(<ref>) are due to Milne <cit.>, Schlosser <cit.>(see also also <cit.>), and Gustafson and Rakha <cit.>, respectively. The p=0 case of the C_n summation (<ref>) was found independently by Denis and Gustafson <cit.>, and Milneand Lilly <cit.>. The p=0 case of the C_n summation (<ref>) is due toSchlosser <cit.>. The p=0 case of the transformation (<ref>) was obtained, again independently, by Denis and Gustafson <cit.>, and Milne and Newcomb <cit.>.Thep=0 cases of (<ref>) and (<ref>) are due to Bhatnagar and Schlosser <cit.> and Kajihara <cit.>,respectively. The p=0 instances of (<ref>), (<ref>), (<ref>), (<ref>)and (<ref>) were not known prior to the elliptic cases.For general p, the A_n summations (<ref>) and (<ref>) were first obtained by Rosengren <cit.> using an elementaryinductive argument. A derivation of (<ref>) from (<ref>) using residue calculus isgiven in <cit.> and a similar derivation of (<ref>) from(<ref>) in <cit.>.The summation (<ref>) was conjectured by Spiridonov <cit.> andproved, independently, by Ito and Noumi <cit.> andby Rosengren <cit.>.As mentioned in the introduction, Warnaar <cit.> conjectured the C_n summation (<ref>). He also proved the more elementary C_n summation (<ref>). Van Diejen and Spiridonov <cit.> showed that theC_n summations (<ref>) and (<ref>) follow from the (at that time conjectural) integral identities (<ref>) and (<ref>).This in particular implied the first proof of (<ref>) for p=0. For general p, the summations (<ref>) and (<ref>) were proved by Rosengren <cit.>, using the case N=1 of Warnaar's identity (<ref>).Subsequent proofs of (<ref>) were given in<cit.>. The proofs in <cit.> establish the more general sum (<ref>) for elliptic binomial coefficients. In <cit.> the identity (<ref>) arises as a special case of the discrete biorthogonality relation (<ref>) for the elliptic biorthogonal functions R̃_λ.The transformations (<ref>) and (<ref>) were obtained by Rosengren <cit.>, together with two more A_n transformations that are not surveyed here.The transformation (<ref>) was obtained independently by Kajihara and Noumi <cit.> and Rosengren <cit.>. Both these papers contain further transformations that can be obtained by iterating (<ref>).The transformation (<ref>) was proved by Rains (personal communication, 2003) by specializing the parameters of <cit.> to a union of geometric progressions.It appeared explicitly in <cit.> using a similarapproach to Rains. The transformation (<ref>) was conjectured by Warnaar <cit.> and established by Rains <cit.> using the symmetry of the expression (<ref>) below. The transformation (<ref>) is stated somewhat implicitly by Rains <cit.>; it includes (<ref>) as a special case.A discrete analogue of the type II A_n beta integral (<ref>) has been conjectured by Spiridonov and Warnaar in <cit.>. Surprisingly, this conjecture contains the C_n identity(<ref>) as a special case.The summation formula (<ref>) can be obtained as a determinant ofone-dimensional summations. Further summations and transformations of determinantal type are given in <cit.>. The special case t=q of (<ref>) and (<ref>) is also closelyrelated to determinants, see <cit.>.Transformations related to the sum (<ref>) are discussed in <cit.>. In their work on elliptic Bailey lemmas on root systems, Bhatnagar andSchlosser <cit.> discovered two further elliptic Jacksonsummations for A_n, as well as corresponding transformation formulas. For none of these an integral analogue is known. Langer, Schlosser and Warnaar <cit.> proved a curious A_n transformation formula, which is new even in the one-variable case. § ELLIPTIC MACDONALD–KOORNWINDER THEORY A function f on (ℂ^∗)^n is said to be BC_n-symmetric if it is invariant under the action of the hyperoctahedral group (ℤ/2ℤ)≀ S_n. Here the symmetric group S_n acts by permuting the variables and ℤ/2ℤ by replacing a variable with its reciprocal. The interpolation functions R_λ^∗(x_1,…,x_n;a,b;q,t;p),introduced independently by Rains <cit.> and by Coskunand Gustafson <cit.>, are BC_n-symmetric ellipticfunctions that generalize Okounkov's BC_n interpolation Macdonald polynomials <cit.> as well as the Macdonaldpolynomials of type A <cit.>.They form the building blocks of Rains' more generalBC_n-symmetric functions <cit.>R̃_λ(x_1,…,x_n;a:b,c,d;u,v;q,t;p).The R̃_λ are an elliptic generalization of the Koornwinderpolynomials <cit.>, themselves a generalization toBC_n of the Askey–Wilson polynomials <cit.>. The price one pays for ellipticity is that the functions R^∗_λ and R̃_λ are neither polynomial nor orthogonal.The latter do however form a biorthogonal family, and for n=1 they reduce to the continuous biorthogonal functions of Spiridonov(elliptic case) <cit.> and Rahman (the p=0 case) <cit.> and, appropriately specialized, to the discrete biorthogonal functionsof Spiridonov and Zhedanov (elliptic case) <cit.> and Wilson (the p=0 case) <cit.>.There are a number of ways to define the elliptic interpolation functions. Here we will describe them via a branching rule. The branching coefficient c_λμ is a complex function on (ℂ^∗)^7, indexed by a pair of partitions λ,μ. It is defined to be zero unless λ≻μ, in which casec_λμ(z;a,b;q,t,T;p) = (aTz^±,pqa/bt)_λ/(aTz^±,pqa/bt)_μ (pqz^±/bt,T)_μ/(pqz^±/b,tT)_λ×∏_(i,j)∈λ λ'_j=μ'_jθ(q^λ_i+j-1 t^2-i-λ'_j aT/b)/θ(p q^μ_i-j+1 t^μ'_j-i)∏_(i,j)∈λ λ'_jμ'_jθ(q^λ_i-j t^λ'_j-i+1)/θ(pq^μ_i+j t^-i-μ'_j aT/b)×∏_(i,j)∈μ λ'_j=μ'_jθ(pq^λ_i-j+1t^λ'_j-i)/θ(q^μ_i+j-1t^1-i-μ'_j aT/b)∏_(i,j)∈μ λ'_jμ'_jθ(pq^λ_i+j t^1-i-λ'_j aT/b)/θ(q^μ_i-j t^μ'_j-i+1).From (<ref>) and the invariance under the substitution z↦ z^-1 it follows that c_λμ is aBC_1-symmetric elliptic function of z. The elliptic interpolation functions are uniquely determined by thebranching ruleR^*_λ(x_1,…,x_n+1;a,b;q,t;p) =∑_μ c_λμ(x_n+1;a,b;q,t,t^n;p)R^*_μ(x_1,…,x_n;a,b;q,t;p),subject to the initial condition R^*_λ(– ;a,b;q,t;p)= δ_λ,0. It immediately follows that the interpolation function (<ref>) vanishes if l(λ)>n. From the symmetry and ellipticity of the branching coefficient it also follows that the interpolation functions are BC_1-symmetric and elliptic in each of the x_i. S_n-symmetry (and thus BC_n-symmetry), however, is not manifest and is a consequence ofthe non-trivial fact that∑_μ c_λμ(z;a,b;q,t,T;p) c_μν(w;a,b;q,t,T/t;p)is a symmetric function in z and w; see also the discussion around (<ref>) below.In the remainder of this section x=(x_1,…,x_n). Comparison of their respective branching rules shows that Okounkov's BC_n interpolation Macdonald polynomials P^∗_λ(x;q,t,s) and the ordinary Macdonald polynomials P_λ(x;q,t) arise in the limit asP_λ^∗(x;q,t,s)= lim_p→ 0(-s^2t^2n-2)^-λ q^-n(λ') t^2n(λ)(t^n)_λ/C^-_λ(t) R_λ^∗(st^δx;s,p^1/2b;q,t;p),and P_λ(x;q,t) =lim_z→∞ z^-λlim_p→ 0(-at^n-1)^-λ q^-n(λ') t^2n(λ)(t^n)_λ/C^-_λ(t)R_λ^∗(zx;a,p^1/2b;q,t;p),where δ=(n-1,…,1,0) is the staircase partition of length n-1, st^δx=(st^n-1x_1,…,st^0 x_n) and zx=(zx_1,…,zx_n).Many standard properties of P_λ(x;q,t) and P_λ^∗(x;q,t,s) have counterparts for the elliptic interpolation functions. Here we have space for only a small selection. Up to normalization, Okounkov's BC_n interpolation Macdonald polynomials are uniquely determined by symmetry and vanishing properties.The latter carry over to the elliptic case as follows:R_μ^∗(aq^λt^δ;a,b;q,t;p)=0if μ⊄λ. For q,t,a,b,c,d∈ℂ^* the elliptic difference operatorD^(n)(a,b,c,d;q,t;p), acting on BC_n-symmetric functions, is given by(D^(n)(a,b,c,d;q,t;p)f)(x) =∑_σ∈{± 1}^nf(q^σ/2x) ∏_i=1^nθ(a x_i^σ_i,b x_i^σ_i, c x_i^σ_i,d x_i^σ_i)/θ(x_i^2σ_i)∏_1≤ i<j≤ nθ(t x_i^σ_i x_j^σ_j)/θ(x_i^σ_i x_j^σ_j),where q^σ/2x=(q^σ_1/2x_1,…,q^σ_n/2x_n). ThenD^(n)(a,b,c,d;q,t;p)R_λ^∗(x;aq^1/2,bq^1/2;q,t;p) =∏_i=1^n θ(abt^n-i,acq^λ_it^n-i, bcq^-λ_it^i-1) · R_λ^∗(x;a,b;q,t;p)provided that t^n-1abcd=p. Like the Macdonald polynomials, there is no simple closed-form expression for the elliptic interpolation functions.When indexed by rectangular partitions of length n, however, they do admit a simple form, viz. R^*_(N^n)(x;a,b;q,t;p) =∏_i=1^n (ax_i^±)_N/(pqx_i^±/b)_N.The principal specialization formula for the elliptic interpolation functions isR^*_λ(zt^δ;a,b;q,t;p) =(t^n-1az,a/z)_λ/(pqt^n-1z/b,pq/bz)_λ.The R^*_λ satisfy numerous symmetries, all direct consequence of symmetries of the branching coefficients c_λμ. Two of the most notable ones are R^*_λ(x;a,b;q,t;p)=R^*_λ(-x;-a,-b;q,t;p) =(qt^n-1a/b)^2λ q^4n(λ') t^-4n(λ)R^*_λ(x;1/a,1/b;1/q,1/t;p).Specializations of R^*_μ give rise to elliptic binomial coefficients. Before defining these we introduce the functionΔ_λ(a| b_1,…,b_k)= (pqa)_2λ^2/C^-_λ(t,pq)C^+_λ(a,pqa/t) (b_1,…,b_k)_λ/(pqa/b_1,…,pqa/b_k)_λ,where the dependence on q,t and p has been suppressed andwhere 2λ^2 is shorthand for the partition(2λ_1,2λ_1,2λ_2,2λ_2,…). Explicitly, for λ such that l(λ)≤ n,Δ_λ(a| b_1,…,b_k)= ((-1)^k a^k-3 q^k-3t/b_1⋯ b_k)^λ q^(k-4)n(λ') t^-(k-6)n(λ) (at^1-n,aqt^-n,b_1,…,b_k)_λ/(qt^n-1,t^n,aq/b_1,…,aq/b_k)_λ ×∏_i=1^n θ(at^2-2iq^2λ_i)/θ(at^2-2i)∏_1≤ i<j≤ n( θ(t^j-iq^λ_i-λ_j, at^2-i-jq^λ_i+λ_j)/θ(t^j-i,at^2-i-j) (t^j-i+1)_λ_i-λ_j(at^3-i-j)_λ_i+λ_j/(qt^j-i-1)_λ_i-λ_j(aqt^1-i-j)_λ_i+λ_j),so thatr+1rn(a;b_1,…,b_r-4)=∑_λ(b_3,…,b_r-4,qt^n-1b_1b_2)_λ/(aq/b_3,…,aq/b_r-4,at^1-n/b_1b_2)_λ Δ_λ(a| t^n,b_1,b_2,at^1-n/b_1b_2). The elliptic binomial coefficientsλμ_[a,b]=λμ_[a,b];q,t;p may now be defined asλμ_[a,b]= Δ_μ(a/b| t^n,1/b)R^*_μ(x_1,…,x_n;a^1/2t^1-n,ba^-1/2;q,t;p) |_x_i=a^1/2q^λ_i t^1-i,where on the right n can be chosen arbitrarily provided thatn≥ l(λ),l(μ).Apart from their n-independence, the elliptic binomial coefficients are also independent of the choice of square root of a. Although λ0_[a,b]=1 they are not normalized like ordinary binomial coefficients, andλλ_[a,b]= (1/b,pqa/b)_λ/(b,pqa)_λ C^+_λ(a)/C^+_λ(a/b).The elliptic binomial coefficients vanish unless μ⊂λ, are elliptic in both a and b, and invariant under the simultaneous substitution (a,b,q,t)↦ (1/a,1/b,1/q,1/t). They are also conjugation symmetric:λμ_[a,b];q,t;p= λ'μ'_[aq/t,b];1/t,1/q;p.A key identity isλν_[a,c]= (b,ce,cd,bde)_λ/(cde,bd,be,c)_λ (1/c,bd,be,cde)_ν/(bcde,e,d,b/c)_ν∑_μ(c/b,d,e,bcde)_μ/(bde,ce,cd,1/b)_μ λμ_[a,b]μν_[a/b,c/b]for generic parameters such that bcde=aq. The c→ 1 limit of λν_[a,c] (c)_λ/(1/c)_ν exists and is given by δ_λν. Multiplying both sides of (<ref>) by (c)_λ/(1/c)_ν and then letting c tend to 1 thus yields the orthogonality relation∑_μλμ_[a,b]μν_[a/b,1/b] =δ_λμ. For another important application of (<ref>)we note that by (<ref>) (N^n)μ_[a,b] =Δ_μ(a/b| t^n,1/b,aq^Nt^1-n,q^-N).Setting λ=(N^n) and ν=0 in (<ref>) and recalling (<ref>) yields the C_n Jacksonsummation (<ref>). Also (<ref>) may be obtained as a special case of (<ref>) but the details of the derivation are more intricate, see <cit.>. As a final application of (<ref>) it can be shown that(b,b'e)_λ/(b'de,bd)_λ (b'de,bf/c)_ν/(b/c,b'de/f)_ν∑_μ(c/b,d,e,bb'de,b'g/c,b'f/c)_μ/(bb'de/c,b'e,b'd,1/b,f,g)_μ λμ_[a,b]μν_[a/b,c/b]is symmetric in b and b', where bb'de=aq and cde=fg. Setting λ=(N^n) and ν=0 results in the transformation formula(<ref>). Now assume that bcd=b'c'd'. Twice using the symmetry of (<ref>) it follows that ∑_μ(c,d,aq/c',aq/d')_λ/(c,d,aq/c',aq/d')_μ (c'/b,d'/b,aq/bc,aq/bd)_μ/(c'/b,d'/b,aq/bc,aq/bd)_ν ×(1/b,aq/b)_ν/(1/b,aq/bb')_μ (b',aq)_μ/(b',aq/b)_λ λμ_[a,b]μν_[a/b,b']is invariant under the simultaneous substitution (b,c,d)↔ (b',c',d'). The branching coefficient (<ref>) may be expressed as an elliptic binomial coefficient asc_λμ(z;a,b;q,t,T;p)= (aTz^±,pqa/bt,t)_λ/(aTz^±,pqa/bt,1/t)_μ (pqz^±/bt,T,pqaT/b)_μ/(pqz^±/b,tT,pqaT/bt)_λ λμ_[aT/b,t],so that, up to a simple change of variables and the use of(<ref>), the z,w-symmetry of (<ref>) corresponds to the b=b' case of the symmetry of (<ref>). To conclude our discussion of the elliptic binomial coefficients we remark that they also arise as connection coefficients between the interpolation functions. Specifically,R^∗_λ(x;a,b;q,t;p) =∑_μλμ_[t^n-1a/b,a/a'] (a/a',t^n-1aa')_λ/(a'/a,t^n-1aa')_μ (pqt^n-1a/b,pq/ab)_μ/(pqt^n-1a'/b,pq/a'b)_λR^∗_μ(x;a',b;q,t;p). Let a,b,c,d,u,v,q,t be complex parameters such that t^2n-2abcduv=pq, and λ a partition of length at most n.Then the BC_n-symmetric biorthogonal functionsR̃_λ are defined asR̃_λ(x;a:b,c,d;u,v;q,t;p) =∑_μ⊂λλμ_[1/uv,t^1-n/av] (pq/bu,pq/cu,pq/du,pq/uv)_μ/(t^n-1ab,t^n-1ac,t^n-1ad,t^n-1av)_μR^*_μ(x;a,u;q,t;p).By (<ref>) this relation between the two families of BC_n elliptic functions can be inverted. We also note that from (<ref>) it follows thatR̃_λ(x;a:b,c,d;u,t^1-n/b;q,t;p) =(pq/au,pqt^n-1a/u)_λ/(b/a,t^n-1ab)_λR^*_μ(x;b,u;q,t;p),so that the interpolation functions are a special case of the biorthogonal functions. Finally, from Okounkov's binomial formula for Koornwinder polynomials <cit.> it follows that in the p→ 0 limit the R̃_λ simplify to the Koornwinder polynomials K_λ(x;a,b,c,d;q,t):K_λ(x;a,b,c,d;q,t)= lim_p→ 0(at^n-1)^-λ t^n(λ)(t^n,t^n-1ab,t^n-1ac,t^n-1ad)_λ/C^-_λ(t)C^+_λ(abcdt^2n-2/q) ×R̃_λ(x;a:b,c,d; up^1/2,vp^1/2,q,t;p). Most of the previously-listed properties of the interpolation functions have implications for the biorthogonal functions. For example, using (<ref>) and (<ref>) one canprove the principal specialization formulaR̃_λ(bt^δ;a:b,c,d;u,v;q,t;p) =(t^n-1bc,t^n-1bd,t^1-n/bv,pqt^n-1a/u)_λ/(t^n-1ac,t^n-1ad,t^1-n/av,pqt^n-1b/u)_λ.Another result that carries over is the elliptic difference equation (<ref>). Combined with (<ref>) it yieldsD^(n)(a,u,b,pt^1-n/uab;q,t;p) R̃_λ^∗(x; aq^1/2:bq^1/2,cq^-1/2,dq^-1/2; uq^1/2,vq^-1/2,q,t;p) =∏_i=1^n θ(abt^n-i,aut^n-i,but^n-i) ·R̃_λ(x;a:b,c,d; u,v,q,t;p). The Koornwinder polynomials are symmetric in the parameters a,b,c,d. From (<ref>) it follows that R̃_λ issymmetric in b,c,d but the choice of normalization breaks the full S_4 symmetry. Instead,R̃_λ(x;a:b,c,d;u,v;q,t;p)= R̃_λ(x;b:a,c,d;u,v;q,t;p) R̃_λ(bt^δ;a:b,c,d;u,v;q,t;p).For partitions λ,μ such that l(λ),l(μ)≤ n the biorthogonal functions satisfy evaluation symmetry:R̃_λ(at^δq^μ; a:b,c,d;u,v;q,t;p)= R̃_μ(ât^δq^λ; â:b̂,ĉ,d̂; û,v̂;q,t;p),whereâ=√(abcd/pq),âb̂=ab,âĉ=ac,âd̂=ad, aû=âu, av̂=vâ. Given a pair of partitions λ,μ such that l(λ),l(μ)≤ n, defineR̃_λμ(x;a:b,c,d;u,v;t;p,q) =R̃_λ(x;a:b,c,d;u,v;p,t;q) R̃_μ(x;a:b,c,d;u,v;q,t;p).Note that R̃_λμ(x;a:b,c,d;u,v;t;p,q) is invariant under the simultaneous substitutions λ↔μ and p↔ q. The functionsR̃_λμ(x;a:b,c,d;u,v;t;p,q) form a biorthogonal family, with continuous biorthogonality relationκ_n^C ∫_C_λν,μωR̃_λμ(z_1,…,z_n; t_1:t_2,t_3,t_4;t_5,t_6;t;p,q)R̃_νω(z_1,…,z_n; t_1:t_2,t_3,t_4;t_6,t_5;t;p,q) ×∏_1≤ i<j≤ nΓ(tz_i^±z_j^±)/Γ(z_i^±z_j^±)∏_j=1^n∏_i=1^6Γ(t_iz_j^±)/Γ(z_j^± 2)z_1/z_1⋯ z_n/z_n=δ_λνδ_μω∏_m=1^n( Γ(t^m)/Γ(t)∏_1≤ i<j≤ 6Γ(t^m-1t_it_j)) ×1/Δ_λ(1/t_5t_6| t^n,t^n-1t_0t_1,t^n-1t_0t_2,t^n-1t_0t_3, t^1-n/t_0t_5,t^1-n/t_0t_6;p,t;q)×1/Δ_μ(1/t_5t_6| t^n,t^n-1t_0t_1,t^n-1t_0t_2,t^n-1t_0t_3, t^1-n/t_0t_5,t^1-n/t_0t_6;q,t;p).Here, C_λν,μω is a deformation of 𝕋^nwhich separates sequences of poles of the integrand tending to zerofrom sequences tending to infinity. The location of these polesdepends on the choice of partitions, see <cit.> for details.Provided t<1 and t_i<1 for 1≤ i≤ 6 we can take C_00,00=𝕋^n so that for λ=μ=ν=ω=0 one recovers the type C_n^(II) integral (<ref>). The summation (<ref>), which is the discrete analogue of (<ref>), follows in a similar manner from the discrete biorthogonality relation∑_μ⊂ (N^n) Δ_μ(t^2n-2a^2|t^n,t^n-1ac,t^n-1ad,t^n-1au,t^n-1av,q^-N) ×R̃_λ(aq^μ t^δ; a:b,c,d;u,v;t;p,q)R̃_ν(aq^μ t^δ; a:b,c,d;v,u;t;p,q) =δ_λν/Δ_λ(1/uv| t^n,t^n-1ab,t^n-1ac,t^n-1ad, t^1-n/au,t^1-n/av)×(b/a,pq/uc,pq/ud,pq/uv)_(N^n)/(pqt^n-1a/u,t^n-1bc,t^n-1bd,t^n-1bv)_(N^n),where t^2n-2abcduv=pq and q^Nt^n-1ab=1. The discrete biorthogonality can also be lifted to the functionsR̃_λμ but since the resulting identity factors into two copies of (<ref>) — the second copy with q replaced by p and N by a second discrete parameter M — this is no more general than the above.The final result listed here is a (dual) Cauchy identity which incorporates the Cauchy identities for the Koornwinder polynomials, BC_n interpolation Macdonald polynomials and ordinary Macdonald polynomials:∑_λ⊂ (N^n)Δ_λ(q^1-2N/uv| t^n,q^-N,q^1-Nt^1-n/av,a/u)×R̃_λ(x;a:b,c,d;q^Nu,q^N-1v;q,t;p) R̃_λ̂(y;a:b,c,d; t^nu,t^n-1v;t,q;p) =(a/u,pq^1-N/au,pq^1-N/bu,pq^1-N/cu,pq^1-N/du, pq^2-2N/uv)_(N^n)/(t^n-1ab,t^n-1ac,t^n-1ad,q^N-1t^n-1av)_(N^n) ×∏_i=1^n∏_j=1^N θ(x_i^± y_j) ∏_i=1^n 1/(ux_i^±)_m∏_j=1^N 1/(p/uy_j,y_j/u;1/t,p)_n,where x=(x_1,…,x_n), y=(y_1,…,y_N), λ̂=(n-λ'_m,…,n-λ'_1) and abcduvq^2m-2t^2n-2=p. §.§ NotesInstead of R_λ^∗(x_1,…,x_n,a,b;q,t;p), Rains denotes the BC_n-symmetric interpolation functionsas R_λ^∗(n)(x_1,…,x_n,a,b;q,t;p), see<cit.>. An equivalent family of functions is defined by Coskun and Gustafson in<cit.> (see also <cit.>).They refer to these as well-poised Macdonaldfunctions, denoted as W_λ(x_1,…,x_n;q,p,t,a,b). The precise relation between the two families is given byW_λ(x_1/a,…,x_n/a;q,p,t,a^2,a/b)=(t^1-nb^2/q^2)^λq^-2n(λ') t^2n(λ)(t^n)_λ/C^-_λ(t) (qt^n-2a/b;q,t^2;p)_2λ/(qa/tb)_λC^+_λ(qt^n-2a/b)R^∗_λ(x_1,…,x_n;a,b;q,t;p).Similarly, Rains writes R̃_λ^(n)(x_1,…,x_n;a:b,c,d;u,v;q,t;p) for the biorthogonal functions instead of R̃_λ(x_1,…,x_n;a:b,c,d;u,v;q,t;p), see again <cit.>.The branching rule (<ref>) is the k=1 instance of<cit.> or the μ=0 case of<cit.>. The vanishing property (<ref>) is <cit.> combined with (<ref>), or <cit.>. The elliptic difference equation (<ref>) is <cit.>. The formula (<ref>) for the interpolation function indexed by a rectangular partition of length n is the λ=0 case of <cit.> or <cit.>. The principal specialization formula (<ref>) is <cit.>. The symmetry (<ref>) is<cit.> and the symmetry (<ref>) is <cit.> or <cit.>. The definition of the elliptic binomial coefficients(<ref>) is due to Rains, see <cit.>. Coskun and Gustafson define so-called elliptic Jackson coefficientsω_λ/μ(z;r;a,b)=ω_λ/μ(z;r,q,p;a,b),see <cit.>. Up to normalization these are the elliptic binomials coefficients:ω_λ/μ(z;r;a,b)= (1/z,az)_λ/(qbz,qb/az)_λ(qbz/r,qb/azr,bq,r)_μ/(1/z,az,qb/r^2,1/r)_μ λμ_[b,r].The value of the elliptic binomials (<ref>) is<cit.>.It is equivalent to <cit.> and also <cit.>. The conjugation symmetry (<ref>) of the ellipticbinomial coefficients is <cit.>. The summation (<ref>) is<cit.>, and is equivalent to the “cocycle identity” <cit.> for the elliptic Jackson coefficients. The orthogonality relation (<ref>) is<cit.> or <cit.>. The symmetry of (<ref>) is <cit.> or <cit.>, and the symmetry of (<ref>) is <cit.>. The expression (<ref>) for the branching coefficients is a consequence of <cit.> or <cit.>. The connection coefficient identity (<ref>) is <cit.>. It is equivalent to the “Jackson sum” <cit.> for the well-poised Macdonald functions W_λ.Definition (<ref>) of the birthogonal functions is <cit.>, its principal specialization (<ref>) is <cit.> and the difference equation (<ref>) is <cit.>. The parameter and evalation symmetries (<ref>) and (<ref>) are <cit.> and<cit.>, respectively. The important biorthogonality relation (<ref>) is a combination of <cit.> and <cit.>. Its discrete analogue (<ref>)is <cit.>, see also <cit.>. Finally, the Cauchy identity (<ref>) is<cit.>.The BC_n-symmetric interpolation functions satisfy severalfurther important identities not covered in the main text, such as a “bulk” branching rule <cit.> which extends (<ref>), anda generalized Pieri rule <cit.>.In <cit.> Coskun applies the elliptic binomial coefficients (elliptic Jackson coefficients in his language) to formulate an elliptic Bailey lemma of type BC_n. The interpolation functions further admit a generalization to skewinterpolation functions <cit.>ℛ_λ/μ([v_1,…,v_2n];a,b;q,t;p), μ⊆λ.These are elliptic functions, symmetric in the variables v_1,…,v_2n, such that <cit.>R^*_λ(x_1,…,x_n;a,b;q,t;p) =(pqa/tb)_λ/(t^n)_λ ℛ^∗_λ/0([t^1/2x_1^±,…,t^1/2x_n^±]; t^n-1/2a,t^1/2b;q,t;p).They also generalize the n-variable skew elliptic Jackson coefficients <cit.>ω_λ/μ(x_1,…,x_n;r,q,p;a,b)of Coskun and Gustafson:ω_λ/μ(r^n-1/2x_1/a,…, r^n-1/2x_n/a;r;a^2r^1-2n,ar^1-n/b) =(-b^3/q^3a)^λ-μq^3n(μ')-3n(λ') t^3n(λ)-3n(μ) r^-nμ(aq/br)_λ/(aqr^-n-1/b)_μ (r)_μ/(r)_λ ×ℛ^∗_λ/μ([r^1/2x_1^±,… r^1/2x_n^±];a,b;q,t;p).A very different generalization of the interpolation functions is given in <cit.> in the form of an interpolation kernel𝒦_c(x_1,…,x_n;y_1,…,y_n;q,t;p). By specialising y_i=q^λ_i t^n-ia/c with c=√(t^n-1ab) for all 1≤ i≤ n one recovers, up to a simple normalising factor,R^∗_λ(x_1,…,x_n;a,b;q,t;p). 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"S. Ole Warnaar"
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"title": "Elliptic hypergeometric functions associated with root systems"
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Penrose junction conditions extended: impulsive waves with gyratonsJ. Podolský^[email protected], R. Švarc^[email protected], R. Steinbauer^[email protected] and C. Sämann^[email protected] ^1 Institute of Theoretical Physics,Charles University, Faculty of Mathematics and Physics, Prague V Holešovičkách 2, 18000 Prague 8, Czech Republic.^2 Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.December 30, 2023 ==================================================================================================================================================================================================================================================================================================================================================================================================================================================================== We generalize the classical junction conditions for constructing impulsive gravitational waves by thePenrose “cut and paste” method. Specifically, we study nonexpanding impulses which propagate inspaces of constant curvature with any value of the cosmological constant (that is Minkowski, de Sitter, or anti-de Sitter universes) when additional off-diagonal metric components are present. Such components encode a possible angular momentum of the ultra-relativistic source of the impulsive wave — the so called gyraton. We explicitly derive and analyze a specific transformation that relates the distributional form of the metric to a new form which is (Lipschitz) continuous. Such a transformation automatically implies an extended version of the Penrose junction conditions. It turns out that the conditions for identifying points of the background spacetime across the impulse are the same as in the original Penrose “cut and paste” construction, but their derivatives now directly represent the influence of the gyraton on the axial motion of test particles. Our results apply both for vacuum and nonvacuum solutions of Einstein's field equations, and can also be extended to other theories of gravity. § INTRODUCTION An impulsive gravitational wave can most intuitively be understood as a limit of a suitable family of sandwich waves with their profiles approaching the Dirac delta. Formally, if d_ε() denotes the sandwich wave profile whereis the retarded time coordinate, it is assumed thatthe supports of the sequence d_ε() shrink to zero as ε→0 (the sandwich waves have “ever shorterduration” ε) but simultaneously their amplitudes become bigger (the waves are “ever stronger” as ε^-1). Such a distributional limit d_ε()→δ() gives a solution, which (at least formally) represents an impulsive wave localized on a single wave-front =0.In the simplest yet important context of vacuum pp -waves propagating in Minkowski space, this procedure was first explicitly considered in <cit.> and later elsewhere (see, e.g., <cit.>). Employing the well-know Brinkmann form of pp -waves <cit.>, one directly obtains the metricṣ^2= 2 η̣ ̣̅η-2+2H(η,η̅) δ() ^2 .Analogously, more general nonexpanding planar impulsive waves of the Kundt type and expanding spherical impulsive waves of the Robinson–Trautman type can be constructed. A nonvanishing cosmological constant Λ can also be considered, so that the impulsive waves may propagate in any space of constant curvature — that is Minkowski, de Sitter, or anti-de Sitter universe. Detailed accounts of these spectimes, various methods of their construction, their mutual relations, the most important examples, and a number of references can be found in <cit.> and in the review parts of the recent works <cit.>.However, although the classical metric (<ref>) has been employed and investigated in many works, it is not the most general form of impulsive pp -waves. As considered already in the original paper by Brinkmann <cit.>, additional off-diagonal terms can be added,[In fact, the most general Brinkmann geometry also admits higher dimensions and the possibility that the transverse Riemannian space is not flat, see <cit.>.] leading to the metricṣ^2=2 η̣ ̣̅η-2+2H(η,η̅) δ() ^2 +2J(η,η̅,) η̣ + 2J̅(η,η̅,) ̣̅η.In vacuum regions, it is a standard and common approach to completely remove these additional metric terms by a suitable coordinate transformation. However, such a gauge freedom is only local and ignores the global (topological) properties of the spacetimes. By neglecting the metric function J in (<ref>), an important physical property of the spacetime is eliminated, namely the possible rotational character of the source of the gravitational wave (its internal spin/helicity).This remarkable fact was first noticed by Bonnor <cit.>, see <cit.>, who studied both the interior and the exterior field of a “spinning null fluid” in the class of axially symmetric pp-waves. Spacetimes with such a localized spinning source moving at the speed of light, whose angular momentum is encoded in the function J, were independently rediscovered in 2005 <cit.>, their physical application as a model of relativistic particles was emphasized, and they were called “gyratons”. These pp-wave-type gyratons were then investigated in greater detail and also generalized to higher dimensions and various nonflat backgrounds in a wider Kundt class which may also include a cosmological constant or an additionalelectromagnetic field, see <cit.> for more details and further references.It is well known that there exist several distinct methods of constructing impulsive pp -waves represented by the metric (<ref>), namely the “cut and paste” method with Penrose junction conditions, explicit construction of continuous coordinates, distributional limits of sandwich waves, boosts of specific initially static sources, and embedding from higher dimensions. These were recently reviewed, e.g., in <cit.>. Surprisingly, apart from the straightforward distributional limit of sandwich waves, the other construction methods have not yet been given for the more general impulsive metric (<ref>) with gyratonic terms given by nonzero J. It is now our goal in this paper to derive such a generalization, that is to find generalized Penrose junction conditions in his “cut and paste” method, and to discover the corresponding continuous metric form of impulsive pp -waves with gyratons. We will also find its relation to the distributional metric form (<ref>).First, however, it is necessary to briefly summarize the main construction methods for the classical (non-gyratonic) impulsive pp -waves (<ref>).§.§ Penrose junction conditions and the “cut and paste” methodPenrose in <cit.> and in a seminal work <cit.> presented a geometrical “cut and paste” method for constructing a class of impulsive waves in flat background, represented by the metric (<ref>). It is based oncutting Minkowski space M along a plane null hypersurface N and then “re-attaching” the two “halves” M^- and M^+ by identification of boundary points with a specific “warp”, see Figure <ref>.More explicitly, the Penrose method first removes (by a “cut”) the plane null hypersurface N given by =0 from flat spacetime in the form ṣ_0^2= 2 η̣ ̣̅η -2 ,and then re-attaches (by a “paste”) the halves M^-(<0) and M^+(>0) by making the identification of boundary points with a “warp” in the coordinatesuch that[ η, η̅, , =0_- ]__ M^-≡[ η, η̅, -H(η,η̅), =0_+ ]__ M^+ ,where H(η,η̅) is any real-valued function of η and η̅. It was shown in <cit.> that these Penrose junction conditions (<ref>) automatically guarantee that the Einstein field equations are satisfied everywhere including on =0. Thus gravitational (plus possibly null-matter) impulsive waves are obtained.§.§ Continuous coordinates for impulsive pp -waves The Penrose “cut and paste” approach is an elegant and general method because, by prescribing the junction conditions (<ref>) in(<ref>), all impulsive gravitational waves in Minkowski space can be constructed. However, the formal identification of points on both sides of the impulsive hypersurface does not directly yield explicit metric forms of the entire spacetimes.It is thus crucial to know a suitable coordinate system for these solutions in which the metric is explicitly continuous everywhere, including on the impulse. Such a metric reads ṣ^2= 2 |Ẓ+(H_,Z̅ ZẒ+H_,Z̅Ẓ̅̅Z)|^2-2 ỤṾ ,where H(Z,Z̅) is an arbitrary real-valued function while≡(U)= 0ifU ≤ 0 ,UifU ≥ 0is the kink-function. Notice that formally =UΘ, where Θ=Θ(U) is the Heaviside step function. Since the kink function is Lipschitz continuous, the metric (<ref>) is locally Lipschitz in the variable U even across the null hypersurface U=0.[But H as a function of Z, Z̅ may have singularities.]This implies that the curvature is a distribution (we are within the “maximal” distributional curvature framework as identified by Geroch and Traschen <cit.>). Indeed, the discontinuity in the derivatives of the metric introducesimpulsive components in the Weyl and curvature tensors proportional to the Dirac distribution, namely Ψ_4 = H_,ZZ δ(U) and Φ_22 = H_,ZZ̅ δ(U), see <cit.>. The metric (<ref>) thus explicitly describesimpulsive waves in Minkowski background: it is of the general Rosen form of impulsive pp -waves <cit.>. Let us note that the continuous coordinate system for the particular Aichelburg–Sexl solution <cit.> was found in <cit.>. In fact, the continuous form of the impulsive metric (<ref>) can be obtained systematically by a suitable transformation of the flat Minkowski metric (<ref>). For U>0(in M^+) the transformation =U, =V+H+UH_,ZH_,Z̅, η=Z+UH_,Z̅ yields ṣ_0^2= 2 |Ẓ+U(H_,Z̅ ZẒ+H_,Z̅Ẓ̅̅Z)|^2-2 ỤṾ. This can now be combined with the metric (<ref>) for U<0(in M^-) in which the identity =U, =V, η=Z is applied. The combined transformation relating both parts of (<ref>) to (<ref>) is thus =U, =V+Θ H+ H_,ZH_,Z̅ , η =Z+ H_,Z̅ .It is clearly discontinuous in the coordinateon =0. Now, using the fact that the global coordinates U,V,Z give rise to the continuous form of the metric (<ref>), we obtain from (<ref>)exactly the Penrose junction conditions (<ref>) for reattaching the two halves of the spacetime M^- and M^+ with the warp →-H. Thisprocedure is thus an explicit Penrose's “cut and paste” construction of impulsive gravitational pp -waves. §.§ Distributional form of impulsive pp -waves Interestingly, the distributional form of the impulsive pp -wave spacetimes is obtained from the continuous form of the metric (<ref>) by applying the combined transformation (<ref>), if we consider also the terms which arise from the derivatives of Θ(U) and (U). Indeed, (<ref>) relates (<ref>) formally to ṣ^2= 2 η̣ ̣̅η-2+2H(η,η̅) δ() ^2 ,which explicitly includes the impulselocated on the wavefront =0. This is a gravitational wave (or an impulse of null matter) in flat spacetime, depending on the specific form of the function H. It is just the Brinkmann form of a general impulsive pp -wave (<ref>). Of course, the discontinuity in the complete transformation (<ref>) which formally relates the continuous and distributional forms ofimpulsive solutionscauses some subtle mathematical problems. In fact, to obtain (<ref>) one is forced to use the distributional identities Θ'=δ and '=Θ, together with the multiplication rules Θ^2=Θ and Θ=. It is well known that in general this leads to inconsistencies, see e.g. <cit.>. However, it was shown in <cit.> that (<ref>) is in facta rigorous example of a generalized coordinate transformation in the sense of Colombeau's generalized functions. Moreover, it is possible to interpret this change of coordinates as the distributional limit of a family of smooth transformations which is obtained by a general regularization procedure, i.e., by considering the impulse as a limiting case of sandwich waves with an arbitrarily regularized wave profile. These results put the formal (“physical”) equivalence ofboth continuous and distributional forms of impulsive spacetimes on a solid ground. Therefore, the full family of impulsive limits (<ref>) of sandwich pp -waves is indeed equivalent to the distributional form of the solutions (<ref>), and consequently to the continuous metric (<ref>) obtained by the explicit “cut and paste” method (<ref>) in flat background (<ref>). Now we will present generalizations of these main methods of construction of impulsive waves to include gyratons. In the next Section <ref> we will concentrate on the family of impulsive pp -waves in Minkowski space. In Section <ref> we then further generalize our results to all nonexpanding impulsive waves in (anti-)de Sitter space with Λ≠0. § GENERALIZATION TO INCLUDE GYRATONSIn order to find a continuous metric form for impulsive pp -waves with additional gyratonic terms, as represented by the distributional metric (<ref>), it is necessary to generalize the transformation (<ref>). We found the following ansatzwhich leads to such a generalization: Set =U, =V+Θ H+ H_,ZH_,Z̅+W , η = (Z+ H_,Z̅) exp(F) ,where H=H(Z,Z̅). The additional functions W=W(Z,Z̅,U) and F=F(Z,Z̅,U) are assumed to be real-valued, and are taken to satisfy the conditions F_,U = J̅/Z+ H_,Z̅ exp(- F) , W_,U =-J J̅ .Let us remark that, in fact, the right-hand side of equation (<ref>) is always real for the standard gyratonic metric of the form (<ref>) which does not involve the off-diagonal term ρ̣. Indeed, as has been shown in <cit.>, this is the most reasonable choice to represent the physically relevant quantities in the metric functions.By substituting these relations into themetric (<ref>), a straightforward calculation leads toṣ^2≡ 2 | ζ +ζF |^2+2 [(δ-Θ') H + ((')^2-') H_,Z H_,Z̅]Ụ^2+2 [('-Θ) H + ('-)(H_,ZH_,Z̅-H_,Z̅H_,Z)+' ( ζ H_,Z -H_,Z̅)F- W]Ụ -2 ỤṾ ,where we have used the convenient shorthandζ≡Z+ H_,Z̅ ,and the symbolstands for the spatial differentials of the functions ζ andH, F, W, that isζ≡Ẓ + (H_,Z̅ ZẒ+H_,Z̅Ẓ̅̅Z),andH≡H_,Z Ẓ+H_,, F≡F_,Z Ẓ+F_,, W≡W_,Z Ẓ+W_,.Finally, employing the standard distributional identitiesΘ'=δ , '=Θ ,and the multiplication rulesΘ^2=Θ , Θ= ,the metric (<ref>) simplifies considerably toṣ^2 = 2 | ζ +ζF |^2+2 [ Θ( ζ H_,Z -H_,Z̅)F- W]Ụ-2 ỤṾ .We immediately observe that the new metric (<ref>) reduces to the continuous metric form (<ref>) without the gyratonic terms: indeedJ=0 allows for the solutions F=0=W. Moreover, (<ref>) is continuous provided the spatial differentials F and W are continuous functions of U, and F is vanishing at U=0. Then (<ref>) is locally Lipschitz, and the transformation (<ref>) is formal precisely in the same way as (<ref>): Indeed, the distributional identities (<ref>) and the multiplication rules (<ref>) which have to be employed to cancel the terms proportional to Ụ^2, H Ụ and H_,ZỤ are precisely the same as when going from (<ref>) to (<ref>), as can explicitly be seen from the metric (<ref>). The additional terms generated from F and W cancel purely due to the differential conditions (<ref>) and (<ref>), and we will comment on regularity issues in the special cases considered below. In particular, in the fundamental case when J is proportional to a step-functionΘ, it is expected that F and W involve the kink function , and they are vanishing for U≤0 (so that =U, =V, η=Z for U≤0).Thus, we have reached our aim to generalize (<ref>) in a natural wayto impulsive pp -waves with gyratons, provided we find appropriate functions F and W satisfying equations (<ref>) and (<ref>) with J≠0. For this, of course, the specific gyraton function J(η,η̅,) must be re-expressed as a function J(Z,Z̅,U) using (<ref>). In the next subsection we will explicitly show that the key example has exactly these properties, and thus the metric (<ref>) will be a continuous metric representation of the gyratonic impulsive waves in that case. §.§ The key exampleAs an illustration, let us now consider the gyratonic extension of standard impulsive pp -waves decribed by the distributional metric (<ref>) with the ^2-term2H(η,η̅) δ() ,and J(η,)=χ/2η Θ() ,where χ is a constant (see <cit.>) as shown in Figure <ref>. In such a case, the explicit integration of equations (<ref>), (<ref>) with H(Z,) gives the functionsF=χ/2(Z H_,Z- H_,) logZ+H_,/Z+ ZH_,Z , W=χ^2/4(Z H_,Z- H_,) logZ+H_,/Z+ ZH_,Z .Interestingly, W=(χ/2) F. It can be seen that both F and W are now indeed real locally Lipschitz continuous functions of U, even on the impulsive surface U=0. The resulting metric (<ref>) becomesṣ^2=2 | Ẓ + (H_,) +(Z+ H_,)F |^2+2[ (Z H_,Z- H_,)ΘF- W ]Ụ -2 ỤṾ ,where we explicitly haveF=-χ/2logZ+H_,/Z+ ZH_,Z×(Z H_,Z)-( H_,)/(Z H_,Z- H_,)^2 +χ/2H_1 +^2 H_2/(Z H_,Z- H_,)(Z+ ZH_,Z)(Z+H_,) , W=χ/2F,withH_1= (Z H_,Z- H_,)(Ẓ -Z) + Z ( ( H_,)-(Z H_,Z)) ,H_2= Z H_,Z ( H_,)- H_, (Z H_,Z).Moreover, F=0=W for all U≤0 since the log term in (<ref>), (<ref>) vanishes for U≤0. Hence the metric (<ref>) is locally Lipschitz continuous (even across U=0), and can be written asṣ^2=2 | Ẓ +(H_,Z̅ ZẒ+H_,Z̅Ẓ̅̅Z)+(Z+ H_,)F |^2+[2(Z H_,Z- H_,)-χ ]F Ụ -2 ỤṾ .In the absence of the gyraton, χ=0 implying F=0, the metric (<ref>) simplifies to the standard metric (<ref>).§.§ Gyratonic Aichelburg–Sexl metricThe expressions (<ref>), (<ref>) are valid for (<ref>) and any metric function H except the case whenZ H_,Z- H_, = 0⇔logZ+H_,/Z+ ZH_,Z =0.This special case involves all axially symmetric geometries with H depending only on Z.In particular, this exceptional situation includes the Aichelburg–Sexl caseH=-μlog (2ηη̅), which is a famous solution to vacuum Einstein's equations. Since ηη̅=ZZ̅ at =U=0, see (<ref>), this is equivalent toH=-μlog (2 ZZ̅).Instead of (<ref>), (<ref>), the functions corresponding to (<ref>) have to be taken asF=-χ/2 /Z-μ, W=-χ^2/4 /Z-μ,and the metric isṣ^2= 2 | Ẓ + (μ /^2+χ/2 Ẓ+Z/(Z-μ ))|^2-χ^2/2ẒỤ +ZỤ/(Z-μ )^2 -2 ỤṾ .This is the new continuous metric form of the class of Frolov–Fursaev gyratons considered in <cit.>, which represent gyratonic extensions of the classic Aichelburg–Sexl solution <cit.>. It describes the impulsive gravitational wave generated by a relativistic monopole point source (located at the spatial origin η=0 on the impulse =0) in which the constant μ isrelated to the mass-energy of the source, while χ Θ() determines the angular momentum density of the gyraton, see <cit.>. It is convenient to introduce polar coordinates by setting Z=1/√(2) ϱexp(ϕ), in which the continuous metric (<ref>) becomesṣ^2 =(1+2μ/ϱ^2)^2ϱ̣^2 +[(1-2μ/ϱ^2)ϱ ϕ̣+2χ /ϱ^2(1-2μ/ϱ^2)^-1ϱ̣]^2 -2χ^2 /ϱ^3(1-2μ/ϱ^2)^-2ϱ̣ Ụ -2 ỤṾ .Notice that for U≤0 this is just ṣ^2 = ϱ̣^2+ϱ^2ϕ̣^2-2 ỤṾ, which is Minkowski space in standard polar coordinates.Without the gyratonic terms (when the source has no spin, i.e., χ=0), the metric (<ref>) reduces to the much simpler formṣ^2= 2 | Ẓ +μ/^2|^2 -2 ỤṾ ,and (<ref>) becomesṣ^2= (1+2μ/ϱ^2)^2ϱ̣^2 +(1-2μ/ϱ^2)^2ϱ^2ϕ̣^2 -2 ỤṾ ,which is identical to the forms found previously in <cit.>.§.§ Alternative distributional and continuous forms of the gyratonic impulsesAlternatively, instead of (<ref>), it is also possible to start from thedistributional form of the pp -wave metric with gyratons expressed in real coordinatesṣ^2 = ρ̣^2+ρ^2 φ̣^2-2+2H(ρ,φ) δ() ^2 +2χ() φ̣.This is obtained from (<ref>) by the transformationη=1/√(2) ρexp(φ) ,withJ=χ()/2η ,where χ() is any real function. Typically, in the Einstein theory in vacuum, H(ρ,φ) is a solution of the Laplace equation H = 0 (see equation (79) in Section VII. of <cit.>, and the discussion of more general solutions therein). The simplest explicit solution of this type is the Aichelburg–Sexl axially symmetric spacetime with H(ρ)=-2μlogρ, which for χ()≠0describes the Frolov–Fursaev gyraton.The corresponding continuous form (<ref>) of the metric (<ref>) is obtained by the transformation analogous to (<ref>), namely =U, =V+Θ H+ H_,ZH_,Z̅+W , ρ = √(2) |Z+ H_,Z̅| ,φ =F+1/2logZ+ H_,Z̅/Z̅+ H_,Z ,in which again H(Z,Z̅) is an arbitrary real-valued function. Indeed, if the functions W(Z,Z̅,U), F(Z,Z̅,U) satisfy the equations F_,U = -χ()/2 ζζ̅ ,W_,U = -χ^2()/4 ζζ̅ ,where ζζ̅=ηη̅ with ζ≡Z+ H_,Z̅ defined by (<ref>), the resulting metric is exactly the continuous metric (<ref>). For the particular gyraton given by the Heaviside step function,χ()=χ Θ() ,χ =constant,(as in the example depicted in Figure <ref>) the real functions F and W take the form (<ref>) and (<ref>), respectively. They are zero for all U≤0, and continuous across U=0. The corresponding continuous metric is (<ref>).§.§ Penrose junction conditions with gyratonsUsing our explicit transformation (<ref>), or alternatively (<ref>), we can now investigate the Penrose junction conditions (<ref>) in order to also include gyratons.These Penrose conditions identify the corresponding points across the null hypersurface N located at =U=0, which separates the two halves M^-(<0) and M^+(>0) of the background Minkowski space (<ref>), see Figure <ref>. The complete transformation relating both parts of (<ref>) to the continuous metric form (<ref>) is (<ref>). Performing its limits U→0^- and U→0^+, and using continuity ofthe coordinates {U,V,Z, Z̅}, as well as the properties of the functions F and W, we immediately obtain ^+_ = ^-_ + H_ , η^+_ = η^-_ ,where the subscript _ indicates the value of the corresponding quantity at =U=0. The only discontinuity across the impulse is thus in the coordinate , in full agreement with the standard Penrose warp →-H across the impulsive surface prescribed by (<ref>). The presence of a gyraton thus makes no difference at all in the Penrose junction conditions (<ref>).Nevertheless, from the physical point of view the two distinct situations — without a gyraton and with a gyraton — must have some measurable effect. It is not contained in the Penrose identification of points, but it manifests itself in the related identification of velocities (tangent vectors to any geodesic) on both sides of the impulse. Such relations are not part of the Penrose junction conditions (<ref>), but are contained in our generalized explicit transformation (<ref>).Indeed, by differentiating the equations (<ref>) with respect to the parameter τ of any geodesic {U(τ),V(τ),Z(τ), Z̅(τ)} crossing U=0, for the important form of J given by (<ref>) and illustrated inFigure <ref>, we obtain[Due to the continuity, η=Z on the impulse, and thus H_,Z=H_,η≡∂/∂ηH(η,η̅), and H_,Z̅=H_,η̅≡∂/∂η̅H(η,η̅).] ^+_ = ^-_ , ^+_ = ^-_ + H_,Z η̇^-_ + H_,Z̅ η̇̅̇^-_ + ( H_,ZH_,Z̅ - χ^2/4 η^-_η̅^-_) ^-_ , η̇^+_ = η̇^-_ + ( H_,Z̅ -χ/2 η̅^-_) ^-_ .Here we have employed that the geodesics of (<ref>) are C^1-curves and are unique, given initial data off the impulse. This fact can be established using the Fillipov solution concept for geodesics in locally Lipschitz continuous spacetimes <cit.>, along the lines of Remark 4.1(2) in <cit.>. For χ=0 (implying F=0=W) we recover the conditions given by equation (4.4) in <cit.>, and also in <cit.>. The formulae (<ref>)–(<ref>) extend these results by involving the gyratonic terms with χ. Clearly, there is an additional jump in the longitudinal and transverse velocities _ and η̇_ across the impulsive wave surface, namelyΔ_= - χ^2/4 η^-_η̅^-_^-_ andΔη̇_ = -χ/2 η̅^-_^-_ .These specific jumps become unbounded as η^-_→ 0, which is physically understandable because the singular gyratonic source is located at the spatial origin η=0 of the impulsive surface. It is also interesting to observe that|Δη̇_|^2 = - ^-_ Δ_ .This identity is related to the conservation of normalization of the four-velocity, which is guaranteed by the C^1-regularity of geodesics and the continuity of the metric. The effect of the gyraton on test particles moving along geodesics is even more nicely seen if we employ real polar coordinates and the explicit transformation (<ref>). This yields^+_ = ^-_ + H_ ,ρ^+_ = ρ^-_ ,φ^+_ = φ^-_ ,so that both the radial position ρ_ and the polar position φ_ arecontinuous (uneffected by the presence of a gyraton), but there is a discontinuity in the corresponding velocities ^+_ = ^-_ + 1/√(2)(e^φ^-_ H_,Z + e^-φ^-_ H_,Z̅) ρ̇^-_ + /√(2)(e^φ^-_ H_,Z - e^-φ^-_ H_,Z̅) ρ^-_φ̇^-_+ ( H_,ZH_,Z̅ - χ^2/2(ρ^-_)^2) ^-_ , ρ̇^+_ = ρ̇^-_ + 1/ρ^-_( H_,Z + H_,Z̅) ^-_ ,φ̇^+_ = φ̇^-_ + [/√(2) ρ^-_(e^φ^-_ H_,Z - e^-φ^-_ H_,Z̅) -χ/(ρ^-_)^2] ^-_ .In particular, the gyraton has no effect on the radial velocity ρ̇_, but it causes a specific jump of the axial velocity φ̇_ given by the term χ /(ρ^-_)^2. This is consistent with the physical interpretation of the gyratonic term: it encodes the additional rotational (spin) character of the source of the impulsive gravitational wave. The gyraton also affects the longitudinal velocity _ via the term χ^2 /2(ρ^-_)^2 in the expression (<ref>). It agrees with the studies presented in <cit.>. § EXTENSION TO ANY COSMOLOGICAL CONSTANT ΛThese results on impulsive pp -waves with gyratons can be generalized to any background of constant curvature, i.e., to gyratonic impulses propagating in de Sitter or anti-de Siter spacetimes.Indeed, it was already shown in <cit.> that the original Penrose “cut and paste” construction method in Minkowski space (summarized here in Subsection <ref>) can be extended to any Λ by applying exactly the same junction conditions (<ref>) to a more general background metric generalizing (<ref>), namely ṣ_0^2= 2 η̣ ̣̅η -2/[ 1+1/6Λ(ηη̅-) ]^2 .This introduces impulsive waves in the de Sitter (Λ>0) or anti-de Sitter (Λ<0) universes.Of course, the geometry of such impulses depends on Λ since the null hypersurface N, given by =0, along which the spacetime is cut into the two halves M^- and M^+ and re-attached with a specific “warp” (<ref>), has the induced 2-metric σ̣^2= 2 ( 1+1/6 Λ η η̅ )^-2 η̣ ̣̅η with the Gaussian curvature K=1/3Λ. Thus, for Λ=0 the impulsive wave surface is a plane, for Λ>0 it is a sphere, while for Λ<0 it is a hyperboloid. The geometry of these nonexpanding impulsive spherical and hyperboloidal waves was described in detail in <cit.> using various coordinate representations.It was also shown in <cit.> that applying the same transformation (<ref>) to the metric (<ref>), the explicit continuous form of the impulsive metric is obtained, namely ṣ^2= 2 |Ẓ+(H_,Z̅ZẒ+H_,Z̅Ẓ̅̅Z)|^2-2 ỤṾ/[ 1+1/6Λ(ZZ̅-UV- G) ]^2 ,where G(Z,Z̅)≡ H-ZH_,Z-Z̅H_,Z̅.Notice that this metric is conformal to the continuous form of impulsive pp -waves (<ref>), to which it reduces in the case Λ=0. Although it is continuous across the null hypersurface U=0, the discontinuity in the derivatives of the metric introducesimpulsive components in the Weyl and curvature tensors proportional to the Diracdistribution <cit.>. The metric (<ref>) thus explicitly describesimpulsive waves in de Sitter, anti-de Sitter or Minkowski backgrounds.Moreover, for any Λ the transformation (<ref>) automatically incorporates the Penrose junction conditions (<ref>) for reattaching the two halves of the spacetime M^- and M^+ with the warp →-H.The corresponding distributional form of these impulsive solutions reads ṣ^2= 2 η̣ ̣̅η-2+2H(η,η̅) δ() ^2/[ 1+1/6Λ(ηη̅-) ]^2 .It is obtained from the continuous form of the impulsive-wave metric (<ref>) by applying the transformation (<ref>) if the distributional terms arising from the derivatives of Θ() and () are also kept (see<cit.>, <cit.> for more details). The impulse, located on the wavefront =0, propagates in a background spacetime of constant curvature (<ref>). For Λ=0 this is the Brinkmann form of an impulsive pp -wave (<ref>) in Minkowski space, while for Λ≠0 it represents an impulse propagating in curved (anti-)de Sitter universe (the metric (<ref>) is conformal to the Brinkmann metric). In fact, the family of impulsive spacetimes (<ref>) contains all nonexpanding impulses in Minkowski, de Sitter or anti-de Sitter universes that can be constructed from the whole Kundt class of type N solutions with a cosmological constant Λ <cit.>. A systematic analysis of such distributional limits of Kundt sandwich waves was performed in <cit.>, for areview and explicit transformations see <cit.>. This recent review also summarizes the most important explicit impulses of this type, namely the axially symmetric Hotta–Tanaka solution <cit.>, which was obtained by boosting the Schwarzschild–de Sitter orSchwarzschild–anti-de Sitter black hole to the speed of light (the analogue of the Aichelburg–Sexl solution <cit.> for Λ≠0), and more general nonexpanding impulsive waves generated by null multipole particles <cit.>. Such solutions are more clearly expressed and analyzed by employing the 5-dimensional representation of the (anti-)de Sitter spacetime <cit.>. §.§ Continous form of impulsive waves with gyratons and Λ Interestingly, an explicitly continuous metric form of gyratonic impulsive waves in de Siter and anti-de Sitter spacetimes are obtained by applying the discontinuous transformation (<ref>)–(<ref>) on the background metric (<ref>), which yieldsṣ^2= 2 | ζ +ζF |^2 +2[ Θ( ζ H_,Z-H_,Z̅)F- W ]Ụ -2 ỤṾ/[ 1+1/6Λ(ZZ̅-UV- G) ]^2 ,whereG(Z,Z̅,U)≡ H-ZH_,Z-Z̅H_,Z̅+W.It is obvious that for Λ=0 this reduces to (<ref>), while in the absence of gyratons (W=0=F implying J=0) one recovers the metric form (<ref>). If both Λ=0 and J=0, the classical metric form (<ref>) is obtained.Indeed, to establish this, note that the numerator of (<ref>) is identical to (<ref>). As demonstrated in Section <ref>, this is equivalent to the distributional metric (<ref>) via the transformations (<ref>)–(<ref>), i.e., to the background (<ref>) for ≡ U≠0. The (anti-)de Sitter background (<ref>) involves the additional conformal factor, namely the square of 1+1/6Λ(ηη̅-) in the denominator. Performing the transformation (<ref>) and using the multiplication rules =UΘ and Θ^2=Θ, this expression becomes1+1/6Λ(ZZ̅-UV- (H-ZH_,Z-Z̅H_,Z̅)-U W ).Now, the key observation is that UW≡ W whenever W=0 for all U≤ 0, as is the case in the explicit examples (<ref>) and (<ref>). Consequently, 1+1/6Λ(ηη̅-)=1+1/6Λ (ZZ̅-UV- G), where G is defined by (<ref>). This gives the denominator in (<ref>), completing the argument.§.§ Distributional form of impulsive waves with gyratons and Λ Combining the results and relations employed in the previous subsection, we may also conclude that the class of spacetimes studied here can be written in the following form ṣ^2= 2 η̣ ̣̅η-2+2H(η,η̅) δ() ^2 +2J(η,η̅,) η̣ + 2J̅(η,η̅,) ̣̅η /[ 1+1/6Λ(ηη̅-) ]^2 .This is a combination of the metric (<ref>) for impulsive pp -waves with the gyratonic off-diagonal terms (generalized Brinkmann metric<cit.>) in the numerator, with the conformal factor [ 1+1/6Λ(ηη̅-) ]^-2. This distributional form of the metric is a general expression for impulsive gyratonicwaves propagating in any spacetime of constant curvature, that is Minkowski, de Sitter or anti–de Sitter universe, according to the sign of the cosmological constant Λ. The metric (<ref>) is clearly conformal to the distributional metric (<ref>) for impulsive gyratonic pp -waves. §.§ Penrose junction conditions with gyratons and Λ Similarly as in Subsection <ref>, we can now obtain and discuss the extended Penrose junction conditions in the presence of gyratons when Λ≠0. They identify the corresponding points across the impulsive null hypersurface N located at =U=0, which separates M^-(<0) and M^+(>0) of the background (anti-)de Sitter manifold (<ref>).The full transformation relating both parts M^- and M^+ in the form (<ref>) to the continuous metric form (<ref>) is again given by (<ref>). Interestingly, the inclusion of the cosmological constant Λ only occurs via the conformal factor [ 1+1/6Λ(ηη̅-) ]^-2=[ 1+1/6Λ (ZZ̅-UV- G) ]^-2 in (<ref>), where G is defined by (<ref>). Moreover, evaluated on the impulse it takes the very simple form[ 1+1/6Λ ηη̅ ]^-2=[ 1+1/6ΛZZ̅ ]^-2. Performing the limits U→0^- and U→0^+ of the transformation (<ref>), employing the continuity of the coordinates {U,V,Z, Z̅}, we obtain ^+_ = ^-_ + H_ , η^+_ = η^-_ . Clearly, these are the same as the relations (<ref>), (<ref>). We have thus recoveredthe Penrose junction conditions (<ref>) for proper identification of points (positions) across the impulse which are valid for any value of the cosmological constant, see <cit.>. Of course, for Λ≠0 the coordinates {, , η, η̅} are not the usual Minkowski double null coordinates, but thecoordinates of the (anti-)de Sitter conformally flat metric (<ref>).These gyratonic impulsive waves propagating in the (anti-)de Sitter universe have a specific effect on the velocities of free test particles. As in the case Λ=0, the corresponding junction conditions are obtained by differentiating the explicit transformation (<ref>) with respect to the parameter τ of any geodesic, and comparing its limitsU→0^- and U→0^+. Due to (<ref>), we obtain the same relations (<ref>), (<ref>), (<ref>) as in the Minkowski background case.§ CONCLUSIONS Let us summarize the main results of our work:* We have derived the new continuous metric form (<ref>) for impulsivepp -waves with gyratons, which naturally generalizes the classical metric (<ref>) without the gyratonic terms. * We have found the transformation (<ref>)–(<ref>) relating the continuous metric form (<ref>) to the distributional metric form (<ref>), which — compared to (<ref>) — contains the additional off-diagonal metric function J representing the gyraton. This is an extension of the classical transformation (<ref>). * We have explicitly presented the continuous form (<ref>) of the impulsive metric for the key example (<ref>) whenJ is proportional to the Heaviside step function of retarded time, including also the exceptional case(<ref>), or (<ref>), of the Frolov–Fursaev gyratons constructed from the axially symmetric Aichelburg–Sexl vacuum solution. * We have proved that the Penrose junction conditions (<ref>) for identifying the corresponding points in the “cut and paste” construction method remain valid even in the presence of gyratons (because the additional continuous functions W and F vanish on the impulsive hypersurface). * Nevertheless, the presence of a gyraton manifests itself through the “derivatives” of the junction conditions, as represented bythe transformation (<ref>)–(<ref>). This leads to a specific jump in the velocities (<ref>), (<ref>) of test particles crossing the gyrating impulse. As clearly demonstrated by (<ref>), (<ref>), the gyraton affects only the axial component of the transverse velocity. * Finally, we have generalized all these new results (obtained for the case of impulsive pp -waves with gyratons, and thus necessarily when Λ=0) to any value of the cosmological constant Λ. In particular, starting from the unified, conformally flat, form of themetric (<ref>) representing the de Sitter (Λ>0), anti-de Sitter (Λ<0) or flat Minkowski(Λ=0) backgrounds, we demonstrated that the transformation (<ref>)–(<ref>) can by applied for any Λ, and it relates the new continuous metric form (<ref>), (<ref>) to the corresponding distributional metric form (<ref>) of gyratonic impulsive waves propagating in de Sitter or anti-de Sitter spacetimes. There are also several possibilities for further extension of the above results. 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Quantum Grav. 33 (2016) 115002.SS:2017 Sämann C., Steinbauer R., Geodesics in nonexpanding impulsive gravitational waves with Λ, part II, arXiv:1704.05383 [math-ph]. | http://arxiv.org/abs/1704.08570v1 | {
"authors": [
"Jiri Podolsky",
"Robert Svarc",
"Roland Steinbauer",
"Clemens Sämann"
],
"categories": [
"gr-qc",
"hep-th",
"83C15, 83C35, 83C10"
],
"primary_category": "gr-qc",
"published": "20170427134746",
"title": "Penrose junction conditions extended: impulsive waves with gyratons"
} |
A New Look at Physical Layer Security, Caching, and WirelessEnergy Harvesting for Heterogeneous Ultra-dense Networks Lifeng Wang, Member, IEEE,Kai-Kit Wong, Fellow, IEEE,Shi Jin, Member, IEEE, Gan Zheng, Senior Member, IEEE,and Robert W. Heath, Jr., Fellow, IEEE L. Wang, and K.-K. Wong are with the Department of Electronic and Electrical Engineering, University College London, WC1E 7JE, London, UK (E-mail: {lifeng.wang, kai-kit.wong}@ucl.ac.uk). S. Jin is with National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China (Email: [email protected]). G. Zheng is with the Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Leicestershire, LE11 3TU, UK (Email: [email protected]). Robert W. Heath, Jr. is with the Department of Electrical and Computer Engineering, The University of Texas at Austin, Texas, USA (E-mail: [email protected]).December 30, 2023 ======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================§ INTRODUCTIONThe inflationary models, which solve successfully the horizon, flatness and relic problems <cit.> and generate the primordial density perturbations finally initiating the formation of galaxies and large-scale structure <cit.>, are the most reasonable models for the evolution of the early Universe. In simplest case inflation is controlled by a single scalar field (the inflaton) with an effective potential that plays a role of the cosmological constant during inflation.A fundamental step towards the unification of physics at all energy scales could be the possibility to describe the inflation using particle physics models. In numerous models (for a review see <cit.>) the role of the inflaton has been performed by the Standard Model (SM) Higgs boson <cit.> or a boson in Grand Unified Theories (GUTs) <cit.> or a scalar boson in supersymmetric (SUSY) models <cit.> (see <cit.> as reviews). A number of advantages of simplified SUSY GUTs in comparison with nonsupersymmetric GUTs such as naturally longer period of exponential expansion and better stability of the effective Higgs potential with respect to radiative corrections due to cancelation of loop diagrams have been noted quite long ago <cit.>.Thus, the implementation of inflationary scenario within a well-defined model of particle physics consistent with collider phenomenology where the inflaton is unambiguously identified is a longstanding problem. The only candidate on the role of the inflaton in the SM is the Higgs boson. The Higgs-driven inflation <cit.> was originally proposed as a single-field model based on the SM in the unitary gauge. This minimal model uses the Higgs isodoublet Φ interaction with gravity of the form ξ R Φ^†Φ (R is the scalar curvature and ξ is a positive constant). The Higgs-driven inflation leads to the spectral index value n_s=0.967 and the tensor-to-scalar ratio r=3· 10^-3 which are in agreement with the Planck Collaboration data <cit.>. However, the effects of Goldstone bosons should be included at an energy scale relevant to inflation in the model which is actually multifield.For the Higgs-driven inflation it was found <cit.> that the multifield effects are negligibly small during inflation and do not influence the observable quantities, such as the spectral index of primordial perturbations and the ratio of squared amplitudes for the tensor and the scalar perturbations (tensor-to-scalar ratio). This fact and the known considerations about the need to extend the SM, which could be an effective limit of GUTs, supersymmetry, supergravity or other beyond the SM theories, lead to the belief that inflation that is compatible with recent observations <cit.> might have been generated by several fields. It has been shown <cit.> that there is a class of inflationary models with two scalar fields non-minimally coupled to gravity that provides good agreement with the Planck data. Hybrid inflation proposed in <cit.> which involves the potential of two scalar fields ensures inflationary expansion, explains the observed spectrum of density fluctuations not requiring the unnatural scalar field amplitudes at the Planck scale. At the same time, a greater degree of uncertainty arises in the theory. Identification of the two scalar fields in the framework of a gauge theory model is not simplified, the initial conditions are not unambiguously fixed <cit.> theoretically in the two-dimensional field space and their tuning is needed to ensure adequate phenomenological consequences.It should be noted that some tension is observed between the experimental data and predictions of the minimal model.In to order to explain cosmic microwave background observables in the Higgs-driven inflationary scenario the parameter of non-minimal coupling should be very large (ξ∼ 10^4). So large value of ξ is not satisfactory from general theoretical backgrounds because it leads to violation of perturbative unitarity at the scale M_Pl/ξ which is smaller than the expected inflationary range above the M_Pl/√(ξ) (M_Pl denotes the Planck mass). In order to restore unitarity above the scale M_Pl/ξ,"new physics" (new particles interacting with the SM ones) should be introduced which modify the SM Higgs potential. The more serious problem of a large ξ value in the SM is the renormalization group evolution (RGE) of meaningful parameters which demonstrates unsatisfactory matching with the measured Higgs boson and top quark masses <cit.> as soon as the inflationary range of the order of M_Pl/√(ξ) or above is concerned. In order to reduce the value of ξ, an extremely small value of the effective quartic coupling λ_eff(μ) near the Planck scale is needed. At the Higgs boson mass m_h=125GeV such value of λ_eff(μ) can be achieved at the top quark mass which is more than 2σ below its observable central value <cit.>. In this case, the value of ξ necessary for a satisfactory inflationary scenario decreases thus allowing to avoid the problem of perturbative unitarity violation below the inflationary scale. Note that the GUT motivated inflationary model <cit.> predicts the same order of the parameter for a non-minimal coupling. However, there are cosmological models (see, for example <cit.>) with the same function of the non-minimal coupling and even polynomial potential of the fourth order that could provide a suitable inflationary parameters at small values of ξ.Apparent tensions arising in connection with parameter matching of the Higgs-driven inflation model increase the popularity of models with new physics at the TeV and multi-TeV scales. New particles consistent with restrictions on the new physics imposed by the LHC data provide extensive opportunities to improve significantly the Higgs-driven inflationary model. Analogue of this single-field model for the multifield scenarios is based on an observation that redefined fields in the Einstein framepractically coincide with primary fields in the Jordan frame at the low energy scale of the order of superpartners mass scale M_SUSY, reproducing the MSSM potential, while at the scale higher than the GUT scale the potential in the redefined fields can be slowly changing respecting the slow-roll approximation of an inflationary scenarios. This observation is sufficiently general.Recent analyses <cit.> of multifield models showed that unlike the single-field models they generically provide density (entropy) perturbations which can induce the curvature perturbation to evolve beyond the cosmological horizon in the process of inflation <cit.>. Evolution of density perturbations in multifield models should be studied in order to analyze new features in the observables such as non-Gaussianities <cit.> which are absent in the single-field inflationary models. Deviations of observable power spectrum calculated in multifield models from predictions of the single-field models could take place in the power of one of the three criteria <cit.>: (i) noncanonical kinetic terms; (ii) violation of slow-roll approximation; (iii) nonstandard initial ground state (different from Bunch–Davies vacuum). The main feature of the multifield models which leads to nonstandard primordial spectrum is the ability of trajectories of slow-roll fields to rotate in the field space, that occurs due to the presence of bumps and ridges in the effective multifield potential. When the slow-roll field trajectories turn in the field space, nonstandard contributions to primordial spectra can be amplified enough to be detectable in the microwave background <cit.>. In this paper we analyze a multifield extension of the standard Higgs-driven inflation inspired by the MSSM. A few general observations let us make first. In the framework of a sypersymmetric model the natural class of cosmological models are those with local supersymmetry (supergravity models). For the case when interactions at an energy scale below M_Pl are described by an effective N=1supergravity, the general form for the effective potential of scalar fields in the Einstein frame was derived in <cit.>. In the notation of <cit.> the Lagrangian can be written asL_B=e^-G [G_k (G^-1)^k_i G^i +3]-1/2ĝ^2 Re f^-1_αβ (G^i T^α j_i z_j) (G^k T^β j_k z_j) + G^i_j D_μ z_i D^μ z^*j - 1/2 Rwhere G_k=∂ G/∂φ^k, ĝ is the gauge coupling constant, T are generators of the groups, D_μ are covariant with respect to gravity and gauge group, (z_i, χ_i) is the chiral supermultiplet.The Kähler potentialG can be written in terms of the function ϕ which transforms as a real vector superfield and the superpotential g_s in the following form:G=3log(-ϕ/3)-log(|g_s|^2). Equation (<ref>) is a consequence of a Lagrangian written in terms of chiral superfields Φ̃:L= -6 ∫d^2 θE [R-1/4 ( D^2-8 R) Φ̃^†Φ̃+g_s ] + h.c.,here R is the superspace curvature and E is chiral density connected with local superspace basis (see <cit.>). One can show <cit.> that minimal coupling to gravity which takes place for ϕ=z^*_i z_i - 3 in the Kähler potential can be modified to a non-minimal coupling R→ R+p(Φ̃)R instead of the first term in eq. (<ref>) by the replacement ϕ=z^*_i z_i - 3 -3 (p(z)+h.c.)/2 for a given polynomial form p.[Taking frequent in the literature point of view that the main qualitative features at the scale of the order of M_Pl are valid despite the loop effects of gravity, there is an opinion that in a simplest case for a single field any polynomial form p(x) can be adjusted by taking dg_s/dx=√((3+p(x)exp(G(x)))/2) (where x is Re z, (z_i, χ_i) is the chiral supermultiplet) with the following extension of a solution in the form of series expansion to complex z <cit.>.]In the case of the MSSM natural choice is p=ξΦ̅_1 Φ̅_2, where Φ̅_1 and Φ̅_2 are chiral Higgs-Higgsino superfields. This choice of p in combination with the general form for the superpotential g_s=Λ+μΦ̅_1 Φ̅_2 (Λ and μ are real constants) leads to problems of achieving a suitable inflationary scenario, see <cit.>. For ξ parameter large enough when different regimes for the 'flat direction' tanβ parameter are taken,either there is no slow roll or the potential takes negative values. Not referring here to the possibility of the MSSM extension with a gauge singlet (non-minimal MSSM or NMSSM) where unsuitable behavior can be cured, we introduce non-minimal couplings in the non-holomorphic form (ξ_1 H^†_1 H_1+ξ_2 H^†_2 H_2)R (here H_1 and H_2 are SU(2) spinors and R is the Ricci scalar) that have no counterparts in supergravity. So only small electroweak quartic couplings g_1 and g_2 in the D-terms of eq. (<ref>) which then appear in the tree-level scalar potential at the SUSY scale provide grounds to speak about 'MSSM-inspired' inflationary scenarios. This sort of model is not a direct extension of models associated with the MSSM which include scalar fields minimally coupled to gravity <cit.>. The inflaton fields are identified as Higgs sector fields, thus, in this case one is talking about the multifield extension of the SM single-field Higgs inflation. Note that other realizations of the inflationary scenario in the MSSM are possible, when the inflaton is a combination of squark and slepton fields <cit.>, while the process of inflation is controlled by flat directions of the MSSM potential which are lifted by non-renormalizeable superpotential terms and soft supersymmetry breaking terms. It is assumed that D-terms in eq. (<ref>) vanish in the hidden sector. A number of other options of the MSSM-inspired inflation can be found in <cit.>. The model which is considered in the following sections includes two Higgs doublets coupled with gravity non-minimally. We focus on the two-Higgs doublet MSSM potential in the mass basis of scalar fields that has been analyzed starting from 1975 <cit.>. This potential includes three massless Goldstone bosons and five massive Higgs bosons. Working in the physical gauge, in this paper we do not take Goldstone bosons into account and consider inflationary scenarios that include Higgs bosons only. We show that inflationary scenarios with suitable parameters n_s and r are possible at the scale corresponding to the Hubble parameter H∼ 10^-5M_Pl. By this way a MSSM-inspired extension of the original Higgs-driven inflation is constructed.The structure of the paper is as follows. In section <ref> we define the MSSM two-Higgs-doublet potential in the basis of mass eigenstates for the five Higgs bosons at the superparticle mass scale. The mixing angles of the SU(2) field eigenstates are chosen in the form which is acceptable for the low-energy Higgs phenomenology. In section <ref> the MSSM-inspired potential taken in the Jordan frame with the polynomial form of the non-minimal coupling function is transformed to the Einstein frame. Equations of motion in the Friedmann–Lemaitre–Robertson–Walker (FLRW) metric are described in section <ref>. Numerical integration of the equations of motion with the initial conditions which are adjusted in a way suitable for reproduction of the observable values for the spectral index n_s and the tensor-to-scalar ratio r is preformed in section <ref>. In section <ref> we discuss briefly the strong coupling (SC) approximation for the MSSM-inspired potential under consideration. Results are summarized in section <ref>.§ THE MSSM-INSPIRED HIGGS POTENTIAL Two Higgs doublets of the MSSM can be parameterized using the SU(2) statesΦ_1 = ([ -i ω_1^+; 1/√(2) (v_1+η_1+i χ_1) ]), Φ_2 = ([-i ω_2^+; 1/√(2) (v_2 +η_2+i χ_2) ]),where ω^+_1,2 are complex scalar fields, η_1,2 and χ_1,2 are real fields, the vacuum expectation values v_1 and v_2 are usually redefined in (v, tanβ) parametrization: v=√(v^2_1+v^2_2) and tanβ=v_2/v_1 (v=246GeV). Two doublets Φ_1 and Φ_2 can be used to form the SU(2) × U(1) invariant and renormalizable effective potential which breaks gauge symmetry.The most general two-doublet effective potential can be written as <cit.>:V(Φ_1,Φ_2)=-μ_1^2 (Φ_1^†Φ_1) -μ_2^2 (Φ_2^†Φ_2) - [ μ_12^2 (Φ_1^†Φ_2) +h.c.] + λ_1 (Φ_1^†Φ_1)^2 +λ_2 (Φ_2^†Φ_2)^2 + λ_3 (Φ_1^†Φ_1)(Φ_2^†Φ_2) + λ_4 (Φ_1^†Φ_2)(Φ_2^†Φ_1) + [λ_5/2(Φ_1^†Φ_2)(Φ_1^†Φ_2)+ λ_6 (Φ^†_1 Φ_1)(Φ^†_1 Φ_2)+λ_7 (Φ^†_2 Φ_2)(Φ^†_1 Φ_2)+h.c.]. Let us consider the action in the Jordan frameS=∫ d^4x√(-g̃)[f(Φ_1,Φ_2)R̃-δ^abg̃^μν∂_μΦ_a^†∂_νΦ_b -V(Φ_1,Φ_2)],where g̃ is the determinant of metric tensor g̃_μν, and R is the scalar curvature. The factor in front of the kinetic term is not dependent on fields, so the case of Brans–Dicke gravity-like models are beyond our analysis. However, δ^ab in front of the kinetic term is not narrowing the generality of consideration, see details in appendix <ref>. Variation of action with respect to metric tensor g̃^μν and isodoublets Φ_a of the fields leads to the following equationsf(Φ_1,Φ_2)[R̃_μν-R̃/2g̃_μν] = (∇_μ∇_ν - g̃_μν∇^α∇_α)f(Φ_1,Φ_2) +2δ^ab[∂_μ(Φ_a)^†∂_νΦ_b-1/2g̃_μν∂_α(Φ_a)^†∂^αΦ_b] - 1/2V(Φ_1,Φ_2)g̃_μν, 2Φ_a = -∂f(Φ_1,Φ_2)/∂Φ_a^†R̃ + ∂V(Φ_1,Φ_2)/∂Φ_a^† ,where a=1,2,∇_μ is a covariant derivative and the d'Alembert operator acting on the scalar fields is denoted by ≡1/√(-g̃)∂_μ(√(-g̃)g̃^μν∂_ν). In the following we are using notations and normalization conventions for the potential V(Φ_1,Φ_2) in the generic basis (with λ_6,7 terms) from <cit.>, where the mass eigenstates for scalars were constructed.Note that the potential in eq. (<ref>) explicitly violates CP invariance if parameters μ_12, λ_5, λ_6, or λ_7 are complex-valued. For simplicity, we are not considering such possibility in the following. At the tree-level λ_5, λ_6, andλ_7 are equal to zero in the MSSM two-doublet potential. Nonzero parameters λ_5,6,7 of the effective Higgs potential can be generated by radiative corrections coming from the sector of soft supersymmetry breaking terms, where scalars couple to quark superpartners. To simplify the analysis we will not consider this possibility remaining with the tree-level potential at the M_SUSY scale. It is well-known that radiative corrections are large and in the context of this simplification (when the upper limit of the light CP-even state mass m_h does not exceed the Z-boson mass m_Z=91.2GeV) it is impossible to describe adequately the spectrum of Higgs boson masses. However, precision fitting of the collider data is not the primary purpose at this stage of consideration.The mass basis of scalars is constructed in a standard way. The SU(2) eigenstates (ω^±_a,η_a and χ_a, a=1,2) are expressed through mass eigenstates of the Higgs bosons h, H_0, A and H^± and the Goldstone bosons G^0, G^±by means of two orthogonal rotations( [ η_1; η_2 ])= O_α( [ H_0; h ]), ( [ χ_1; χ_2 ])= O_β( [ G^0; A ]), ( [ ω_1^±; ω_2^± ])= O_β( [ G^±; H^± ]),where the rotation matrixO_X=( [cos X -sin X;sin Xcos X ]),X=α, β.Masses of the CP-even scalars h and H_0 are m_h and m_H_0, the charged scalar mass is m_H^± and the CP-odd scalar mass is m_A. At the superpartners mass scale M_SUSY the m_A and tanβ can be chosen as the input parameters which fix the dimension-two parameters μ^2_1,μ^2_2 and μ^2_12 of the Higgs potential, whilethe dimensionless factors λ_i (i=1,...,7) at the tree level are expressed, using the SU(2) and U(1) gauge couplings g_2 and g_1, as follows <cit.>λ_1,2^ tree(M_SUSY) = g_1^2+g_2^2/8, λ_3^ tree(M_SUSY)=g_2^2-g_1^2/4, λ_4^ tree(M_SUSY) = -g_2^2/2, λ_5,6,7^ tree(M_SUSY)=0.The dimension-two parameters μ^2_1,μ^2_2 and μ^2_12 are fixed using the minimization conditions:μ_1^2=-m_A^2 sin^2(β)+m_Z^2/2cos(2 β),μ_2^2=-m_A^2 cos^2(β)-m_Z^2/2cos(2 β),μ_12^2=m_A^2 sin(β) cos(β),where m_Z=v√(g_1^2+g_2^2)/2. Then the potential given by eq. (<ref>) can be rewritten in the mass basis of scalar bosons, which are massless Goldstone bosons G^0, G^+, G^- and massive Higgs bosons h, H_0, A, H^+, H^-:V(h,H_0,A,H^±,G^0,G^±)=m_h^2/2 h^2+m^2_H_0/2 H_0^2+m_A^2/2 A^2+m_H^±^2 H^+ H^- +I_3+I_4,wherem_h^2=m_Z^2 sin^2 (α+β)+m_A^2 cos^2(α-β), m_H_0^2=m_Z^2 cos^2 (α+β) +m_A^2 sin^2 (α-β), m_H^±^2=m_A^2+m_W^2.Explicit forms of the interaction terms I_3 and I_4 are presented in the appendix <ref>. The mixing angles α and β at the M_SUSY scale are connected by the following equationtan(2α)=m_A^2+m_Z^2/m_A^2-m_Z^2tan (2 β).The scalar resonance with mass 125GeV which is experimentally observed at the LHC <cit.> has properties consistent with the SM. However, MSSM identifications are still possible with limited experimental statistics. Experimental data of the LHC Run I demonstrates the SM-like couplings of observed Higgs boson to fermions and vector bosons at the level of statistical significance only on the level slightly better than 2σ <cit.>. In the following consideration, the CP-even state h of the MSSM, when it is overridden mass, which is determined by the radiation corrections from the squark sector, will be identified as the 125GeV resonance. In the presence of other scalars H_0, A, H^+ and H^-, which are not experimentally observed, such identification is possible for the two specific features in the MSSM parameter space: (i) the decoupling regime <cit.> and/or (ii) the alignment limit <cit.>. In the decoupling regime masses of scalars H_0, A, and H^± are very large (they are at multi-TeV scale where also the lightest superpartners can be found), so their contributions to the observables at the top quark scale are strongly suppressed, while in the alignment limitH_0, A, and H^± are not necessarily extremely heavy. The alignment limit will be used in the following consideration. In this limit β-α≈π/2and the potential in eq. (<ref>) can be simplified by a special choice of mixing angles α and β. After rotation of scalar isodoubletsΦ^'_1=-Φ_1 sinβ+ Φ_2 cosβ, Φ^'_2=Φ_1 cosβ+ Φ_2 sinβto so-called Higgs basis <cit.> and the choice of mixing angles β=π/2 and α=0, the SU(2) components of isodoublets and the vacuum expectation values areη_1=H_0, η_2=h, v_1=0, v_2=v.So, in the unitary gauge G^0=G^±=0, we getχ_1=-A, χ_2=0, ω_1^±=-H^±, ω_2^±=0and the isodoublet convolutions are given by(Φ_1^†Φ_1) =H^-H^+ +A^2/2+H^2_0/2≡1/2(Ω^2_±+Ω^2_0), (Φ_2^†Φ_2)=h_v^2/2, (Φ_1^†Φ_2)=h_v/2(H_0+iA), (Φ_2^†Φ_1)=h_v/2(H_0-iA),where h_v=h+v, Ω^2_0=H_0^2+A^2, and Ω^2_±=2 H^+ H^-. The kinetic terms have canonical form ∂_μΦ_1^†∂^μΦ_1 =∂_μ H^- ∂^μ H^+ +1/2(∂ A)^2 +1/2(∂ H_0)^2, ∂_μΦ_2^†∂^μΦ_2=1/2(∂ h)^2. It follows from eq. (<ref>) that μ^2_12=0 and the potential in eq. (<ref>) becomesV(h_v,Ω_0,Ω_±) = - m^2_1 h^2_v + m^2_2 ( Ω^2_0 +Ω^2_±) + ν_1 ( h^4_v +Ω^4_0 + Ω^4_± ) - 2ν_1 h^2_v Ω^2_0 + 2ν_2 h^2_v Ω^2_± + 2ν_1Ω^2_0 Ω^2_± ,wherem^2_1=m^2_Z/4,m^2_2=m^2_A/2+m^2_Z/4, ν_1=g^2_1+g^2_2/32, ν_2=g^2_2-g^2_1/32. The potential in eq. (<ref>) qualitatively corresponds to the MSSM potential at the scale M_SUSY. It is invariant under two-dimensional rotations in (H_0, A) space and (H^+, H^-) space, what is the consequence of the specific choice of the mixing angles α and β in the alignment limit. This property allows reducing the number of five physically significant fields h, H_0, A, H^+ and H^- to the three field combinations, h^2_v, Ω^2_0 and Ω^2_±. Note that at h_v=0 the potential given by eq. (<ref>) is invariant under rotations in the four-dimensional field space. Tree-level quartic couplings λ_i, eq. (<ref>), are expressed through the gauge couplings g_1,2 which are fixed by collider data, since the gauge boson masses at tree level m_Z=v√(g^2_1+g^2_2)/2, m_W=vg_2/2 and cross sections of W^±, Z production are precisely measured (v=√(v^2_1+v^2_2)=(G_F√(2))^-1/2, G_F is the Fermi constant). Substituting m_Z=91.2GeV and m_W=80.4GeV, we obtain g_1=0.36 and g_2=0.65, which are used in numerical calculations of section <ref>.§ THE MSSM-INSPIRED MODEL WITH NON-MINIMAL INTERACTIONGeneric action which is dependent on N scalar fields ϕ^I, I=1, ..., N with the standard kinetic term and non-minimal coupling to gravity can be written asS_J=∫ d^4x√(-g̃)[f(ϕ^I)R̃-1/2δ_IJg̃^μν∂_μϕ^I∂_νϕ^J- V(ϕ^I) ],where tilde denominates the metric tensor and curvature in the Jordan frame. In our case V(ϕ^I) depends on five real scalar fieldsϕ^1=H^+ +H^-/√(2), ϕ^2=H^+ -H^-/√(2)i, ϕ^3=A,ϕ^4=H_0, ϕ^5=h_v.This action can be transformed to the following action in the Einstein frame <cit.> (see also <cit.>):S_E=∫ d^4x√(-g)[M^2_Pl/2R-1/2𝒢_IJg^μν∂_μϕ^I∂_νϕ^J-W],where𝒢_IJ=M^2_Pl/2f(ϕ^K)[δ_IJ +3f_, I f_, J/f(ϕ^K)], W= M^4_PlV/4f^2,the reduced Planck mass M_Pl≡1/√(8π G), f_,I = ∂ f/∂ϕ^I. Metric tensors in the Jordan and the Einstein frames are related by the equationg_μν=2/M^2_Plf(ϕ^I)g̃_μν.In the single-field Higgs-driven inflation the function f has been chosen as a sum of the Hilbert–Einstein term and the induced gravity term. We choose the function f in an analogous form:f(Φ_1,Φ_2)=M^2_Pl/2+ξ_1Φ_1^†Φ_1+ξ_2Φ_2^†Φ_2,where ξ_1 and ξ_2 are positive dimensionless constants. This form of function f follows from the requirement of renormalizability for quantum field theories in curved space-time <cit.>, where non-minimal couplings appear as renormalization counterterms for scalar fields. We also assume that vacuum expectation values for scalar fields are negligibly small in comparison with M_Pl.Note that non-minimal interaction in the form of eq. (<ref>) was considered <cit.> in the framework of the (nonsuperymmetric) two-Higgs-doublet model, when the boundary condition eq. (<ref>) is not used and the Higgs potential includes seven quartic couplings. Arbitrariness of the choice of λ_i is constrained imposing exact or approximate Z_2 symmetry (discrete symmetry whose breaking results in the appearance of the axion) on the generic two-Higgs-doublet potential which takes a specific functional form different from the 'MSSM-inspired', eq. (<ref>). It is assumed that the Higgs doublets Φ_1 and Φ_2 in this simplified potential are (0,v_1/√(2)) and (0,v_2/√(2)), what happens if the fields ω_1,2, η_1,2 and χ_1,2 in eq. (<ref>) are taken to be zero, so the MSSM mass eigenstates h, H, A and H^± are not specified. In our case the function f depends on the five scalar fields:f(Φ_1,Φ_2)=M^2_Pl/2+ξ_1/2(Ω^2_±+Ω^2_0)+ξ_2/2h^2_v. § PROPERTIES OF THE EQUATIONS OF MOTION IN THE FLRW METRIC Let us consider a spatially flat FLRW universe with metric intervalds^2=-dt^2+a^2(t)(dx_1^2+dx_2^2+dx_3^2),where a(t) is the scale factor. Varying the action in eq. (<ref>) with respect to g_μν and fields we get the following equations for the FLRW metricH^2=1/3M_Pl^2(σ̇^2/2+W), Ḣ= -1/2M_Pl^2σ̇^2,where the Hubble parameter H=ȧ/a, σ̇^2=𝒢_IJϕ̇^I ϕ̇^J, and dots mean the time derivatives. Field equations have the following form <cit.>ϕ̈^I+3Hϕ̇^I + Γ^I_ JKϕ̇^J ϕ̇^K + 𝒢^IK W'_, K = 0 ,where Γ^I_ JK is the Christoffel symbol for the field-space manifold, calculated in terms of 𝒢_IJ, W'_,K = ∂ W/∂ϕ^K. Hereafter, primes denote derivatives with respect to the fields. Due to the relationship of inflationary evolution in the Jordan and the Einstein frames, eqs. (<ref>) and (<ref>) are equivalent to eqs. (<ref>) and (<ref>) after transformation of the latter to Einstein frame.During inflation the Hubble parameter is positive and the scalar factor is a monotonically increasing function. To describe the evolution of scalar fields during inflation we use the number of e-foldings N_e=ln(a/a_e), where a_e is the value of the scalar factor at the end of inflation, as a new measure of time. The notation N^*_e=-N_e will be also used for convenience.Using d/dt=H d/dN_e one can write eqs. (<ref>) and (<ref>) in the formH^2=2W/6M_Pl^2-(σ^')^2, dlnH/dN_e= -1/2M_Pl^2(σ^')^2, dϕ^I/dN_e=ψ^I,dψ^I/dN_e= -(3+dlnH/dN_e)ψ^I - Γ^I_ JKψ^Jψ^K - 1/H^2𝒢^IK W^'_,K ,where (σ^')^2 = H^2 (σ̇)^2. After substitution of eqs. (<ref>) and (<ref>) a system defined by eqs. (<ref>) and (<ref>) includes ten first order equations which are suitable for numeric integration. Integration was performed by means of built-in subroutines of several computer algebra systems with cross-checks of results. Note that so far in this section and in the previous section <ref> we have not made any approximations.In order to calculate the observables, spectral index n_s and tensor-to-scalar ratio r, slow-roll parameters are introduced analogously to the single-field inflationϵ = -Ḣ/H^2,η_σσ = M_Pl^2M_σσ/W,whereM_σσ≡σ̂^K σ̂^J ( D_KD_J W),σ^I=ϕ̇^̇İ/σ̇ is the unit vector in the field space and D denotes a covariant derivative with respect to the field-space metric, D_Iϕ^J=∂_I ϕ^J+ Γ^J_IKϕ^K. Then the spectral index n_s and tensor-to-scalar ratio r at the time when a characteristic scale (50–65 e-foldings before the end of inflation) is of the order of the Hubble radiusin the course of inflation, can be calculated using the single-field equations valid to lowest order in slow-roll parameters <cit.>n_s=1-6 ϵ+2 η_σσ,r=16 ϵ. § NUMERICAL SOLUTIONS OF THE EQUATIONS OF MOTIONThe isosurfaces for the potential W(h_v,Ω_0,Ω_±) (one of the three variables is fixed) are shown in figure <ref>. At the fixed value of h_v, Ω_0 or Ω_± of the order of 0.1 (in Planck units) the saddle configuration of the surface is observed, as shown in figure <ref>(a). A characteristic feature of W which demonstrates ridges and gullies is shown in figure <ref>(b). In the gullies evolution of the field system looks as an infinite expansion at the constant Hubble parameter. One can see that the slow-roll inflation is possible if the initial condition for h_v or Ω_0 is chosen in the vicinity of zero, which is equivalent to four nonzero values of the fields (A,H_0,H^±) or three nonzero values of the fields (h_v,H^±). Initial conditions for a number of successful inflationary scenarios of this sort are presented in table <ref>. The evolution of fields superimposed on the Einstein-frame potential for the inflationary scenarios A and B, see table <ref>, is shown in figure <ref>, where the dashed fragments of field trajectories correspond to the inflationary stage when 0 ≤ N_e^*≤ 65. If we assume that the number of e-foldings during inflation N_e^*=65, then we get the initial conditions for inflationary trajectories presented in tables <ref> and <ref>. For scenarios A_1, A_2, A_3, B_3, and B_4 (see table <ref>) the value of the Hubble parameter in the beginning of inflation is H_init<3.6·10^-5M_Pl. Note that this value of the Hubble parameter is found to be in good agreement with the observational data <cit.>. 0mm0mm One can see that for the type A inflationary scenarios the field system rolls slowly down to the potential minimum (see also figures <ref>(a) and <ref>(b)), while for the B type inflationary scenarios, except B_4, all nonzero fields demonstrate rapidly damped oscillations going to zero h_v for the number of e-foldings before the end of inflation N_e^* ≫ 65, see figures <ref>(c) and <ref>(d). At the same time, significant nonzero value of h_v in the initial field configuration is suitable for inflation in the case B_4 when ξ_1=ξ_2. Note that inflationary scenarios with initial conditions denoted by A and B in tables <ref>, <ref> demonstrate remarkable stability of slow-roll parameters ϵ, η_σσ and observables r and n_s. In different cases with the Hubble parameter H ∼ 10^-5 M_Pl the values of n_s and rcoincide up to three digits. Such "attractor behavior" when over a wide range of initial conditions the system evolves along the same trajectory in the course of inflation is known for single-field models <cit.>, but it is not an obvious observation, generally speaking, for multifield models. In this sense the phenomenological stability inherent to the single-field Higgs inflation is preserved for the multifield MSSM-inspired model under consideration.The problem of perturbative unitarity violation at a large values of ξ parameters mentioned in the Introduction may persist in the MSSM although an order of magnitude smaller values of ξ appear in comparison with the SM Higgs inflation (except A_4 scenario, see Table <ref>). While in the SM for the Higgs inflation a simple unitarity bound can be derived E<M_Pl/ξ on the general basis of power-counting formalism for effective theory (for example, <cit.>), in the MSSM-inspired models with several fields such a simple criteria is not reliable and the situation with partial wave unitarity is much more difficult. Recent analysis <cit.> for the case of a general two-Higgs-doublet model without any discrete symmetry imposed on the scalar potential leads to non-trivial constraints on the masses and mixings which may depend on the scenario of new physics at a high energy scale. § THE STRONG COUPLING APPROXIMATION It follows from configurations shown in tables <ref> and <ref> that quite different initial values of scalar fields and parameters ξ_1 and ξ_2 appear in all cases in combination with a very small value of h_v. In all cases, but A_4, the inflationary parameters practically coincide, see figure <ref> and table <ref>. In this section, we show that such a pattern can be explained in the framework of the so-called "strong coupling approximation". It has been shown in a large number of analyses <cit.> that there are several classes of the single-field inflationary models such that within a given class all models predict the same values of observable parameters n_s and r in the leading 1/N_e approximation. These classes are known as cosmological attractors. A similar analysis of two-field inflationary models has been made in <cit.>. For string-motivated supergravity theory in which both the field-space metric and the potential usually have poles at the same points, the inflationary dynamic and the corresponding attractor have been studied <cit.>. The idea of a cosmological attractor is based on an observation that the kinetic term in Jordan frame practically does not affect the slow-roll parameters if the "strong coupling regime" is respected during inflation. In the case of multifield modelsthe field system is in the SC regime if the following inequality is respected:δ_IJ∂_μϕ^I∂_νϕ^J≪3/f(ϕ^K)f_,If_,J∂_μϕ^I∂_νϕ^J.In the approximation of eq. (<ref>) the action given in eq. (<ref>) can be written asS_E=M^2_Pl/2∫ d^4x√(-g)[R-3g^μν/2f^2(ϕ^K)f_,If_,J∂_μϕ^I∂_νϕ^J -M^2_PlV(ϕ^I)/2f^2(ϕ^I)]and rewritten in the equivalent formS_E=∫ d^4x√(-g)[M^2_Pl/2R-g^μν/2∂_μ[√(3/2)M_Plln(f/f_0)] ∂_ν[√(3/2)M_Plln(f/f_0)] -M^4_PlV/4f^2],where f_0 is a positive constant with the same dimension as f. The role of inflaton in the strong coupling approximation is performed by the "effective field"Θ=√(3/2)M_Plln(f/f_0),in terms of which the action S_E includes the standard kinetic term of Θ and does not include kinetic terms of any other scalar fields which can be interpreted as model parameters. This circumstance allows one to calculate the inflationary parameters in the SC approximation using the single-field model. If we adjust Θ in such a way that Θ=0 corresponds to Ω_0=0 and Ω_±=0, then f_0=M^2_Pl/2.The single-field model consistent with the above-mentioned scenarios A and B (see tables <ref> and <ref>) can be easily defined. In the scenario A we set ϕ^5=h_v=0 during inflation, while in the scenario B (except B_4) one can observe that inflation starts whenh_v^2≪∑_I=1^4(ϕ^I)^2.We do not consider the case B_4 here. In all other cases we can neglect h_v and write the potential in the formV_sc=m^2_2 (Ω^2_0 +Ω^2_±)+ν_1 (Ω^2_0 + Ω^2_±)^2.The function f is approximated byf_sc=M^2_Pl/2+ξ_1/2(Ω^2_±+Ω^2_0)and therebyV_sc=m^2_2/ξ_1(2f-M^2_Pl)+ν_1/ξ_1^2(2f-M^2_Pl)^2,so the Einstein framepotential can be written as followsW_sc=M^4_Pl(M^2_Pl-2f_sc)[(M^2_Pl-2f_sc)ν_1-m_2^2ξ_1]/4f_sc^2ξ_1^2.Using m_2^2ξ_1≪ M^2_Plν_1 we get from eq. (<ref>)W_sc≃M^4_Plν_1/ξ_1^2(M^2_Pl/2f_sc-1)^2=M^4_Plν_1/ξ_1^2(1-M^2_Pl/2f_0e^-√(6)Θ/(3M_Pl))^2.The slow-roll parameters areϵ = M^2_Pl/2(W^'_Θ/W)^2=4/3(e^√(6)Θ/(3M_Pl)-1)^-2, η=M^2_PlW^''_Θ/W =4(e^√(6)Θ/(3M_Pl)-2)/3(e^√(6)Θ/(3M_Pl)-1)^2.With these analytic expressions for the slow-roll parameters in the SC approximation the inflationary parameters can be easily calculated. It is convenient to express the inflationary parameters as a functions of f_scn_s=1-8M^2_Pl(M^2_Pl+2f_sc)/3(M^2_Pl-2f_sc)^2, r=64M^4_Pl/3(M^2_Pl-2f_sc)^2 .Straightforward numerical cross-checks demonstrate that the ratioC_sc=|f(ϕ^K)δ_IJϕ̇^Iϕ̇^J/3f_,If_,Jϕ̇^Iϕ̇^J |is less than 7× 10^-5 in the scenario A and 2× 10^-4 in the scenario B, so the SC approximation is meaningful. It is demonstrated in table <ref> that in all cases the values of inflationary parameters r and n_s calculated using eq. (<ref>) are close to the parameter values that have been found numerically in section <ref>. Note in this connection that the primordial non-Gaussianities which do not arise in the single-field inflationary models should be very small in the case under consideration as soon as the reduction to a single-field scenario is precise enough. It should be mentioned that f_in/M^2_Pl close to 44 is not a sufficient condition for an inflationary scenario with suitable values of n_s and r. For a large number of initial data with such values of f_in/M^2_Pl, but beyond the abovementioned A and B type scenarios, acceptable inflationary evolution is not observed[The initial conditions Ω_±=0 and both h_v and Ω_0 nonzero lead to exotic situation when the field trajectory rapidly(after ∼ 0.05 e-foldings) rolls into the gully h_v^2 = Ω_0^2 (see figure <ref>(b)). This direction is not absolutely flat (the case when critical points are degenerate and not isolated <cit.>), but so close to flat that cannot be analysed by numerical methods. Simple estimate with f_in = 45 and ξ_1+ξ_2 ∼ 2· 10^3 gives an extremely long slow-roll with the number of e-foldings of the order of 10^12.].§ SUMMARYIn this paper, we constructed a MSSM-inspired extension of the original Higgs-driven inflation <cit.> using the two-Higgs-doublet potential of the MSSM which is simplified in a way suitable for calculation of transparent symbolic and numerical results for the main observables, the spectral index n_s and the tensor-to-scalar ratio r. The shape of the MSSM potential surface in the Einstein frame where ridges and bumps influence the trajectory in the fields space is different from the usual form in models of hybrid inflation. The model under consideration incorporates multiple non-minimally coupled scalar fields and non-canonical kinetic terms in the Einstein frame which are induced by the curvature of the field-space manifold. For these reasons, the evolution of fields is generically different from slow-roll, at least for some time interval during inflation.The analysis of the background inflation dynamics demonstrated that after setting up the initial conditions for the five-dimensional field configuration such simplified MSSM-inspired model successfully describes the Higgs-driven inflation consistently with the observations of the Planck and BICEP2 collaborations. Two types of consistent inflationary scenarios are found with the initial conditions denoted as A and B, see table <ref>, which demonstrate the remarkable stability of the observables with respect to the shift of the initial field system configuration. The main difference between these two cases is the presence of rapid field oscillations in the initial phase of case B before the beginning of inflation, while oscillations are absent in case A. During the period of cosmological evolution which determines the observables, h_v field is negligibly small so the value of ξ_2 parameter practically does not influence the result and in the MSSM-inspired model degenerate values of ξ_1 and ξ_2 are always meaningful. Inflation occurs for field values much smaller than the Planck scale, although no suitable expansion scenario was found for initial state when h_v, Ω_0 and Ω_± are very small at the same time. In all cases trajectories of the system do not turn steeply in the field space, so specific features of the potential like bumps and ridges are not expected to induce primordial non-Gaussianities with a magnitude large enough to be detectable in the cosmic microwave background.Multifield model under consideration demonstrates rather strong attractor behavior and can be mapped to the single-field model with the effective inflaton field defined by eq. (<ref>). Such models share very close results for the spectral index and the tensor-to-scalar ratio in combination with negligible non-Gaussianity, which are in good agreement with the latest experimental data.In conclusion let us also note that an important point beyond our analysis is the stability of results with respect to radiative corrections. The flatness of the effective potential in the region of the field amplitudes of the order of M_Pl is an essential property for a suitable slow-roll. While the quantum gravity corrections are expected to be rather small of the order of V/M^4_Pl∼ g^2_p/ ξ^2, the corrections induced by the SM fields and the superpartner fields involved in the F and D soft supersymmetry breaking Lagrangian terms require careful analysis which is dependent on the MSSM parametric scenario under consideration. For example, in the "natural MSSM scenario" which is used for LHC analyses the superpartners of quarks show up at the multi-TeV scale, while gauginos decouple. At the one-loop resummed level the superpartner threshold corrections to the two-doublet MSSM potential are expressed by Coleman-Weinberg terms Δ V = 1/(64π^2) Sp[(V^”(ϕ))^2 (log (V^”(ϕ)/μ^2)-3/2)], where second derivatives taken at the local minimum of Higgs potential are equal to masses of scalars. Nontrivial significant contributions are provided in the higher orders of perturbation theory by nonrenormalizable operators <cit.>. Fermionic and bosonic loops give contributions of different signs which could partially compensate each other. Contributions of the SM vector bosons and fermions are smaller than the MSSM ones because of small gauge and Yukawa couplings, so main corrections from the third generation of quark superpartners interacting with Higgs isodoublets must not spoil a small slope of the potential. Important correction can be provided also by the renormalization group (RG) evolution of ξ non-minimal couplings from the top quark scale to the M_Pl scale. RG evolution gives at least a factor of two for the value of ξ in the framework of SM Higgs-driven inflation, but moderate changes of the order of ten percent in the inflationary region. Models which are described by the RG-improved effective action <cit.> should provide an improved precision for observables. Careful MSSM evaluations which are beyond our analysis are appropriate in order to ensure stability of results.This work was partially supported by Grant No. NSh-7989.2016.2. The research of E.O.P. and E.Yu.P. was supported in part by Grant No. MK-7835.2016.2.§ GENERAL ACTION FOR NON-MINIMAL HIGGS INTERACTIONS IN THE MSSMIn the general case one can write the action for non-minimal interaction of the MSSM Higgs doublets with gravity in the form (here we redefine f(Φ̃_1,Φ̃_2)=M^2_Pl/2[1+ϱ(Φ̃_1,Φ̃_2)])S = ∫d^4x√(-g){M^2_Pl/2[1+ϱ(Φ̃_1,Φ̃_2)]R - g^μνG^IJ∂_μΦ̃_I^†∂_νΦ̃_J-𝒱(Φ̃_1,Φ̃_2)},whereG^IJ = [ G^11 G^12; G^21 G^22 ],ϱ(Φ̃_1,Φ̃_2) = ∑_a,bξ̂_ab(Φ̃_a^†Φ̃_b) + ∑_a,b,c,dẑ_abcd(Φ̃_a^†Φ̃_b)(Φ̃_c^†Φ̃_d)+..., 𝒱(Φ_1,Φ_2) = -∑_a,bμ̂_ab(Φ̃_a^†Φ̃_b) + ∑_a,b,c,dλ̂_abcd(Φ̃_a^†Φ̃_b)(Φ̃_c^†Φ̃_d).One can find some (may be non-unitary) transformation Φ̃_a →Φ_a = U_abΦ̃_b to diagonalize G^IJ→δ^IJ, so U_ab isU_ac^†G^cdU_db = δ_ab. After such transformation the action can be written asS = ∫d^4x√(-g){M^2_Pl/2[1+ρ(Φ_1,Φ_2)]R - g^μνδ^IJ∂_μΦ_I^†∂_νΦ_J-V(Φ_1,Φ_2)},whereρ(Φ_1,Φ_2) = ∑_a,bξ_ab(Φ_a^†Φ_b)+ ∑_a,b,c,dz_abcd(Φ_a^†Φ_b)(Φ_c^†Φ_d)+..., V(Φ_1,Φ_2) = -∑_a,bμ_ab(Φ_a^†Φ_b) + ∑_a,b,c,dλ_abcd(Φ_a^†Φ_b)(Φ_c^†Φ_d).Thus one can always start with the action in the form (<ref>) or (<ref>) without loss of generality. § HIGGS POTENTIAL IN THE MASS BASISThe potential given in eq. (<ref>) can be written in terms of the mass eigenstates, which are massless Goldstone fields G_0, G_+, G_- and massive Higgs bosons h, H_0, A, H_+, H_-,[For convenience, we rewrite H^±,G^0, and G^± as H_±,G_0, and G_±, correspondingly.] in the following formV(h,H_0,A,H_±,G_0,G_±)=m_h^2/2 h^2+m_H^2/2 H_0^2+m_A^2/2 A^2+m_H^±^2 H_+ H_- + I_3+I_4,whereI_3= v/8( g_p^2 { s_α+β [c_2 α h^3+c_2 βh(A^2-G_0^2-2G_- G_+)] . . +c_α+β [c_2 α H_0^3-c_2 βH_0 (A^2-G_0^2-2 G_-G_+)] + h H_0/2[(c_α-β-3 c_3α+β)h-(s_α-β+3s_3α+β)H_0]+ . 2 s_2 β A G_0(s_α+β h-c_α+βH_0) }+ 2 i g_2^2 A(H_+G_–H_-G_+) + h[(g^2s_α-β+g_p^2s_α+3 β)H_+H_–(g_m^2c_α-β+g_p^2 c_α+3 β)(H_+G_-+H_-G_+)] - . H_0[(g^2c_α-β+g_p^2c_α+3 β)H_+H_-+(g_m^2s_α-β+g_p^2 s_α+3 β)(H_+G_-+H_-G_+)] ), I_4= g_p^2/8{ -s_4 β [ AG_0(G_+G_–H_+H_-)+H_+G_+(G_-^2-H_-^2) . + H_-G_-(G_+^2-H_+^2)+G_0^2-A^2/2(AG_0+H_+G_-+H_-G_+)] - 2 c_4 βH_+H_-G_+G_-+s_2 β^2(G_+^2H_-^2+G_-^2H_+^2) +c_2 β^2 [ G_0^4+A^4/4+G_+G_-(G_+G_-+G_0^2)+H_+H_-(H_+H_-+A^2) ] + [c_2 β(A^2-G_0^2)+2s_2 βAG_0][c_2 α(h^2-H_0^2)/2+s_2 α hH_0]+ 1/4.[(1-3c_4 α)h^2 H_0^2+(1-3c_4 β)A^2 G_0^2 +c_2 α^2(h^4+H_0^4)+2s_4 αhH_0(h^2-H_0^2)]}+ ig_2^2/4(H_-G_+-H_+G_-)[s_α-β(hA+H_0G_0)+c_α-β(hG_0-H_0A)] + 1/4[(g_1^2s_2β^2-g_2^2c_2β^2)AG_0(H_-G_++H_+G_-)-(g_1^2c_2βs_2α+g_2^2s_2 βc_2 α)hH_0G_+G_-] + 1/16[(2g_2^2+g_m^2c_2(α-β)-g_p^2c_2(α+β))(h^2G_+G_-+H_0^2H_+H_-) +(2g_2^2-g_m^2c_2(α-β)+g_p^2c_2(α+β))(H_0^2G_+G_-+h^2H_+H_-) -(g^2+g_p^2 c_4 β)(H_+H_-G_0^2+G_+G_-A^2)] + 1/8 (g_1^2 s_2 β c_2 α+g_2^2 c_2 β s_2 α)(h^2-H_0^2)(H_+G_-+H_-G_+) + hH_0/8[(g_p^2 s_2(α+β)-g_m^2s_2(α-β))H_+H_- -(g_p^2c_2(α+β)+g_m^2 c_2(α-β))(H_-G_++H_+G_-)], m_h^2 = m_Z^2 s_α+β^2+m_A^2 c_α-β^2,m_H^2 = m_Z^2 c_α+β^2+m_A^2 s_α-β^2,m_H_±^2 = m_A^2+m_W^2, m_W=v/2 g_2,sinα=s_α, etc., andg_p^2=g_1^2+g_2^2,g_m^2=g_2^2-g_1^2,g^2=g_1^2-3g_2^2.99general1_0 A. 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"authors": [
"M. N. Dubinin",
"E. Yu. Petrova",
"E. O. Pozdeeva",
"M. V. Sumin",
"S. Yu. Vernov"
],
"categories": [
"hep-ph",
"astro-ph.CO",
"gr-qc",
"hep-th"
],
"primary_category": "hep-ph",
"published": "20170526152956",
"title": "MSSM-inspired multifield inflation"
} |
addressref=1,2,corref,[email protected]]T.S. Travis S. Metcalfe addressref=3,4,corref,[email protected]]J.L. Jennifer van Saders[id=1]Space Science Institute, 4750 Walnut Street, Suite 205, Boulder CO 80301 USA [id=2]White Dwarf Research Corp., 3265 Foundry Place, Unit 101, Boulder CO 80301 USA [id=3]The Observatories of the Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena CA 91101 USA [id=4]Dept. of Astrophysical Sciences, Princeton University, Princeton NJ 08544 USAMetcalfe & van Saders Disappearance of Sun-like CyclesAfter decades of effort, the solar activity cycle is exceptionally wellcharacterized but it remains poorly understood. Pioneering work at theMount Wilson Observatory demonstrated that other sun-like stars also showregular activity cycles, and suggested two possible relationships betweenthe rotation rate and the length of the cycle. Neither of theserelationships correctly describe the properties of the Sun, a peculiaritythat demands explanation. Recent discoveries have started to shed light onthis issue, suggesting that the Sun's rotation rate and magnetic field arecurrently in a transitional phase that occurs in all middle-aged stars.Motivated by these developments, we identify the manifestation of thismagnetic transition in the best available data on stellar cycles. Wepropose a reinterpretation of previously published observations to suggestthat the solar cycle may be growing longer on stellar evolutionarytimescales, and that the cycle might disappear sometime in the next0.8-2.4 Gyr. Future tests of this hypothesis will come from ground-basedactivity monitoring of Kepler targets that span the magnetic transition,and from asteroseismology with the TESS mission to determine precisemasses and ages for bright stars with known cycles.§ ASTROPHYSICAL CONTEXTThe periodic rise and fall in the number of sunspots every 11 years wasfirst noted by <cit.>, and the detailed patterns of spotorientation and migration throughout this solar activity cycle havesubsequently been characterized with exquisite observations spanning manydecades. Stellar dynamo theory attempts to understand these patterns byinvoking a combination of convection, differential rotation, andmeridional circulation to modulate the global magnetic field<cit.>. Observations of other sun-like stars arenecessarily more limited because in most cases we cannot spatially resolvespots on their surfaces. However, the solar activity cycle is clearlydetectable without spatial resolution from observations of the intensityof emission in the Ca ii H (396.8 nm) and K (393.4 nm) spectrallines (hereafter Ca HK). These lines have long been used as a proxy forthe strength and filling factor of magnetic field because the emissiontraces the amount of non-radiative heating in the chromosphere<cit.>. The most comprehensive spectroscopic survey forCa HK variations in sun-like stars was conducted over more than 30 yearsfrom the Mount Wilson Observatory <cit.>,yielding the first large sample of stars with measured rotation rates andactivity variations to help validate stellar dynamo theory.Initial results from the Mount Wilson sample suggested that both thestellar cycle period and the mean activity level depend on the Rossbynumber, the rotation rate normalized by the convective turnover time<cit.>. Cycle periodswere shortest for the most rapidly rotating young stars, while they werelonger for older stars with slower rotation. <cit.>suggested that there were actually two distinct relationships between therotation rate and the length of the cycle, with an upper sequence of starsshowing a cycle every 300-500 rotations, and a lower sequence of shortercycles requiring fewer than ∼100 rotations. At moderate rotationrates (10-22 days), some stars exhibited cycles simultaneously on bothsequences. <cit.> interpreted this dual pattern asevidence for two stellar dynamos operating in different shear layers,possibly at the bottom of the outer convection zone (the tachocline), orin the near-surface regions as suggested by helioseismic inversions<cit.>.One of the most perplexing results from the Mount Wilson survey is thatneither of the stellar-based relationships between the length of the cycleand the rotation rate correctly describe the properties of the Sun. With amean cycle period of 11 years and a sidereal rotation period of 25.4 days(P_ cyc/P_ rot∼160), the Sun falls between the twostellar sequences <cit.>. Recent work may have identifiedthe reason why the solar activity cycle does not fit the patternestablished by other stars: the Sun's rotation rate and magnetic field maybe in a transitional phase that occurs in all middle-aged stars<cit.>. In the following section, we reviewthe evidence for a magnetic transition and the underlying mechanisms. InSection <ref>, we examine previously published observations toidentify the manifestation of this transition in stellar activity cycles,and we propose a new scenario for the evolution of the solar cycle. Wediscuss the potential for future observational tests of this hypothesis inSection <ref>.§ MAGNETIC METAMORPHOSIS The idea of using rotation as a diagnostic of stellar age dates back to<cit.>, and a decade of effort has gone into calibrating themodern concept of gyrochronology <cit.>. Although starsare formed with a range of initial rotation rates, the stellar windsentrained in their magnetic fields lead to angular momentum loss frommagnetic braking <cit.>. The angular momentum lossscales strongly with the angular rotation velocity dJ/dt∝ω^3,which forces convergence to a single rotation rate at a given mass afterroughly 500 Myr in sun-like stars <cit.>. The evidencefor this scenario relies on studies of rotation in young clusters atvarious ages, and until recently the only calibration point for agesbeyond ∼1 Gyr was from the Sun.The situation changed after the Kepler space telescope provided new datafor older clusters and field stars. The initial contributions from Keplerincluded observations of stellar rotation in the 1 Gyr-old clusterNGC 6811 <cit.> and the 2.5 Gyr-old cluster NGC 6819<cit.>, extending the calibration of gyrochronologysignificantly beyond previous work. The first surprises emerged whenasteroseismic ages became available for Kepler field stars with measuredrotation periods <cit.>. Initial indications ofa possible conflict between asteroseismology and gyrochronology were notedby <cit.>, who found that no single mass-dependent relationshipbetween rotation and age could simultaneously describe the cluster andfield populations. Although they used low-precision asteroseismic agesfrom grid-based modeling <cit.>, the tension was stillevident.§.§ Breakdown of Magnetic Braking The source of disagreement between the age scales from asteroseismologyand gyrochronology came into focus after <cit.> scrutinizedKepler targets with precise ages from detailed modeling of the individualoscillation frequencies <cit.>.They confirmed the existence of a population of field stars rotating morequickly than expected from gyrochronology. They discovered that theanomalous rotation became significant near the solar age for G-type stars,but it appeared ∼2-3 Gyr for hotter F-type stars and ∼6-7 Gyrfor cooler K-type stars. This dependence on spectral type suggested aconnection to the Rossby number, because cooler stars have deeperconvection zones with longer turnover times. They postulated that magneticbraking may operate with a dramatically reduced efficiency beyond acritical Rossby number, and they reproduced the observations with modelsthat eliminated angular momentum loss beyond Ro ∼ 2. This value isderived from a model-dependent estimate of the convective turnover timeone pressure scale-height above the base of the outer convection zone.Although the specific value obtained by <cit.> depends onmixing-length theory, the observed trend for stars of various masses andages is robust. The anomalous rotation discovered by <cit.> is illustratedfor sun-like stars in Figure <ref>. A standard rotational evolutionmodel (solid line) and the modified model that eliminates angular momentumloss beyond a critical Rossby number (dashed line) are from the originalpaper, which also used hotter and cooler stars to constrain the fit. Notethat the solar age and rotation rate (marked with a ⊙ symbol) wereused to calibrate the standard model beyond the 0.5-2.5 Gyr age range ofclusters. Asteroseismic ages for the Kepler sample (black points and16 Cyg) have been updated with values from <cit.>. The shadedregion represents the expected dispersion due to the range of masses andmetallicities within the sample (e.g., the two high points are lowermetallicity stars, giving them thinner convection zones that reach thecritical Rossby number at faster rotation rates). The asteroseismicrotation rates and ages for a ∼3 Gyr-old solar analog binary system<cit.> have been overplotted, validating asteroseismic rotationmeasurements and the age scale for sun-like Kepler stars. A fewwell-characterized solar analogs are shown with yellow points, including18 Sco <cit.>, α Cen A<cit.>, and 16 Cyg A & B <cit.>. Although some uncertainties remain for 18 Sco andα Cen A, these bright stars appear to follow the same pattern ofanomalous rotation observed in the Kepler sample.<cit.> suggested that magnetic braking might become lessefficient in older stars from a concentration of the field into smallerspatial scales. <cit.> demonstrated that the dipole componentof the global field is responsible for most of the angular momentum lossdue to the magnetized stellar wind <cit.>. TheAlfvén radius is greater for the larger scale components of the field,and because both the open flux and the effective lever-arm increase withincreasing Alfven radius, low-order fields consequently shed more angularmomentum. The inverse of this process may be responsible for the onset ofefficient magnetic braking in very young stars <cit.>.§.§ Triggering the Magnetic Transition <cit.> identified a magnetic counterpart to the rotationaltransition discovered by <cit.>. They compiled publishedCa HK measurements for the Kepler sample and compared them to a selectionof sun-like stars from the Mount Wilson survey <cit.>. Such a comparison requires the Ca HK measurements to beconverted to a chromospheric activity scale (log R'_ HK) thataccounts for the bolometric flux of different spectral types.The relationship between chromospheric activity and rotation isillustrated in Figure <ref>. The Kepler targets are plotted byspectral type, including F-type (triangles), G-type (circles), and K-typestars (squares), while the Mount Wilson targets are shown as star symbols.Several rotational evolution models from <cit.> are shown,converted from Rossby number to chromospheric activity using therotation-activity relation of <cit.>. The activity levels thatcorrespond to key Rossby numbers are shown as shaded regions on eitherside of the Vaughan-Preston gap <cit.>. Thedotted line connects some well-characterized solar analogs, including thesame stars shown with yellow points in Figure <ref>.The magnetic evolution of sun-like stars appears to change dramaticallywhen they reach the critical Rossby number (Ro∼2) identified by<cit.>. The shutdown of magnetic braking near the activitylevel of 18 Sco <cit.> keepsthe rotation rate nearly constant as the activity level continues todecrease with age toward α Cen A <cit.> and 16 Cyg <cit.>. A similar transition occurs at fasterrotation rates for hotter stars like HD 143761 (blue triangle), and atslower rotation rates for cooler stars like HD 219834A (red square). Theinfluence of this magnetic transition on stellar activity cycles isdescribed in Section <ref>.<cit.> proposed that a change in the character ofdifferential rotation is the mechanism that ultimately disrupts thelarge-scale organization of magnetic fields in sun-like stars. The processbegins at Ro ∼ 1, where the rotation period becomes comparable tothe convective turnover time. Differential rotation is an emergentproperty of turbulent convection in the presence of Coriolis forces, and<cit.> showed that many global convection simulations exhibita transition from solar-like to anti-solar differential rotation nearRo ∼ 1 <cit.>. The Vaughan-Preston gap canthen be interpreted as a signature of rapid magnetic evolution triggeredby a shift in the character of differential rotation. <cit.> usedactivity measurements of stars in several open clusters to constrain theage of F-type stars crossing the gap to be between 1.2 and 1.4 Gyr. Thetwo most active F-type stars in the Kepler sample have ages of 0.94 and1.64 Gyr and fall on opposite sides of the gap, again validating theasteroseismic age scale.Emerging from the rapid magnetic evolution across the Vaughan-Preston gap,stars reach the Ro ∼ 2 threshold where magnetic braking operateswith a dramatically reduced efficiency, possibly due to a shift inmagnetic topology. The rotation period then evolves as the star undergoesslow expansion and changes its moment of inertia as it ages. At the sametime, the activity level decreases with effective temperature as the starexpands and mechanical energy from convection largely replaces magneticenergy driven by rotation as the dominant source of chromospheric heating<cit.>.§ MANIFESTATION IN STELLAR ACTIVITY CYCLES The new picture of rotational and magnetic evolution provides a frameworkfor understanding some observational features of stellar activity cyclesthat have until now been mysterious. An updated version of a diagrampublished in <cit.> is shown in Figure <ref>, usingdata from <cit.>. More recent data have been added from<cit.>, <cit.>, <cit.>, <cit.>,<cit.>, <cit.>, <cit.>,and <cit.>. We do not include marginal detections of stellarcycles that may obscure the relationships suggested by the best availabledata <cit.>.The stellar sequence along the bottom of Figure <ref> has threedistinct regimes. For faster rotators (P_ rot<22 days), thissequence is dominated by short cycles for stars that also show longercycles on the upper sequence (vertical dotted lines). Many of the MountWilson targets in this regime appeared to have “chaotic variability” intheir chromospheric activity. This may be due to the ubiquity of shortperiod cycles on the lower sequence, combined with seasonal data gaps thatfailed to sample these timescales adequately. F-type stars are expected tobegin the magnetic transition at rotation periods ∼15 days, but thereare very few hot stars with well determined cycles. The oldest cyclingF-type star in the Mount Wilson sample is HD 100180<cit.>, which has P_ rot=14 daysand shows normal cycles on both sequences. The more evolved starHD 143761 has P_ rot=17 days, and shows flat activity at logR'_ HK=-5.04 for 25 years <cit.>. The age fromgryochronology implied by this rotation period is 2.5±0.4 Gyr<cit.>, which agrees with the age of F-type stars observed byKepler that have reached the critical Rossby number (blue dashed line)where the rotation period subsequently evolves much more slowly<cit.>.The transition across Ro ∼ 2 for G-type stars occurs at rotationperiods comparable to the Sun (P_ rot∼ 23–30 days). Beforereaching this threshold, magnetic braking continues in these stars andtheir cycle periods evolve along the two sequences as their rotationslows. When they reach the critical Rossby number, the rotation ratechanges much more slowly and we postulate that the cycle period respondsto the magnetic transition. If we consider the evolutionary sequencedefined by 18 Sco <cit.>, the Sun(4.6 Gyr), and α Cen A <cit.>, thedata suggest that a normal cycle on the lower sequence may grow longeracross the transition (yellow dashed line). Eventually stars reach a lowactivity state like 16 Cyg A & B <cit.>, where cyclic activity is no longerdetected <cit.>. The Sun falls to the right of thisevolutionary sequence because it is slightly less massive than the otherstars (with a longer convective turnover time), so it does not reach thecritical Rossby number until its rotation is a bit slower. Consideringother sun-like stars, we propose that the solar cycle may be growinglonger on stellar evolutionary timescales, and that the cycle mightdisappear sometime in the next 0.8-2.4 Gyr (between the ages ofα Cen and 16 Cyg).All of the slowest rotators with cycles (P_ rot>30 days) areK-type stars, which is now understandable—magnetic braking shuts down inmore massive main-sequence stars before they reach these long rotationperiods. Depending on the effective temperature, K-type stars reach thecritical Rossby number at rotation periods longer than 35 days. Thehottest cycling K-type star in our sample is HD 219834A, and it appearsto be well along the magnetic transition (red dashed line). All of thestars to the right of HD 219834A are significantly cooler, so they havenot yet reached the critical Rossby number <cit.>. Theslightly hotter star HD 182572 <cit.> appears to have already completed themagnetic transition like 16 Cyg A & B, showing flat activity at logR'_ HK=-5.10 for 13 years <cit.>.Although α Cen A appears in the same region of Figure <ref>as several K-type stars, the broader evolutionary scenario suggests thatthe current cycle evolved from a shorter period on the lower sequence near18 Sco. The expected cycle period on the upper sequence for G-type starsat the rotation period of α Cen A is ∼35 years, much longerthan the observed cycle <cit.>. In addition, there is noevidence of a shorter cycle in α Cen A <cit.>,even though 18 Sco shows a cycle on the lower sequence at essentially thesame rotation period.We can understand the evolution of the cycle toward longer periods duringthe magnetic transition by considering the variation of convectivevelocity with depth. The velocity is larger in the outer regions of theconvection zone, and becomes progressively smaller in deeper layers<cit.>. Consequently, a star will initially exceedthe critical Rossby number in the outer layers and the condition will onlybe met later in the deeper layers as the character of differentialrotation shifts and the local rotation rate continues to slow<cit.>. If the lower sequence in Figure <ref>represents a dynamo operating closer to the surface, while the uppersequence is the result of a dynamo driven in deeper layers<cit.>, we speculate that the cycle period may growlonger as the magnetic transition proceeds and pushes the dynamo intodeeper layers. The size of the convection zone might then set the overalltimescale for completing the transition, when cycles disappear (or becomeextremely long) as in HD 143761, 16 Cyg A & B and HD 182572.However, other identifications of the underlying dynamos that areresponsible for the two stellar sequences may not support thisinterpretation.§ DISCUSSION AND FUTURE OUTLOOKMotivated by the recent discoveries of a rotational and magnetictransition in middle-aged stars <cit.>, wehave identified the corresponding evolution of stellar activity cycles. Areinterpretation of previously published observations suggests that cycleperiods grow longer along two sequences as magnetic braking slows thestellar rotation, but at a critical Rossby number (Ro∼2) the surfacerotation rate changes more slowly while the cycle gradually grows longerbefore disappearing. Evidence for this scenario exists for a range ofspectral types, from the hotter F-type stars (HD 100180, HD 143761), towell-characterized solar analogs (18 Sco, α Cen A,16 Cyg A & B), to the cooler K-type stars (HD 219834A, HD 182572).The Sun appears to have aleady started this transition, and the solarcycle is expected to grow longer on stellar evolutionary timescales beforedisappearing sometime in the next 0.8-2.4 Gyr (between the ages ofα Cen and 16 Cyg).The greatest obstacle to understanding how the magnetic transitioninfluences stellar activity cycles is the paucity of suitableobservations. The bright sample of stars that were monitored for decadesby the Mount Wilson survey have well-characterized long activity cyclesand rotation periods, but their basic stellar properties are uncertain. Inparticular, the precise masses and ages that would allow us to identifyevolutionary sequences are currently available for just a few stars<cit.>. This situation will soon improve, after the TransitingExoplanet Survey Satellite <cit.> yieldsasteroseismic data for bright stars across the sky during a two yearmission (2018–2020). Although the time-series photometry will span only27 days for most TESS targets, this was sufficient to detect solar-likeoscillations in hundreds of Kepler stars down to V∼12<cit.>. Similar detections are expected from TESS down toV∼7 <cit.>, particularly in F-type and hotter G-typestars with larger intrinsic oscillation amplitudes.Although the basic stellar properties of the fainter Kepler stars arewell-constrained from asteroseismology, chromospheric activity data havenot been collected for long enough to detect stellar cycles. About a dozenstars in the <cit.> sample were monitored in Ca HK severaltimes per year during the Kepler mission <cit.>.The cadence was insufficient to detect the shortest activity cycles, andthe limited duration hindered the identification of longer cycles. So far,the only credible cycle in a Kepler target was detected usingasteroseismic and photometric proxies of activity <cit.>,revealing a 1.5-year cycle on the lower sequence at P_ rot∼ 11days. Most of the Kepler sample has already made the transition acrossRo ∼ 2, so we might expect them to be “flat activity” stars like16 Cyg A & B <cit.>, but this remains to be seen. Futureobservations with the Las Cumbres Observatory (LCO) global telescopenetwork promise to probe the onset and duration of the magnetic transitionthat drives the evolution and eventual disappearance of sun-like activitycycles.We would like to thank Axel Brandenburg, Ricky Egeland, Ed Guinan, JeffHall, Phil Judge, Savita Mathur and Benjamin Shappee for helpfuldiscussions. 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"authors": [
"Travis S. Metcalfe",
"Jennifer van Saders"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170526180104",
"title": "Magnetic Evolution and the Disappearance of Sun-like Activity Cycles"
} |
Optimum Transmission Window for EPONs with Gated-Limited Service Huanhuan Huang, Tong Ye, Member, IEEE, Tony T. Lee, Fellow, IEEE, and Weisheng Hu, Member, IEEE This work was supported in part by the National Science Foundation of China under Grant 61571288, Grant 61671286, and Grant 61433009, and in part by the Open Research Fund of Key Laboratory of Optical Fiber Communications (Ministry of Education of China).The authors are with the State Key Laboratory of Advanced Optical Communication Systems and Networks, Shanghai Jiao Tong University, Shanghai 200240, China (email: [email protected]; [email protected]; [email protected]; [email protected]). May 24, 2017 ===================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================== Data-driven workflows, of which IBM's Business Artifacts are a prime exponent,have been successfully deployed in practice, adopted in industrial standards, and have spawned a rich body of research in academia, focused primarily on static analysis.In previous work, theoretical results were obtained on the verification of a rich modelincorporating core elements of IBM's successful Guard-Stage-Milestone (GSM) artifact model.The results showed decidability of verification of temporal properties of a largeclass of GSM workflows and established its complexity. Following up on these results,the present paper reports on the implementation of SpinArt, a practical verifier based on the classical model-checking tool Spin. The implementation includes nontrivial optimizations and achieves good performance on real-world businessprocess examples. Our results shed light on the capabilities and limitations of off-the-shelf verifiers in thecontext of data-driven workflows.§ INTRODUCTIONThe past decade has witnessed the evolution of workflow specification frameworks from the traditional process-centric approach towards data-awareness. Process-centric formalisms focus on control flowwhile under-specifying the underlying data and its manipulations by the process tasks, often abstracting them away completely. In contrast, data-aware formalisms treat data as first-class citizens. A notable exponent of this class is IBM's business artifact modelpioneered in <cit.>, successfully deployed in practice <cit.> and adopted in industrial standards.In a nutshell, business artifacts (or simply “artifacts”) model key business-relevant entities, which are updated by a set of services that implement business process tasks, specified declaratively by pre-and-post conditions.A collection of artifacts and services is called an artifact system.IBM has developed several variants of artifacts, of which the most recent isGuard-Stage-Milestone (GSM) <cit.>.The GSM approach provides rich structuring mechanisms for services, includingparallelism, concurrency and hierarchy, and has been incorporated in the OMG standardfor Case Management Model and Notation (CMMN) <cit.>.Artifact systems deployed in industrial settings typically specify complex workflowsprone to costly bugs, whence the need for verification of critical properties. Over the past few years, the verification problem for artifact systems was intensively studied. The focus of the research community has been to identify practically relevant classes of artifact systems andproperties for which fully automatic verification is possible. This is an ambitious goal, since artifacts are infinite-state systems due to the presence of unbounded data.Along this line, complexity results were shown for different versions ofthe verification problem with various expressiveness of the artifact models and properties. In particular, a previous work <cit.> studied Hierarchical Artifact Systems (HAS),a model capturing core elements of the GSM model, and established the complexity ofverifying a rich class of linear-time temporal properties for various fragments of HAS. The present paper follows up on the theoretical results of <cit.> bystudying the practical implementation of SpinArt, a fully automatic verifier for artifact systems. The goal in this work is to explore the feasibility of using existing off-the-shelf tools to implement such an artifact verifier. We focus specifically on Spin <cit.>, the mainmodel checker used in the verification community and the natural candidate for a verifier implementation.We begin by defining a core fragment of the HAS model, called Tuple Artifact System (TAS),that can potentially be handled by Spin. At a high level, a tuple artifact system consists of a read-only database, a tuple of updatable artifact variables and a set of services specifying transitions of the system using pre-and-post conditions.This fragment remains very expressive, as demonstrated by our experiments showing that a large set of realistic business processes can be specified as TAS's. The properties of TAS's to be verified are specified using an extension of Linear-Time Temporal Logic (LTL). Our model is expressive enough to allow data of unbounded domain and size,which are features not directly supported by Spin or other state-of-the-art model checkers.Therefore, a direct translation into Spin requires setting limits on the size of the data and its domain, resulting in an incomplete verifier. To address this challenge, we exploit the symbolic verification techniques establishing the decidability results in <cit.> and develop a simple algorithm for translating TAS specifications and properties into equivalent problem instances that can be verified by Spin,without sacrificing either the soundness or the completeness of the verifier. However, a naive use of Spin still results in poor performance even with the translation algorithm. Therefore, we develop an array of nontrivial optimizations techniques to render verification tractable. To the best of our knowledge, SpinArt is the first implementation of anartifact system verifier that preserves decidability under unbounded data while being based on off-the-shelf model checking technology.The main contributions are summarized as follows. =0in=0in* We define Tuple Artifact System (TAS),a core fragment of HAS that permits efficient implementation of a Spin-based verifier. By exploiting the symbolic verification approach from previous work <cit.>,we show a simple algorithm for translating the verification problem into an equivalent instance in Spin. This algorithm forms the basis of our implementation of SpinArt.* We implement SpinArt with two nontrivial optimization techniques to achieve satisfactory performance. The first consists of a more efficient translation algorithm avoiding a quadratic blowup in the size of the specification due to keys and foreign keys,so that it shortens significantly the compilation and execution time for Spin. The second optimization is based on static analysis, and greatly reduces the size of the search space by exploiting constraints extracted from the input specificationduring a pre-computation phase. Although these techniques are designed with Spin as the target tool, we believe that they can be adapted to implementations based on other off-the-shelf model checkers.* We evaluate the performance of SpinArt experimentally using both real-world and synthetic data-driven workflows and properties. We created a benchmark of artifact systems and LTL-FO propertiesfrom existing sets of business process specifications and temporal properties by extending themwith data-aware features. The experiments highlight the impact of the optimizations and various parameters of the specifications and properties on the performance of SpinArt. The paper is organized as follows. We start by reviewing in Sect. <ref> the HAS model andformally defining TAS, a core fragment of HAS.We also review LTL-FO, the temporal logic for specifying properties of TAS's.In Sect. <ref> we first review the theory developed in <cit.>, thendescribe the initial direct implementation of SpinArtbased on the symbolic representation technique introduced there.We next present the specialized optimizations, essential for achieving acceptable performance.The experimental results are shown in Sect. <ref>.Finally, we discuss related work in Sect. <ref> and conclude in Sect. <ref>.§ THE MODELIn this section, we present the variant of artifact systems supported by our verifier, as well as the temporal logic LTL-FO used to specify the properties to be verified. §.§ Tuple Artifact SystemsThe model is a variant of the Hierarchical Artifact System (HAS) model presented in <cit.>. In brief, a HAS consists of a database and a hierarchy (rooted tree) of tasks. Each task has associated to it local evolving data consisting of a tuple of artifact variables and an updatableset of tuples called the artifact relation. It also has an associated set of services. Each application of a service is guarded by a pre-condition on the database and local data and causes an update of the local data, specified by a post condition (constraining the next artifact tuple) and an insertion or retrieval of a tuple from the artifact relation. In addition, a task may invoke a child task with a tuple of parameters, and receive back a result when the child task completes. A run of the artifact system is obtained by any valid interleaving of concurrently running task services. The implemented model restricts the HAS model as follows:=0pt=0pt* it disallows evolving relations in artifact data* it does not use arithmetic in service pre-and-post conditions* the underlying database schema uses an acyclic set of foreign keys[Foreign keys and acyclic schemasare standard database notions, reviewed in Definition <ref>.]As shown by the real-life examples used in the experimental evaluation,the implemented model is powerful enough to capture a wide variety of business processes, and is a good vehicle for studying the implementation of a Spin-based verifier.The implemented model retains the hierarchy of tasks present in HAS. However, for simplicity of exposition, weonly define formally the core of the model, consisting of a single task in which a tuple of artifact values evolves throughout the workflow under the action of services.For clarity, we also describe the algorithms in terms of the core model. The exposition can be easily extended to a hierarchy of tasks. We now present the syntax and semantics of the core model, which we call Tuple Artifact System (TAS).The formal definitions below are illustrated with an intuitive example of the TAS specification of an order fulfillment business process originally written in BPMN <cit.>. Intuitively, the workflow allows customers to place orders and the supplier company to process the orders.We begin with the underlying database schema.A database schemais a finite set of relation symbols, whereeach relation R ofhas an associated sequence of distinct attributescontaining the following: =0pt=0pt* a key attribute ID (providing a unique identifier for tuples in R), * a set of foreign key attributes {F_1, …, F_m}, and* a set of non-key attributes {A_1, …, A_n} disjoint from {ID, F_1, …, F_m}.To each foreign key attribute F_i of R is associated a relation R_F_i ofand the inclusion dependency R[F_i] ⊆ R_F_i[ID], stating that every value of attribute F_i occurring in R is the ID of a tuple in R_F_i.It is said that the foreign key F_i references relation R_F_i.Intuitively, a foreign key F of relation R referencing relation R_F acts as a pointer from the tuples of R to tuples of R_F. The assumption that the ID of each relation is a single attribute is made for simplicity, and multiple-attribute IDs can be easily handled.A database schemais acyclic if there are no cycles in the references induced by foreign keys. More precisely, consider the directed graph FK whose nodes are the relations of the schema and in which there is an edge from R_i to R_j if R_i has a foreign key attribute F referencing R_j. The schemais acyclic if the graph FK is acyclic. All database schemas considered in this paper are acyclic. Note that acyclic schemas include the Star (and Snowflake) schemas <cit.> widely used in business process data management. The order fulfillment workflow has the following database schema: =0pt=0pt* (𝙸𝙳, 𝚗𝚊𝚖𝚎, 𝚊𝚍𝚍𝚛𝚎𝚜𝚜, 𝚛𝚎𝚌𝚘𝚛𝚍),(𝙸𝙳, 𝚒𝚝𝚎𝚖_𝚗𝚊𝚖𝚎, 𝚙𝚛𝚒𝚌𝚎)(𝙸𝙳, 𝚜𝚝𝚊𝚝𝚞𝚜)The IDs are key attributes,𝚙𝚛𝚒𝚌𝚎, 𝚒𝚝𝚎𝚖_𝚗𝚊𝚖𝚎, 𝚗𝚊𝚖𝚎, 𝚊𝚍𝚍𝚛𝚎𝚜𝚜, 𝚜𝚝𝚊𝚝𝚞𝚜 are non-key attributes, and 𝚛𝚎𝚌𝚘𝚛𝚍 is a foreign key attribute satisfying the dependency [record] ⊆[ID]. Intuitively, thetable contains customer information witha foreign key pointing to the customers' credit records stored in .Thetable contains information on the items.Note that the schema is acyclic as there is only one foreign key reference fromto . We assume two infinite, disjoint domains of IDs and data values,denoted by DOM_id and DOM_val, and an additional constantwhere ∉DOM_id∪DOM_val ( is useful as a special initialization value). The domain of all non-key attributes isDOM_val.The domain of each key attribute ID of relation R is an infinite subsetDom(R.ID) of DOM_id, and Dom(R.ID) ∩ Dom(R'.ID) = ∅ for R ≠ R'.The domain of a foreign key attribute F referencing R is Dom(R.ID).Intuitively, in such a database schema, each tuple is an object with a globally unique id. This id does not appear anywhere else in the database except as foreign keys referencing it. An instance of a database schemais a mapping D associating to each relation symbol R a finite relation D(R) of the same arity of R, whose tuples provide, for each attribute, a value from its domain, such that no distinct tuples agree on the key ID. In addition, D satisfies all inclusion dependencies associated with the foreign keys of the schema.removed the active domain since it is not used in this paper. Figure <ref> shows an example of an instance ofthe acyclic schema of the order fulfillment workflow. Note that the domains of .ID, .ID and .ID and the domain for non-key attributes are mutually disjoint. The domain of .record is included in Dom(.ID) since 𝚛𝚎𝚌𝚘𝚛𝚍 is a foreign key attribute referencing .ID. We next proceed with the definition of artifacts and services. Similarly to the database schema, we consider two infinite, disjoint setsof ID variables andof data variables. We associate to each variable x its domain Dom(x). If x ∈, then Dom(x) = DOM_id∪{},and if x ∈, then Dom(x) = DOM_val∪{}.An artifact variable is a variable in ∪. If x̅ is a sequence of artifact variables, a valuation of x̅is a mapping ν associating to each variable x in x̅ an element in Dom(x).An artifact schema is a pair = ⟨, x̅⟩ with an acyclic database schemaand x̅⊆∪ a set of artifact variables. The domain of each variable x ∈x̅ is eitherDOM_val∪{} or dom(R.ID) ∪{} for some relation R ∈. In the latter case we say that the type of x is(x) = R.ID. An instance ρ ofis a pair (D, ν) whereD is a finite instance ofand ν is a valuation of x̅.The artifact schema of the order fulfillment example consists ofthe acyclic database schema described in Example <ref> and the following artifact variables: =0pt=0pt* ID variables:of type .ID andof type .ID* Non-ID variables:and Intuitively,andstore the ID of the customer andthe ID of the item ordered by the customer. Variableindicatesthe different stages of the order, namely “Init”, “OrderPlaced”, “Passed” (passed the credit check), “Shipped” or “Failed”. Variableindicates whether the ordered item is in stock. For a given artifact schema = ⟨, x̅⟩ and a sequence y̅ of variables, a condition on y̅ isa quantifier-free first-order (FO) formula over𝒟ℬ∪{=} whose variables are included in y̅. In more detail, a condition over y̅ isa Boolean combination of relational or equality atoms whose variables are included in y̅. A relational atom over relationR(ID, A_1, …, A_m, F_1, …, F_n) ∈,is of the formR(x, y_1, …, y_m, z_1, …, z_n),where {x, z_1, …, z_n}⊆ and {y_1, …, y_m}⊆. An equality atom is of the form x = z, where x is variable and z is a variable of the same type, or x ∈ and z ∈DOM_val.The special constantcan be used in equalities. If α is a condition on y̅⊆x̅,D an instance of 𝒟ℬ and ν a valuation of x̅,we denote by D α(ν) the fact that D satisfies α with valuation ν, with standard semantics.For an atom R(z̅) in α where R ∈,if ν(z) = for some z ∈z̅, then R(ν(z̅)) is false (since the databaseinstances do not contain ). Although conditions are quantifier-free, conditions with existentially quantified variables (denoted ∃FO)can be easily simulated by adding variables to x̅, so we use them as shorthand whenever convenient. The following ∃FO conditionstates that the customer with IDhas good credit:∃ n ∃ a ∃ r(, n, a, r) (r, “Good”).We next define services in TAS. Let = ⟨, x̅⟩ be an artifact schema. A service σ ofis a tuple ⟨π, ψ, y̅⟩ where: =0pt=0pt* π and ψ, called pre-condition and post-condition,respectively, are conditions over x̅, and* y̅ is the set of propagated variables, where y̅⊆x̅. Intuitively, π and ψ are conditions which must be satisfied by the previous and the next instance respectively when σ is applied.In addition, the values stored in y̅ are propagated to the next instance.The order fulfillment TAS has the following five services:EnterCustomer, EnterItem, CheckCredit, Restock and ShipItem. Intuitively, for each order, the workflow first obtains the customer anditem information by applying the EnterCustomer service and the EnterItem service.Then the credit record of the customer is checked by the CheckCredit service.If the record is good, ShipItem can be called to ship the item to the customer.If the requested item is unavailable, then Restock must be called before ShipItem to procure the item.Next, we illustrate each service in more detail.The EnterCustomer and EnterItem allow the customer to enterhis/her information and the ordered item's information. Theandtables are queried to obtain the customer ID and item ID.When EnterItem is called, the supplier also checks whether the item is currently in stock and sets the variableto “Yes” or “No” accordingly. This step is modeled as an external service so we use the post-condition to enforce that the two values are chosen nondeterministically. In both services, if bothandhave been entered, the current status of the order is updated to “OrderPlaced” (otherwise it remains “Init”). The two services can be called multiple times toallow the customer to modify previously entered data.The propagated variables of EnterCustomer areandsince their values are not modified when the service is applied.Similarly, the only propagated variable of EnterItem is . The two services are formally specified in Fig. <ref>, and Fig. <ref> shows transitions that result from applying the two services consecutively.Need to make sure the figure are in the right place We describe in brief the rest of the services. The CheckCredit service can be called if = “OrderPlaced”. It checks the credit record of the customer using the condition IsGood() in Example <ref>. If the credit record is good, then it updatesto “Passed” otherwise to “Failed”. The Restock service can be called if = “Passed” which means that the credit check is passed. The service simply updatesto “Yes”,indicating that ordered item is now in stock. Finally, the ShipItem can be called if = “Passed” and = “Yes”. It updatesto “Shipped”, meaning that the shipment is successful.We can now define TAS's.A Tuple Artifact System (TAS) is a triple Γ = ⟨, Σ, Π⟩,where 𝒜 is an artifact schema, Σ is a set of services over 𝒜,and Π, called the global pre-condition, is a condition over x̅. We next define the semantics of TAS. Intuitively, a run of a TAS on a database D consists ofan infinite sequence of transitions among artifact instances (also referred to as configurations, or snapshots),starting from an initial artifact tuple satisfying pre-condition Π.We begin by defining single transitions. Let Γ = ⟨𝒜, Σ, Π⟩ be a tuple artifact system, where 𝒜 = ⟨x̅, 𝒟ℬ⟩. We define the transition relation among instances ofas follows. For two instances (ν, D), (ν', D') and service σ = ⟨π, ψ, y̅⟩, (ν, D) σ (ν', D') if D = D', D π(ν), D ψ(ν'), and ν'(y) = ν(y) for each y ∈y̅. Then a run of the TAS Γ = ⟨𝒜, Σ, Π⟩on database instance D is an infinite sequence ρ = { (I_i, σ_i) }_i ≥ 0, where each I_i is an instance (ν_i, D) of , D Π(ν_0),and for each i > 0, I_i-1σ_i I_i. In the run, σ_0 is a special initializing service init, whose role is to produce the instance I_0. §.§ Specifying Properties of TAS's with LTL-FO In this paper we focus on verifying temporal properties of runs of a tuple artifact system.For instance, in the business process of the example above, we would like to specify properties such as: (†) If an order is taken and the ordered item is out of stock, then the item must be restocked before it is shipped. In order to specify such temporal properties we use, as in previous work, an extension of LTL (linear-time temporal logic). LTL is propositional logic augmented with temporal operators such as G (always), F (eventually), X (next) and U (until) (e.g., see <cit.>).An LTL formula φ with propositions prop(φ) defines a property of sequences of truth assignments to prop(φ).For example, Gp says that p always holds in the sequence, Fp says that p will eventually hold, pU q says that p holds at least until q holds,and G(p → Fq) says that whenever p holds,q must hold later in the sequence. An LTL-FO property of a tuple artifact systemis obtainedstarting from an LTL formula using some set P ∪Σ of propositions. Propositions in P are interpreted as conditions over the variables x̅together with some additional global variables y̅, shared by different conditions and allowing to refer to the state of the task at different moments in time. The global variables are universally quantified over the entire property.A proposition σ∈Σ indicates the application of service σ in a given transition. LTL-FO formulas are defined as follows.Let Γ = ⟨, Σ, Π⟩ be a TAS where = (x̅, ). Let y̅ be a finite sequence of variables in ∪ disjoint from x̅, called global variables. An LTL-FO formula for Γ is an expression ∀y̅φ_f, where: =0pt=0pt* φ is an LTL formula with propositions P ∪Σ, where P is a finite set of proposition disjoint from Σ* f is a function from P to conditions over x̅∪y̅* φ_f is obtained by replacing each p ∈ P with f(p) For example, suppose we wish to specify property (†).The property is of the form φ =G (p → ( q Ur)), which means: if p happens, then in the future q will not happen until r is true. Here p says thatthe EnterItem service is called and chooses an out-of-stock item, q states thatthe ShipItem service is called with the same item, and r states thatthe service Restock is called to restock the item. Since the item mentioned in p, q and r must be the same,the formula requires using a global variable i denoting the ID of the item.This yields the following LTL-FO property: ∀ i 𝐆 (( 𝙴𝚗𝚝𝚎𝚛𝙸𝚝𝚎𝚖 = i= “No” ) →((𝚂𝚑𝚒𝚙𝙸𝚝𝚎𝚖 = i) U (𝚁𝚎𝚜𝚝𝚘𝚌𝚔 = i))) A correct specification can enforce (†) simply by requiring in the pre-condition of ShipItem that the item is in stock. One such pre-condition is (=“Yes”=“Passed”), meaning that the item is in stock and the customer passed the credit check. However, in a similar specification where =“Yes” is not tested in the pre-condition but performed in the post-condition of ShipItem(i.e. the post-condition requires that if =“Yes”, thenstays unchanged so the item is not shipped), the LTL-FO property (†) is violated because ShipItem can still be called withoutfirst calling the Restock service. The verifier would detect this error and produce a counter-example illustrating the violation. We say that a run ρ = {(I_i,σ_i)}_i ≥ 0 satisfies∀y̅φ_f, where prop(φ) = P ∪Σ, if φ is satisfied, for all valuations of y̅ in DOM_id∪ DOM_val∪{}, by the sequence of truth assignments to P ∪Σ induced by f on the sequence {(I_i,σ_i)}_i ≥ 0.More precisely, for p ∈ P, the truth value induced for p in (I_i,σ_i) is the truth value of the condition f(p) in I_i; a proposition σ∈Σ holds in (I_i,σ_i) if σ_i = σ. A TAS Γ satisfies ∀y̅φ_f(y̅) if for every run ρ of Γ and valuation ν of y̅, ρ satisfies φ_f(ν(y̅)).It is easily seen that for given Γ with artifact variables x̅ and LTL-FO formula∀y̅φ_f(y̅),one can construct Γ' with artifact variables x̅∪y̅ such that Γ∀y̅φ_f(y̅) iff Γ' φ_f.Indeed, Γ' simply adds y̅ to the propagated variables in each service. Therefore, we only consider in the rest of the paper quantifier-free LTL-FO formulas. § THE SPIN-BASED VERIFIERIn this section we describe the implementation of SpinArt. The implementation is based on Spin, the widely used model checker in software verification. A brief review of Spin and Promela, the specification language for Spin, is provided in Appendix <ref>.Building an artifact verifier based on Spin is a challenging task due to limitations of Spin and Promela. In Promela, one can only specify variables with bounded domains (𝐛𝐲𝐭𝐞, 𝐢𝐧𝐭, etc.) and bounded size (i.e. arrays with dynamic allocation are not allowed),but in the TAS model, the domains of the artifact variables and the database are unbounded and the database instance can have arbitrary size, so a direct translation is not possible. In addition, Spin cannot handle Promela programs of large size becausethe generated verifier V would be too large for the C compiler. Spin could also fail due to space explosion in the course of verification. Thus, our implementation requires a set of nontrivial translations and optimizations, discussed next.§.§ Symbolic VerificationThe implementation makes use of the symbolic representation technique developed in <cit.> to establish decidability and complexity results for HAS. With the symbolic representation, the verification of TAS's is reduced to finite-state model checkingthat Spin can handle.Intuitively, given a TAS specification Γ and an LTL-FO property φ, we use isomorphism types to describe symbolicallythe structure of the portion of the database reachable from the current tuple of artifact variables by navigating the foreign keys. An isomorphism type fully captures the information needed to evaluate any condition in Γ and φ. In addition, we can show, similarly to <cit.>, that to check whether Γφ, it is sufficient to check that all symbolic runs of isomorphism types satisfy φ, or equivalently, that no symbolic run satisfies φ. We define symbolic runs next.We start by defining expressions, which denote variables, constants and navigation via foreign keysstarting from id variables. An expression is either: =0pt=0pt* a constant c in 𝚌𝚘𝚗𝚜𝚝(Γ,φ), the set of all constants that appear in Γ or φ, or* a sequence ξ_1.ξ_2.…ξ_m, where ξ_1 = x for some id variable x,ξ_2 is an attribute of R ∈ where R.ID = (x),and for each i, 2 ≤ i < m, ξ_i is a foreign key and ξ_i+1 is an attribute in the relation referenced by ξ_i. For a set of variables y̅, we denote by (y̅) the set of expressions { y.w | y ∈y̅, |w| ≥ 0}∪𝚌𝚘𝚗𝚜𝚝.Such (y̅) for y̅⊆x̅ is called a navigation set. Note that the length of expressions is bounded because of acyclicity of the foreign keys, so (y̅) is finite. We can now define isomorphism types.Let Γ be a TAS with variables x̅, and φ an LTL-FO property of Γ. An isomorphism type τ for Γ, φ, and variables y̅⊆x̅ consists of a navigation set (y̅) together with an equivalence relation∼_τ over (y̅) such that: =0pt=0pt* c ≁_τ c' for constants c ≠ c' in 𝚌𝚘𝚗𝚜𝚝(Γ,φ), and* if u ∼_τ v and u.f, v.f ∈(y̅) then u.f ∼_τ v.f.We call an equivalence relation ∼_τ as above an equality type for τ. The relation ∼_τ is extended to tuples componentwise. Intuitively, the second condition guarantees satisfaction of the key and foreign key dependencies.Figure <ref> shows an isomorphism types τ of variables {x, y, z}, where R(ID, A) is the only database relation, {x, y, z} are 3 variables of typeR.ID and there is only one non-ID constant c_0.Each pair of expressions (e, e') are connected with an solid line (=-edge) if e ∼_τ e' otherwise a dashed line (≠-edge).The ≠-edges between {x, y, z} and {x.A, y.A, z.A, c_0} are omitted in the figure for clarity. Note that since (x, y) is connected with an =-edge, (x.A, y.A) must also be connected with =-edge as enforced by the key dependency. Note that when y̅ = x̅, τ provides enough information to evaluate conditions over x̅. Satisfaction of a condition φ by an isomorphism type τ, denoted τφ, is defined as follows: =0pt=0pt* x = y holds in τ iff x ∼_τ y, * R(x, y_1, …, y_m) holds in τ for relation R(ID, A_1, …, A_m) iff(y_1, …, y_m) ∼_τ (x.A_1, …, x.A_m), and* Boolean combinations of conditions are standard. Let τ be an isomorphism type with navigation set (y̅) and equality type ∼_τ. The projection of τ onto a subset of variables z̅ of y̅, denoted as τ | z̅,is (∼_τ | z̅, (z̅)) where ∼_τ|z̅ is the projection of ∼_τ onto (z̅).We define the symbolic transition relation among isomorphism types as follows: for a service σ =(π, ψ, y̅) in Σ, τστ' iffτπ, τ' ψ and τ | y̅ = τ' | y̅. A symbolic run of Γ = ⟨, Σ, Π⟩is a sequence ρ̃ = {(τ_i, σ_i)}_i ≥ 0such that for each i ≥ 0, τ_i is an isomorphism type, σ_i ∈Σ, σ_0 = init,τ_0 Π and τ_i σ_i+1 τ_i+1. Figure <ref> shows an example of applying a symbolic transition onan isomorphism type. The previous isomorphism type τ (top-left) satisfies the pre-condition, the next isomorphism type τ' (bottom) satisfies the post-condition, and they are consistent in their projection to the propagated variables {x, z} (top-right). Satisfaction of a quantifier-free LTL-FO property on a symbolic run is defined in the standard way.One can show the following, similarly to <cit.>.Given a TAS Γ and LTL-FO property φ of Γ, Γφ ifffor every symbolic run ρ̃ of Γ, ρ̃φ.§.§ Implementation of SpinArtUsing Theorem <ref>, one can implement a verifier that constructs a Promela program to simulate the non-deterministic execution of symbolic transitions. The programspecifies (x̅) as its variables. Each condition ψ in Γ and φ is translated into a Promela condition f(ψ) as follows. =0pt=0pt* if ψ = (x = y), then f(ψ) = ψ;* if ψ= R(x, y_1, …, y_m) for relation R(ID, A_1, …, A_m), then f(ψ) = ⋀_i = 1^m (x.A_i = y_i);* Boolean connectives are handled in the standard way.removed one example Thensimulates the following process of executing symbolic transitions. First,initializes the constant expressions with distinct values andother expressions with non-deterministically chosen values that satisfy f(Π).Then for each service σ = (π, ψ, y̅), we construct a non-deterministic option with guard f(π) that executes the following: =0pt=0pt(i) For each expression e ∈(x̅) - (y̅), assign to e a non-deterministically chosen value from {0, …, |(x̅)| - 1}.(ii) Proceeds if f(ψ) is 𝚃𝚛𝚞𝚎 and for each pair of expressions e and e', e = e' implies that for every attribute A where {e.A, e'.A}⊆(x̅), e.A = e'.A. Otherwise the run is blocked and invalidated. First, each TAS condition is translated into a condition in Promela. For example, the pre-condition in Example <ref> is translated into. Then, to construct the Promela program , we first have a do-statement toensure that the options constructed according to Sect. <ref> are repeatedly chosen non-deterministically and executed.For example, the service in Example <ref> is translated into the fragment ofa Promela program shown in Fig. <ref>,where thestatement is a built-in macro forassigning a variable with a value non-deterministically chosen from a range (here N is a constant equal to |(x̅)|).Intuitively, each valid valuation v to (x̅) corresponds toa valid isomorphism type τ of x̅ where e ∼_τ e' iff v(e) = v(e'). The guard ensures that the pre-condition holds. Part (i) ensures that the set of next valuations covers all possible valid successors of isomorphism types. Finally, the conditions in (ii) ensure that the post-condition holds and the keys and FKs dependencies are satisfied in the next isomorphism type.Finally, the LTL-FO formula φ is translated into a LTL formula φ̃ in Promela byreplacing each FO component c with f(c) defined above.The universally quantified variables of φ are translated into extra variables added to the Promela program. Small modifications to the LTL formula are also needed to skipthe internal steps for assigning values and testing conditions in the run such that the Spin verification only considers the snapshots right after complete service applications. We can show the following.Every symbolic run ρ̃ = {(τ_i, σ_i)}_i ≥ 0 satisfies φ iff φ̃. The intuition of the above Lemma is thateach valid valuation v to (x̅) incorresponds to an unique isomorphism type τ. The translated transitions in Promela guarantees that the set of runs ofcaptures the set of all symbolic runs. So to check whether Γ satisfies φ,it is sufficient to translate (Γ, φ) into (, φ̃) andverify whether φ̃.However, this approach is inefficient in practice for the following reasons.In part (ii), the size of the tests to ensure satisfaction of the key and foreign key dependencies is quadratic in the number of expressions,so the compilation ofand the generated verifier is slow or simply fails. In (i), assigning to each e values from {0, …, |(x̅)| - 1} is also infeasiblebecause it leads to state explosion when the actual search is performed by the verifier. As shown by the experiments, this leads to either slow execution or memory overflow.To overcome these two major obstacles, we introduce two key optimizations.§.§ Optimization with Lazy Dependency Tests In the first optimization, we reduce the size of the generated Promela program by eliminating the tests of key and foreign key dependencies in step (ii) of the above approach. Instead, we introduce tests of the dependencies in a lazy manner, only when two expressions are actually tested for equality.Formally, instead of performing the tests in (ii), we translate each condition ψ of (Γ, φ) into f(ψ) then add the following additional tests: for every atom (e = e')in the negation normal form[ With negations pushed down and merged with the = and ≠ atoms, the only remaining Boolean operators areand .] of f(ψ), we replace (e = e') with ( ⋀_w : {e.w, e'.w}⊆(x̅)e.w = e'.w ) where w is a sequence of attributes. The size of the tests in the resulting Promela programisO((|π| + |ψ|) ·max_x ∈x̅ |(x) | ) for each service, while the original size is O(|(x̅)|^2 · a)where a is the maximum arity in the database schema . Typically, the size of a condition is much smaller than the number of expressions andmax_x ∈x̅ |(x) | is also smaller than |(x̅)|.We can see that the lazy dependency significantly reduces the size of the tests.Consider the database schema = {R(ID, A, B), S(ID, C, D)}where A and B are foreign key attributes referencing the ID of S and C, D are non-key attributes. A condition R(x, y, z) is translated into without the optimization and if the lazy dependency tests optimization is applied.The additional terms in the conditions are added so that the tests for keys and FKs in the translation can be removed. Consider the service and the translation shown in Fig. <ref>. With lazy dependency tests, the translated pre-condition becomeswith one additional term . The translated post-condition is unchanged and the tests for keys and FKs are removed (lines 12-14). The overall size of the translation is reduced.Correctness.The modified translation using lazy dependency tests preserves correctness. The intuition is the following. With the lazy tests, in some snapshot with valuation v in the execution of , there could be two expressions e, e' where v(e) = v(e') and for some attribute A, v(e.A) ≠ v(e'.A), but this does not matter because e = e' is never tested during the current lifespan of e and e'(the segment of the symbolic run where e and e' are propagated), and neither are any of the prefixes of e and e'. So within the same lifespan, we are free to replace v(e) and v(e')with different values and the run ofremains valid.Thus, there is no need to enforce the equality e.A = e'.A. §.§ Optimization with Assignment Set MinimizationIn the naive approach, assigning expressions with values chosen from a set of size |(x̅)| guarantees correctness by covering all possible isomorphism types,but it results in a large search space for Spin, which can lead to poor performance or memory overflow.The goal of this optimization is to reduce the size of the search space byminimizing the set of values used in the assignments while preserving the correctness of verification. We denote by A(e) the assignment set of a non-constant expression e, which is the set from which the Promela programchooses non-deterministically values for e. The technique relies on static analysis ofand the translated property φ̃, aiming to reduce the size of the assignment sets as much as possible. The intuition behind the optimization is the following.We notice that searching for an accepting run in the generated Promela programcan be regarded as searching for a sequence of sets of constraints {C_i}_i ≥ 0, where each C_i consists of the (in)equality constraints imposed on the current snapshot by the history of the run. More precisely, the statements executed incan be divided into two classes: (1) testing a condition π and (2) assigning new values to some expressions. At snapshot i, executing an (1)-statement can be viewed as adding π to C_i while C_i should remain consistent (no contradiction implied by the =-or-≠ constraints in C_i), and a (2)-statement assigning a value to e can be viewed as projecting away from C_i constraints that involve e. When we construct the assignment set A(·), it is sufficient for correctness that the valuations generated with A(·) can witness the set of all reachable C_i's, which can be a small subset of all the possible isomorphism types. Thus, the resulting A(·) can be much smaller.Computing all reachable C_i's can be as hard as the verification problem itself.So instead, we over-approximate them with the constraint graph G of(, φ̃) obtained by collecting all (in)equalities from (, φ̃), so that all C_i's are subgraphs of G.Formally, the constraint graph G is an undirected labeled graph with (x̅) as the set of nodes, where an edge (e, e', ∘) is in G for ∘∈{=, ≠} if (e ∘ e') is an atom in any condition ofand φ̃ with all conditions converted in negation normal form. A subgraph G' of G is consistent if its edges do not lead to a contradiction (i.e., two nodes connected in G' by a sequence of =-edges are not also connected by an ≠-edge). Observe that G itself is generally not consistent, since it may contain mutually exclusive constraints that never arise in the same configuration. On the other hand, each C_i as above corresponds to a consistent subgraph of G.Intuitively, the approach to minimizing the assignment sets proceeds as follows. First, consider the connected components of G with respect to its equality edges.Clearly, distinct connected components can be consistently assigned disjoint sets of values.Next, within each connected component, all expressions can be provided with the same assignment set, which we wish to minimize subject to the requirement that it must provide sufficiently many values to satisfy each of its consistent subgraphs.More precisely, we can show the following.Let ' be the Promela program obtained from (, φ̃) by replacing the assignment sets with any A(·) that satisfies:=0pt=0pt* for every (e, e', =) ∈ G, A(e) = A(e'), and* for every consistent subgraph G' of G, there exists a valuation v such that for every e ∈(x̅), v(e) ∈ A(e) and for ∘∈{=, ≠}, v(e) ∘ v(e') if (e, e', ∘) ∈ G'. Then φ̃ iff ' φ̃. Note that constants are not taken into account in the above lemma but can be included in a straightforward way. Condition 2 implies that whenever a new valuation v' is generated from a previous valuation v, regardless of the previous and next constraint sets C and C', there exists a v' that is consistent with v, C and C'.We next consider minimizing the assignment sets within each connected component. It turns out that computing the minimal A(·) that satisfies the above conditions is closely related to computingthe chromatic number of a graph <cit.>. Recall that the chromatic number χ(G) of an undirected graph G is the smallest number of colors needed to color G such that no two adjacent nodes share the same color.If the subgraph G' in condition 2 is fixed,then the minimal |A(·)| is precisely the chromatic number of G' restricted to only ≠-edges andwith connected components of the =-edges merged into single nodes. We illustrate it with an example.Consider the constraint graph G in the left of Fig. <ref>.The solid lines represent =-edges and the dashed lines represent ≠-edges. The entire graph consists of a single connected component of =-edges.To find the minimal A(·), we need to find the largest chromatic number over all consistent subgraphs of G. Consider two consistent subgraphs G_1 (middle) and G_2 (right). The chromatic number of G_1 is 3 because (e_2, e_3) (and (e_4, e_5)) must share the same color,so G_1 is in fact a triangle. The chromatic number of G_2 is 2 as it no long requires e_2 and e_5 to have different colors. In fact, G_1 is the subgraph with the largest chromatic number, so setting A(e_i) = {0, 1, 2} for every i minimizes the assignment sets.As computing the chromatic number is np-hard, it is not difficult to show that computing A(·) with minimal size is also np-hard. (We conjecture that it is Π_2^P-hard.)So computing the minimal A(·) can be inefficient.In the implementation, we use a simple algorithm that approximates the maximal chromatic number with the straightforward boundχ(G) (χ(G) - 1) ≤ 2m where m is the number of ≠-edges within the connected component.The algorithm ensures satisfaction of the two conditions and produces reasonably small assignment sets in practice because the constraint graph is likely to be very sparse and contains few ≠-edges.This is confirmed by our experiments.§ EXPERIMENTAL RESULTSIn this section we describe the experiments evaluating the performance of SpinArt.Benchmark.The benchmark used for the experiments consists of a collection of32 artifact systems modeling realistic business processes from different application domains. Because of the difficulty in obtaining fully specified real-world data-driven business processes, we constructed the benchmark starting from business processes specified in the widely used BPMN model, that are provided by the official BPMN website <cit.>.We rewrote the BPMN specifications into artifact systems by manually adding the database schema, variables and service pre-and-post conditions.Table <ref> provides some characteristics of the benchmark. LTL-FO Properties.On each workflow in the benchmark, we run SpinArt on a collection of 12 LTL-FO properties constructed using templates of real propositional LTL properties, yielding a total of 384 runs. The LTL properties are all the 11 examples of safety, liveness and fairness properties collected froma standard reference paper <cit.> and an additional property 𝙵𝚊𝚕𝚜𝚎 used as a baseline when comparing the performance of SpinArt on different classes of LTL-FO properties. We list all the templates of LTL properties in Table <ref>. We choose 𝙵𝚊𝚕𝚜𝚎 as a baseline because it is the simplest property verifiable by Spin. By comparing the running time for a property with the running time for 𝙵𝚊𝚕𝚜𝚎 on the same specification, we obtain the overhead for verifying the property.For each workflow, we generate an LTL-FO property corresponding to each template by replacing the propositions with FO conditions chosen from the pre-and-post conditions of all the services and their sub-formulas. Note that by doing so, the generated LTL-FO properties on the real workflows are combinations of real propositional LTL properties and real FO conditions, and so are close to real-world LTL-FO properties.Setup.We implemented SpinArt inwith Spin version 6.4.6.All experiments were performed on a Linux server with a quad-core Intel i7-2600 CPU and 16G memory. To allow larger search space, Spin was run with the state compression optimization turned on. For faster execution, the Spin-generated verifier was compiled with 𝚐𝚌𝚌 and the -O2 optimization. The time and memory limit of each run was set to 10 minutes and 8G respectively.Performance.In addition to running the full verifier (SpinArt-Full), we also ran the verifier with the lazy dependency tests optimization (LDT) turned off (SpinArt-NoLDT) and with assignment set minimization (ASM) turned off (SpinArt-NoASM). For all the verifiers, we compare their number of failed runs (timeout or memory overflow), the average compilation time[All averages (running times and #States) are taken over the successful runs.] for generating the executable verifier (Compile-Time), the average execution time of the generated verifier (Verify-Time), the average total running time (Verify-Time + Compile-Time), and the average number of reached states as reported by Spin. The results are shown in Table <ref>. We can see that the performance of SpinArt is promising. Its average total running time is within 3 seconds and there are only 3/384 failed runs (<1%) due to memory overflow. This is a strong indication that the approach is sufficiently practical for real-world workloads. The full verifier is also significantly improved compared to SpinArt-NoLDT and SpinArt-NoASM. Without ASM, the the verifier failed on 12.5% (48/384) of all runs and the average running time is >7x times faster when the optimization is turned on. Without LDT, most of the runs are still successful, but the average total running time is >4 times faster with the optimization turned on. Both optimizations significantly reduce the size of the state space (>95% in total), resulting in much shorter verification time. We next discuss the effect of each optimization in more detail.Effect of Lazy Dependency Tests.From Table <ref>, we observe that for the successful runs, compilation time accounts for a large fraction of the total running time, so minimizing the size of the Promela program is critical to improve the overall performance of a Spin-based verifier. Figure <ref> shows the changes in the compilation time as the size of the input specification (#Variables + #Services) increases, for runs with or without the LDT optimization. Each point in the figure corresponds to one specification and the compilation time is measured by the average compilation time of all runs of the specification. The figure shows that with LDT, the compilation time grows not only slower as the input size increases, but in some cases it can compile >10 times faster than compilation without LDT. Overall, LDT leads to an average speedup of 3.2x in compilation. Effect of Assignment Set Minimization.We show the effectiveness of Assignment Set Minimization (ASM) by comparing the approximation algorithm for ASM with a naïve approach (NoASM) where the size of the assignment set of each expression e is simply set to the number of expressions having the same type as e. Figure <ref> shows the growth of the average size of the assignment sets as the size of the input specification increases. For ASM, the average size stays very low (2.05 in average) as the input size grows. This shows that our algorithm is near-optimal in practice. Compared to the naive approach where the average size increases linearly with the input size, our approach produces much smaller assignment sets. In some cases, the assignment set generated by the algorithm is >30 times smaller than the ones generated by the naive approach. Effect of the Structure of LTL-FO Properties.Next, we measure the performance on different classes of LTL-FO properties. Table <ref> lists all the LTL templates used in generating the LTL-FO properties and their intuitive meaning, as in <cit.>. For each template, we measure the average running time over all runs with LTL-FO properties generated using the template. In addition, we measure the overhead of verifying a LTL-FO property by comparing with its running time for the property 𝙵𝚊𝚕𝚜𝚎, the simplest non-trivial property for SpinArt. The overhead of a class of LTL-FO properties is obtained by the average overhead of all properties of the same class. The result in Table <ref> shows that the average running time stays within 2x of the average running time for 𝙵𝚊𝚕𝚜𝚎 and the maximum average overhead is about 70%. The overhead increases as the LTL property becomes more complex, but is within a reasonable range. Note that this is much better than the theoretical upper bound, which is exponential in the size of the LTL formula.I think the following can be moved to the appendix.Results on Synthetic Workflows.Finally, we stress-test the performance of SpinArt by running it on a set of 120 randomly generated TAS specifications. All components of each specification were generated fully at random for a specified size.Each specification has 5 relations in the DB schema, 75 variables and 75 services with randomly generated pre-and-post conditions. The ones with empty search space due to unsatisfiable conditions were removed from this benchmark. On each workflow, we ran SpinArt to verify 12 LTL-FO properties generated from the templates in Table <ref>, resulting in 1440 runs in total. Among these runs, SpinArt succeeded in 1000/1440 (∼70%) runs with an average running time of 83.983s. The remaining runs failed due to timeout or memory overflow.As preformance remains acceptable on the much larger synthetic workflows,the results suggest that SpinArt is scalable to complex workflows.Note that the two optimizations are essential to the above results,since almost all runs failed due to compiler crash if either optimization is turned off.§ ADDITIONAL RELATED WORK The artifact verification problem has been studied mainly from a theoretical perspective. As mentioned in Sect. <ref>, fully automatic artifact verification is a challenging problem due to the presence of unbounded data. To deal with the resulting infinite-state system,a symbolic approach was developed in <cit.> allowing a reduction to finite-state model checking and yielding a pspace verification algorithm for the simplest variant of the model (no database dependencies and uninterpreted data domain). <cit.> extended this approach to allow for database dependencies andnumeric data testable by arithmetic constraints.The symbolic approach developed in <cit.> and revisited in HAS <cit.>provides the theoretical foundation of our Spin-based implementation.Another line of work considers the verification problem for runs starting from a fixed initial database. During the run, the database may evolve via updates, insertions and deletions.Since inputs may contain fresh values from an infinite domain,this verification variant remains infinite-state. The property languages are fragments of first-order-extended μ-calculus <cit.>. Decidability results are based on sufficient syntactic restrictions <cit.>.<cit.> derives decidability of the verification variant by also disallowing unbounded accumulation of input values,but this condition is postulated as a semantic property(shown undecidable in <cit.>).<cit.> takes a different approach, in which decidability is obtained for recency-bounded artifacts,in which only recently introduced values are retained in the current data.On the practical side of artifact verification,<cit.> specifies business processes ina Petri-net-based model extended with data and process components,in the spirit of the theoretical work of<cit.>,which extends Petri nets with data-carrying tokens. The verifier of <cit.> differs fundamentally from ours in thatproperties are checked only for a given initial database, whereasour verifier checks properties regardless of the initial database. <cit.> implementeda verifier for artifact systems specified directly in the GSM model. While the above models are expressive, the verifiers require restrictions strongly limiting modeling power <cit.>,or predicate abstraction resulting in loss of soundness and/or completeness <cit.>.Lastly, the properties verified in <cit.> focus ontemporal-epistemic properties in a multi-agent finite-state system.Thus, the verifiers in these works have a different focus and are incomparable to ours.Practical verification has also been studied in business process management (see <cit.> for a survey). The considered models are mostly process-driven(BPMN, Workflow-Net, UML etc.), with the business-relevant data abstracted away. The implementation of a verifier for data-driven web applications was studied in <cit.> and <cit.>. The model is similar in flavor to the artifact system model but incomparable due to the different application domains. An attempt to build a verifier based on Spin was made in <cit.> but failed due to search space explosion, confirming that the optimizations used in our implementation of SpinArt are essential. § CONCLUSION, RELATED WORK AND DISCUSSIONWe reported on our implementation of SpinArt, a verifier for data-driven workflows usingthe widely used off-the-shelf model checker Spin. With a translation based on the symbolic representation developed in <cit.> enhanced with nontrivial optimizations, SpinArt achieves good performance on a realistic business process benchmark. We believe this is a first successful attempt to bridge the gap between theory and practicein verification of data-driven workflows, with full support for unbounded data and relying on an off-the-shelf model checker. The following paragraph can be shortened. Discussion. The focus of our work is on sound and completeartifact verifiers, in contrast to incomplete verifiers (e.g. based on theorem provers). Within this scope,SpinArt establishes a practical trade-off point on the spectrum ranging from using off-the-shelf general software verifiers to developing dedicated verifiers from scratch.On the one hand, off-the-shelf tools share a number of limitations which are inherited by verifiers based on them (including ours). For instance, general-purpose model checkers have limited support for unbounded data.While our work mitigates this limitation by supportingthe unbounded read-only database with symbolic representation, our model does not support other ingredients of the HAS (and GSM) model, such asdynamically updatable artifact relations, because they require an enhanced symbolic representation countingthe number of tuples of different isomorphism types, which exceeds the capabilities of Promela/Spin. On the other hand, from-scratch implementation is costly as it duplicates functionality already present in mature tools such as Spin. More importantly, the initial implementation cost is typically outweighed by maintenance cost over the verifier's lifetime. In contrast, verifiers based on off-the-shelf model checkers feature lower development and maintenance cost.splncs§ REVIEW OF SPIN AND PROMELAlanguage=Promela The implementation of our artifact verifier relies on Spin, a widely used model checker in software verification. Spin supports the verification of LTL properties of models specified in Promela,a C-like modeling language for parallel systems.At a high level, a single-process Promela program can be viewed as a non-deterministic C program, where one can specify variables of fixed bit-length (e.g. 𝐛𝐲𝐭𝐞, 𝐬𝐡𝐨𝐫𝐭, 𝐢𝐧𝐭) and statements that manipulate the variables (e.g. assignments, goto, etc.). Non-determinism is specified using the if- and do-statements illustrated in Fig. <ref>.When the if-statement is executed, one of its options with no guard or with its guard evaluating to 𝚃𝚛𝚞𝚎 is chosen non-deterministically and executed. Each option is a sequence of one or more statements. If no option can be chosen, then the run blocks the is not considered as a valid run when Spin is executed. The do-statement is similar to the if-statement, with the difference thatthe execution is repeated after an option is completed. Nesting is allowed within the if- or do-statements.Developers can verify LTL properties of a Promela program using Spin. Given a Promela program , a developer can write LTL propertieswhere the propositions are Boolean conditions over the variables of , such as:“”.To check satisfaction of a LTL property φ,Spin first produces the source code of a problem-specific verifier V in C. Then V is compiled with a C-compiler (e.g. 𝚐𝚌𝚌) and executed to produce the result. | http://arxiv.org/abs/1705.09427v3 | {
"authors": [
"Yuliang Li",
"Alin Deutsch",
"Victor Vianu"
],
"categories": [
"cs.DB",
"cs.FL"
],
"primary_category": "cs.DB",
"published": "20170526041317",
"title": "SpinArt: A Spin-based Verifier for Artifact Systems"
} |
=1 | http://arxiv.org/abs/1705.09103v1 | {
"authors": [
"Dmitry A. Abanin",
"Zlatko Papić"
],
"categories": [
"cond-mat.dis-nn",
"cond-mat.mes-hall",
"cond-mat.stat-mech"
],
"primary_category": "cond-mat.dis-nn",
"published": "20170525092751",
"title": "Recent progress in many-body localization"
} |
FERMILAB-PUB-17-173-A 0.2in ORCID: http://orcid.org/0000-0001-8837-4127 [email protected]@jhu.eduORCID: http://orcid.org/0000-0002-3805-6478 ORCID: http://orcid.org/0000-0001-9888-0971 [email protected] [a]Fermi National Accelerator Laboratory, Center for Particle Astrophysics, Batavia, IL 60510 [b]University of Chicago, Department of Astronomy and Astrophysics, Chicago, IL 60637 [c]University of Chicago, Kavli Institute for Cosmological Physics, Chicago, IL 60637 [d]Department of Physics and Astronomy, The Johns Hopkins University, Baltimore, Maryland, 21218 [e]Ohio State University, Center for Cosmology and AstroParticle Physics (CCAPP), Columbus, OH43210 Measurements of the nearby pulsars Geminga and B0656+14 by the HAWC and Milagro telescopes have revealed the presence of bright TeV-emitting halos surrounding these objects. If young and middle-aged pulsars near the Galactic Center transfer a similar fraction of their energy into TeV photons, then these sources could plausibly dominate the emission that is observed by HESS and other ground-based telescopes from the innermost ∼10^2 parsecs of the Milky Way. In particular, both the spectral shape and the angular extent of this emission is consistent with TeV halos produced by a population of pulsars, although the reported correlation of this emission with the distribution of molecular gas suggests that diffuse hadronic processes also must contribute. The overall flux of this emission requires a birth rate of ∼100-1000 neutron stars per Myr near the Galactic Center, in good agreement with recent estimates.TeV Gamma Rays From Galactic Center Pulsars and Tim Linden^e December 30, 2023 ===========================================§ INTRODUCTION The HESS, VERITAS and MAGIC Collaborations have each reported the detection of very high-energy (VHE) gamma-ray emission from the direction of the Galactic Center, extending to energies of ∼30-50 TeV <cit.>. When this emission was initially identified, it was suggested that it may originate from the Milky Way's central supermassive black hole, Sgr A^* <cit.>. More recent measurements, however, have revealed that this emission includes a component that is extended to ∼10^2 parsecs in radius <cit.>. In light of this, it has been proposed that cosmic rays originating from Sgr A^* may be responsible for the observed VHE emission. Because multi-TeV electrons would lose energy through inverse-Compton scattering and synchrotron processes too rapidly to account for the observed extension, this emission has instead been interpreted as evidence that Sgr A^* accelerates cosmic-ray protons up to ∼PeV energies, which then propagate outward and generate the observed VHE gamma-ray emission through pion production <cit.>.In this paper, we revisit the origin of the VHE gamma rays observed from the Inner Galaxy (not to be confused with the GeV excess observed by the Fermi Telescope <cit.>) and offer an alternative interpretation in terms of the inverse-Compton scattering of VHE electrons/positrons generated by a population of centrally located pulsars (see also Refs. <cit.>). In order to calculate the intensity, spectrum and spatial morphology of TeV gamma-ray emission from pulsars in the Galactic Center, we consider the nearby and well-characterized pulsars Geminga and B0656+14 (i.e. the Monogem pulsar) and treat them as representative systems. To this end, we utilize observations of these pulsars as reported by the the HAWC <cit.> and Milagro <cit.> Collaborations. The angular extension observed by these telescopes strongly favor an inverse-Compton origin of this emission <cit.>, and the observed flux indicates that a significant fraction of the spin-down power from these pulsars is transferred into the production of VHE leptons (between 7.2% and 29% in the case of Geminga, across the range of models considered in Ref <cit.>). This conclusion is further supported by the detections of TeV halos around young pulsars by HESS <cit.>.The large number of massive stars and low-mass X-ray binaries present in the Galactic Center indicates that that this region is likely to host a large population of neutron stars <cit.>. The authors of Ref. <cit.>, for example, estimate that ∼10^2-10^3 radio pulsars should be located within the innermost 0.02 parsecs around Sgr A^*. Intriguingly, none of these pulsars have been detected. Until recently, it had been argued that the absence of observed radio pulsars near the Galactic Center was likely due to a large free electron density in the central parsec, which creates significant dispersion in the radio pulse <cit.>. However, the 2013 observation of pulsations from the magnetar SGR J1745-29 <cit.>, located only ∼0.1 parsecs from Sgr A^*, has forced a revision of this view, suggesting that there may be fewer radio pulsars in the Galactic Center than previously expected. The authors of Ref. <cit.>, for example, conclude that this information constitutes a “missing pulsar problem”, and suggests that a possible resolution could be the efficient formation of magnetars (rather than ordinary pulsars). In contrast, the authors of Ref. <cit.> argue that it is premature to conclude that the number of Galactic Center pulsars is small, and derive a conservative upper limit of ∼200 potentially observable pulsars located within in innermost parsec. More generally speaking, it remains widely anticipated that there are many pulsars located near the Galactic Center <cit.>. In this paper, we operate under the assumption that the VHE emission from Geminga and B0656+14 is typical of that from pulsars, including those located in the Inner Galaxy. We find that the observations of the Galactic Center region by HESS and other ground-based telescopes can easily be accommodated by a population of young and middle-aged pulsars. In particular, the spectral shape and angular extent of the observed VHE emission is consistent with a population of pulsars that are born in the innermost parsec and which subsequently migrate outward as a result of pulsar natal kicks. The overall normalization of the observed emission requires a recent average birth rate of ∼100-1000 neutron stars per Myr near the Galactic Center. § THE GAMMA RAY SPECTRUM FROM ELECTRONS AROUND PULSARS High-energy electrons and positions undergo energy losses through a combination of inverse-Compton and synchrotron processes, at a rate given by <cit.>: -dE_e/dt(r)= ∑_i 4/3σ_T ρ_i(r) S_i(E_e) (E_e/m_e)^2 + 4/3σ_T ρ_ mag(r) (E_e/m_e)^2,where σ_T is the Thomson cross section. The quantity S_i(E_e) quantifies the suppression of inverse-Compton scattering in the Klein-Nishina regime (E_em^2_e/2T), and is given by: S_i (E_e) ≈45 m^2_e/64 π^2 T^2_i/(45 m^2_e/64 π^2 T^2_i)+(E^2_e/m^2_e). The sum in Eq. <ref> is carried out over the various components of the radiation backgrounds, consisting of the cosmic microwave background (CMB), infrared emission (IR), starlight (star), and ultraviolet emission (UV). In our previous study focusing on the nearby Geminga and B0656+14 pulsars <cit.>, we adopted the following parameters: ρ_ CMB=0.260 eV/cm^3, ρ_ IR=0.60 eV/cm^3, ρ_ star=0.60 eV/cm^3, ρ_ UV=0.10 eV/cm^3, ρ_ mag=0.224 eV/cm^3 (corresponding to B=3 μG), and T_ CMB =2.7 K, T_ IR =20 K, T_ star =5000 K and T_ UV =20,000 K. In the region surrounding the Galactic Center, however, we expect the energy densities of the radiation and magnetic fields to be significantly higher than those found in the local environment.In Fig. <ref>, we plot the gamma-ray spectrum that results from the inverse-Compton scattering of VHE electrons and positrons from pulsars in the region surrounding the Galactic Center. In each frame, we have parameterized the injected electron spectrum using the form dN_e/dE_e ∝ E_e^-α exp(-E_e/E_c), and show results for parameter values chosen to match the observed spectrum from Geminga (α=1.9, E_c=49 TeV) <cit.>, and for values chosen to provide the best-fit to the spectra shown (α=2.2, E_c=100 TeV). In the left frame, we compare the predicted spectrum to that of the central point source as reported by the HESS Collaboration, while in the right frame we show the spectrum reported by HESS in a 0.2^∘ to 0.5^∘ (partial) annulus around the Galactic Center <cit.>. For the central point source (extended annulus), we calculate the spectrum of inverse Compton emission assuming energy densities of starlight, IR and UV radiation and magnetic fields that are 1000 (10) times higher than in the local interstellar medium. While these energy densities represent a very approximate estimate, we consider it to be reasonable for the inner volume of the Milky Way <cit.>. The main impact of this choice is to reduce the role of scattering with the CMB, and the precise values of these quantities does not strongly impact our results or conclusions. For each curve, the overall normalization was independently chosen to to provide the best-fit to the gamma-ray spectrum reported by HESS.§ MODELING THE GALACTIC CENTER PULSAR POPULATIONIn the previous section, we demonstrated that the spectrum of the gamma-ray emission observed from the Inner Galaxy by HESS is consistent with the TeV halo emission observed from Geminga and B0656+14. We have not yet, however, discussed the normalization of the flux of VHE gamma-rays from the Galactic Center pulsar population, which depends on the evolution of these sources. The spin-down power of a given pulsar (the rate at which it loses rotational kinetic energy through magnetic dipole braking) is given by <cit.>: Ė =-8π^4 B^2 R^6/3 c^3 P(t)^4≈1.0 × 10^35erg/ s×(B/1.6 × 10^12G)^2(R/15 km)^6(0.23 s/P(t))^4, where B is the strength of the magnetic field at the surface of the neutron star, R is the radius of the neutron star, and the rotational period evolves as follows: P(t) = P__0 (1+t/τ)^1/2, where P__0 is the initial period, and τ is the spindown timescale: τ = 3c^3 I P__0^2/4π^2B^2 R^6≈9.1 × 10^3 years × (1.6× 10^12G/B)^2 (M/1.4 M_⊙)(15 km/R)^4(P__0/0.040 sec)^2.To model the population of pulsars born in and around the Galactic Center, we adopt the distribution of initial periods and magnetic fields described in Ref. <cit.>. More specifically, for the initial period we adopt a normal distribution with ⟨ P_0 ⟩ =0.15 s andσ=0.3 s, while for the magnetic field we adopt a log-normal distribution with ⟨log_10 B (G) ⟩ =12.65 andσ=0.55.Focusing on the population of pulsars that originate near the Galactic Center, we assume that each pulsar in our model is formed at a location within a few parsecs (well within the point spread function of HESS) around Sgr A^*. Once formed, however, each pulsar obtains a natal kick velocity which continuously carries it away from the Galactic Center, broadening the angular profile of the resulting gamma-ray emission. For simplicity, we adopt a uniform kick velocity of 400 km/s for each pulsar in our model, and an initial position of 1 parsec from Sgr A^*.Of course not all pulsars originate near the Galactic Center, and much of the diffuse VHE emission observed from elsewhere in the sky could also originate from pulsars. Although we do not explore this possibility here, we consider it plausible that the diffuse TeV-scale emission observed from the Galactic Plane <cit.> could be generated by a population of such objects <cit.>. Pulsars that do not originate in the innermost parsecs of the Galaxy could produce VHE emission that is significantly more spatially extended than that presented here. § THE NUMBER OF PULSARS REQUIRED TO GENERATE THE TEV EMISSION OBSERVED FROM THE GALACTIC CENTER In this section, we will use the spectrum reported by HESS, in conjunction with the measurements of Geminga by HAWC and Milagro, to estimate the number of pulsars located in the region surrounding the Galactic Center (generating the emission associated with both the central source and surrounding diffuse emission, as shown in Fig. <ref>) . In carrying out this estimate, we implicitly assume that the pulsars near the Galactic Center deposit the same fraction of their spin-down power into electron-positron pairs as Geminga. Drawing from the distribution of initial periods and magnetic field strengths described in Sec. <ref>, we find that the average total spin-down power of the modelled pulsar population is Ė≈ 6.34 × 10^37erg/s × (R/1000), where R is the birth rate of pulsars per Myr. Comparing this to the value for Geminga (Ė_ Geminga≈ 3.2× 10^34 erg/s) and correcting for the relative distances (we adopt d_ Geminga = 250^+230_-80 pc <cit.> and d_ GC =8250 pc), we estimate that the total gamma-ray flux from the Galactic Center pulsar population should be equal to (0.84-6.71) × (R/1000) times that of Geminga. From HAWC's measurement of the VHE flux from Geminga, and after correcting for the higher energy densities in radiation and magnetic fields near the Galactic Center, this translates to a flux of (3.25^+8.72_-1.75)× 10^-12 TeV cm^-2 s^-1 × (R/1000) at an energy of 7 TeV.We compare this predicted flux to that reported by the HESS Collaboration. At 7 TeV, the flux observed by HESS from the combination of the central source and the surrounding annulus (as shown in the left and right frames of Fig. <ref>) is 1.59^+0.36_-0.34× 10^-12 TeV cm^-2 s^-1. For an average birth rate of R ≃ 490^+580_-370 new pulsars per Myr, this gamma-ray flux can be fully accounted for by the VHE electrons and positrons injected from pulsars.One should keep in mind that most of these pulsars will produce radio beams that are not aligned toward the Solar System, and will thus be impossible to detect with radio telescopes. For an estimated beaming fraction of 25% <cit.>, we predict the Galactic Center to contain between ∼25-190 pulsars younger than 1 Myr with radio beams oriented in our direction. Assuming that a typical pulsar remains radio bright for ∼10 Myr, this would imply that ∼250-1900 potentially observable radio pulsars should be present within 70 parsecs of the Galactic Center.[The uncertainty on the number of potentially observable pulsars near the Galactic Center is dominated in our calculation by the distance to Geminga. Future refinements of this quantity will enable us to more reliably predict the number of pulsars present.] For comparison, Ref. <cit.> estimate that as many as ∼200 such sources could be present within the Galaxy's innermost parsec.§ THE ANGULAR DISTRIBUTION OF VERY HIGH ENERGY EMISSION FROM PULSARS NEAR THE GALACTIC CENTER In Fig. <ref>, we plot the morphology of the VHE gamma-ray emission from pulsars originating near the Galactic Center for six randomly chosen realizations, each with a neutron star birth rate of 200 per Myr. In addition to calculating the trajectory of each simulated pulsar, we have assumed that the emission from each pulsar has a physical extent equal to that observed for Geminga (a Gaussian with a width of 2^∘× 250pc/d∼ 0.06^∘) and have convolved the predicted emission by the HESS point spread function (which we approximate by a 0.06^∘ Gaussian).[For clarification, it is a coincidence that the physical extent of a Geminga-like TeV halo and the point spread function of HESS are each described by a Gaussian of width 0.06^∘.] We note that the physical extent of the TeV halos surrounding Galactic Center pulsars might differ substantially from those surrounding Geminga and B0656+14, owing to differences in the density of the interstellar medium, and in diffusion and energy loss processes. Broadly speaking, the main features of this simulated emission are consistent with those reported by the HESS collaboration, although a detailed comparison is made difficult by the fact that the predicted emission varies considerably depending on the brightest few pulsars that happen to be present at this particular point in time. That being said, from among the six randomly chosen realizations shown in Fig. <ref>, the ratio of the flux from within the HESS point spread function (0.06^∘) around the Galactic Center to that from within the 0.2-0.5^∘ (partial) annulus varies from between 0.3 and 1.1, consistent with the ratio of fluxes reported by HESS (as inferred by comparing the left and right frames of Fig. <ref>).We note that the ratio of the gamma-ray fluxes attributed to the central point source and to the surrounding annulus depends (to a similar degree) on four parameters: the velocity distribution of neutron star natal kicks, the evolution of the pulsar spin-down luminosity, the physical size of TeV halos, and the point-spread function of HESS. In each case, we have selected well-motivated values which were not fit to the HESS data. However, we stress that alterations in these parameters can affect the ratio of the central and diffuse fluxes.§ MILLISECOND PULSARS Thus far, we have focused on young pulsars, in contrast to recycled pulsars with millisecond-scale periods. The main reason for this is that, to date, no TeV-halos have been observed around any millisecond pulsars (MSPs), and thus we do not know what fraction of the spin-down power of these sources goes into the production of VHE electrons and positrons. It is widely believed, however, that MSPs are indeed likely to generate such emission <cit.>, as the modelling of their light curves favor the abundant production of multi-TeV electron-positron pairs <cit.>.Although no MSPs have been detected near the Galactic Center, the number of such objects present in the Inner Galaxy is not well constrained and could plausibly be large. The subject of MSPs in the Inner Galaxy has received a great deal of attention in recent years, as it has been argued <cit.> that a large population of such objects could plausibly be responsible for the Galactic Center gamma-ray excess observed by the Fermi Telescope <cit.>.The total luminosity of the Galactic Center gamma-ray excess is L_γ∼ 2× 10^36 erg/s above 100 MeV, integrated within 0.5^∘ of the Galactic Center <cit.>. Given that the gamma-ray efficiency (averaged over 4π steradians) measured for the vast majority of MSPs observed by Fermi is between a few percent and unity <cit.>, this implies that the total spin-down power of this MSP population is required to be at least Ė_ total∼ (2-70) × 10^38 erg/s, which exceeds the total spin-down power of centrally located young pulsars by a factor of ∼10-500. Thus, while little is currently known about the VHE gamma-ray emission from MSPs, if these objects transfer more than a few percent of their total spin-down power into VHE pairs, the resulting inverse-Compton emission would exceed that observed by HESS from the Inner Galaxy. § DISCUSSION AND SUMMARY The spectrum and morphology of the very high-energy (VHE) gamma rays observed from the Inner Galaxy have been interpreted as evidence that the Galactic Center (and Sgr A^*, in particular) accelerates protons up to ∼PeV energies, which then propagate outward and generate the observed emission through pion production. This scenario is further supported by the reported correlation between this emission and the distribution of molecular gas. In this article, we argue that the VHE emission from the Inner Galaxy is also likely to receive sizable contributions from pulsars, motivated by observations of nearby pulsars by HAWC and Milagro. In particular, HAWC's measurements of the spectrum and angular distribution of multi-TeV emission from Geminga and B0656+14 indicate that these sources deposit a significant fraction of their spin-down power into VHE electrons and positrons. If we assume that pulsars located in and around the Galactic Center also deposit a similar fraction of their energy into VHE pairs, then one can account for a large fraction of the VHE gamma-ray emission from the Inner Galaxy as observed by HESS and other ground-based telescopes.The contribution of TeV halos to the VHE emission observed from the Inner Galaxy by HESS is well-motivated by three factors. First, the spectrum of the observed gamma-ray emission is consistent with that observed from the TeV halos surrounding the Geminga and B0656+14 pulsars. Second, the intensity of the observed emission is consistent with the expected pulsar population of the Galactic Center, based on the numbers of massive stars and low-mass X-Ray binaries observed in the region. In particular, based on a simple pulsar distribution model, we estimate that the observed emission requires a total birth rate of 490^+580_-370 neutron stars per Myr, some ∼25-190 of which will constitute pulsars with radio beams directed toward the Solar System. Finally, the spatial extension of the observed TeV emission is consistent with a scenario where pulsars are transported out of the Galactic Center by neutron star natal kicks. While previous studies <cit.> have argued that leptonic gamma-ray models cannot produce the spatially extended emission observed from the Galactic Center, our model naturally avoids this constraint by transporting the sources of the mulit-TeV electrons outward and away from Sgr A^*. In the years ahead, we expect a number of observations to refine and clarify this situation. Firstly, we anticipate that HAWC will measure the spectrum and angular extension of a significant number of pulsars <cit.>, allowing us to determine whether the emission observed from Geminga and B0656+14 is representative of the larger pulsar population. And although existing imaging atmospheric Cherenkov telescopes have not yet reported any significant detection of TeV-scale emission from Geminga or B0656+14 <cit.>, next generation telescopes (and in particular the Cherenkov Telescope Array) are expected to be sensitive to extended sources such as these. In the foreseeable future, we anticipate these VHE gamma-ray telescopes to accumulate a sizable catalog of both pulsars and pulsar wind nebulae <cit.>. Future observations of the Inner Galaxy by the Cherenkov Telescope Array will also build upon and expand our knowledge of this region of the sky. Furthermore, during the same period of time, deep large-area radio surveys are anticipated to detect the first pulsars from the inner parsecs around the Galactic Center <cit.>. Acknowledgments. We would like to thank Alice Harding for valuable discussions. DH is supported by the US Department of Energy under contract DE-FG02-13ER41958. Fermilab is operated by Fermi Research Alliance, LLC, under contract DE- AC02-07CH11359 with the US Department of Energy. IC acknowledges support from NASA Grant NNX15AB18G and from the Simons Foundation. TL acknowledges support from NSF Grant PHY-1404311.JHEP | http://arxiv.org/abs/1705.09293v2 | {
"authors": [
"Dan Hooper",
"Ilias Cholis",
"Tim Linden"
],
"categories": [
"astro-ph.HE",
"astro-ph.GA",
"hep-ph"
],
"primary_category": "astro-ph.HE",
"published": "20170525180001",
"title": "TeV Gamma Rays From Galactic Center Pulsars"
} |
National Institute of Information and Communications Technology, Koganei, Tokyo 184-8795, JapanMax Planck Institute for the Science of Light, 91058 Erlangen, GermanyHearne Institute for Theoretical Physics and Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USANational Institute of Information and Communications Technology, Koganei, Tokyo 184-8795, JapanHearne Institute for Theoretical Physics and Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA National Institute of Information and Communications Technology, Koganei, Tokyo 184-8795, JapanSophia University, 7-1 Kioicho, Chiyoda-ku, Tokyo 102-8554, JapanHearne Institute for Theoretical Physics and Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USAIn the lore of quantum metrology, one often hears (or reads) the following no-go theorem:If you put vacuum into one input port of a balanced Mach-Zehnder Interferometer, then no matterwhat you put into the other input port, and no matter what your detection scheme, the sensitivity can never be better than theshot noise limit (SNL). Often the proof of this theorem is cited to be in Ref.[C. Caves, Phys. Rev. D 23, 1693 (1981)], but upon further inspection, no such claim is made there. Aquantum-Fisher-information-based argument suggestive of this no-go theorem appears in Ref. [M. Lang and C. Caves, Phys. Rev. Lett. 111, 173601 (2013)], but is not stated in its full generality. Here we thoroughly explore this no-go theorem and give the rigorous statement: the no-go theorem holds whenever the unknown phase shift is split between both arms of the interferometer, but remarkably does not hold when only one arm has the unknown phase shift. In the latter scenario, we provide an explicit measurement strategy that beats the SNL. We also point out that these two scenarios are physically different and correspond to different types of sensing applications. Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum Jonathan P. Dowling December 30, 2023 ========================================================================================================== Introduction.—In the field of quantum metrology<cit.>,a Mach-Zehnder interferometer (MZI) is a tried and trueworkhorse that has the additional advantage that any result obtained for it also applies to a Michelson interferometer (MI) and hence has a potential application to gravitational wave detection. In most current implementations of gravitational wave detectors, the MI is fed with a strong coherent state of light in one input port and vacuum in the other (Fig. <ref>).It was in this context that Caves in 1981 <cit.> showed that such a design would always only ever achieve the shotnoise limit (SNL).Then he showed if you put squeezed vacuum into the unused port, you could beat the SNL. Several implementations of this squeezed vacuum scheme have already been demonstrated in the GEO 600 gravitational detector, and plans are underway to utilize this approach in the LIGO and VIRGO detectors in the future <cit.>.It then appeared, that in the lore of quantum metrology, this result was extended — without proof — to the following no-go theorem: If you put quantum vacuum into one input port of a balanced MZI, then no matter what quantum state of light you put into the other input port, and no matter what your detection scheme, the sensitivity can never be better than the SNL. Often the proof of this theorem is cited to be the original 1981 paper by Caves <cit.>, but upon further inspection, no such general claim is madethere. A quantum-Fisher-information-based argument suggestive of this no-go theorem appeared in Ref. <cit.> by Lang and Caves, but it does not explore the statement in adequate generality.In this work, we give a full statement of the no-go theorem. The statement proved here is the following: if the unknown phase shifts are in both of the two arms of the MZI, then the no-go theorem holds no matter whether the MZI is balanced or not. However, in the case where the unknown phase shift is in only one arm of the MZI, then the no-go theorem does not necessarily hold. The former is a multiparameter measurement and the latter is a single parameter. For the latter, we show an explicit scheme with a probe and measurement that can beat the SNL in the sense that its classical Fisher information (CFI) is proportional to the square of the total photon number used at the input and the measurement. The underlying issue is that two different models for the unknown phase shift unitary operation in the MZI can give different values of the QFI <cit.>. Since only the phase difference is utilized in both models, it has been thought that this discrepancy is a flaw in the interpretation of the QFI <cit.> or is related to the assumptions of the input states and the measurements <cit.>. By contrast, here we point out that the different unitaries correspond to physically different types of sensors, and their choice should depend on the concrete application scenarios.Also for the former scenario (i.e. unknown phase shifts in two arms), we show that one has to carefully considerthe phase sum (often regarded as the “global phase” though) whereas only phase difference is the quantity of interest.In other words, it is intrinsically a two-parameter estimation problem.Related to the above, we also point out the pitfalls of using only the quantum Fisher information (QFI), or the closely related quantum Cramér-Rao (QCRB) bound <cit.>, to make claims of a quantum metrological advantage, without explicitly providing a detection scheme that would actually achieve that advantage <cit.>. Before the QFI approach came into vogue in recent years, often theorists would try to optimize the input state and the detection scheme simultaneously. This often led to input states and detection schemes difficult to implement. The QFI approach freed us from having to optimized over all detection schemes, more accurately over all Positive Operate Valued Measures (POVM), but that freedom, carried a very high cost. The issue is that the optimal POVM that achieves the QFI may be difficult to implement or contain hidden resources, such as a strong local oscillator, that are not fairly counted as far as a quantum advantage is concerned <cit.>.Quantum Fisher information approach to phase sensing— A schematic of the Mach-Zehnder (MZ) interferometer-type sensingwe consider here is illustrated in Fig. <ref>(a).Two input modes A and B are interfered via a beam splitterwith transmittance T, and then put into the phase shift unitaryoperation Û_ϕ followed by some measurement.In addition to this standard setting, we restrict one of the input statesto always be the quantum vacuum state, whereas the other input can be an arbitrary quantum state (possibly mixed). A similar setup to the one we consider here has recently usedby Lang and Caves <cit.>, where the two input portsare assumed to be in an arbitrary pure state |χ⟩and a coherent state |α⟩, and the beam splitter transmittanceis chosen to be 50/50 (i.e. T=1/2). The phase shift unitary operator isÛ_ϕ = e^i ĝ_s ϕ_s e^i ĝ_d ϕ_d,where ϕ_s and ϕ_d are the phase sum and differenceof the two modes, respectively,ĝ_s = (â^†â + b̂^†b̂)/2, ĝ_d = (â^†â - b̂^†b̂)/2.These two phase shift parameters reflect the unknown phase shifts inthe two arms of the MZI, ϕ_1 and ϕ_2,as ϕ_s = ϕ_1 + ϕ_2, ϕ_d = ϕ_1 - ϕ_2(see Fig. <ref>(b)).â^† (b̂^†) andâ (b̂) are creation and annihilation operators inmode A (B), respectively. Then the authors showed that for a coherent state inputwith α = 0, i.e. for the vacuum input,the quantum Fisher information (QFI) for the phasedifference turns out to be the average photon number of the input:F_Q(|χ⟩, ĝ_d)= ⟨χ| n̂ |χ⟩ = n̅_χ , where n̂=â^†â.This result suggests that the precision ofthe phase sensing is shot-noise limited,when one of the input ports contains only vacuum(and the other mode contains any pure state), since the QCRB is Δ^2ϕ≥1/F_Q. However, the above result does not answer questions such as whether the no-go theorem still holds when the interferometer is not balanced, or when the phase shift unitary operator is chosen differently. Firstly, for the phase shift unitary operator ĝ_d, when T deviates from 1/2, the QCRB already appears to beat the SNL. Keeping T as a free parameter, and using the fact that the QFI of a pure state in estimating a phase shift generated by a generator ĝ is given by4(⟨ĝ^2⟩ -⟨ĝ⟩ ^2), we arrive at F_Q(|χ⟩, ĝ_d, T)= {1-(1-2T)^2}n̅_χ + (1-2T)^2 V_χ. (See Supplemental Material 1 for the derivation.) This beats the SNL for any non-50/50 beam splitter quite spectacularly. For example, with T→ 0, the QCRBapproaches Δ^2ϕ=1/V_x<1/n̅_χ for some inputs such as squeezed vacuum <cit.>.Secondly, as pointed out and rigorously discussed in Ref. <cit.>, a different choice of the phase shift unitary can give a differentvalue for the QFI.For example, in lieu of the phase shift operator ĝ_d, one can instead choose Û_ϕ = e^ i ĝ_1 ϕ_A,where ĝ_1 = â^†â, such that phase shift is generated only in one arm. The QFI for the phase shift unitary operator ĝ_1 is found to be F_Q(|χ⟩, ĝ_1)= n̅_χ + V_χ , where V_χ = ⟨χ| n̂^2 |χ⟩- ⟨χ| n̂ |χ⟩^2 is the photon numbervariance of |χ⟩. (See Supplemental Material 1 for the derivation.)This is obviously different from Eq. (<ref>), and againimplies a sub-SNL result, since V_χ > n̅_χis possible for some inputs, as mentioned above.These results extrapolated from Ref. <cit.> are thus perplexing, sinceseemingly, both Eqs. (<ref>) and(<ref>) suggest the possibility ofsub-SNL precision phase sensing even with the vacuum input into one of the input ports. Phase shift in both arms vs. in one arm in the MZI sensing— We point out that the above phase-shift unitary operators(Figs. <ref>(b) and (c))have different physical meanings and their choiceshould depend on what type of application scenario is in your mind.For the gravitational wave detection application, ĝ_d andĝ_s should be chosen since the two arms of the (Michelson)interferometer both have unknown phase shifts induced by the gravitational waves(Fig. <ref>(b)). Also some commonly used sensing devices such asa differential interference contrast microscope <cit.>should be modeled in the same way. (See also its quantum version <cit.>.)On the other hand, the most primitive use of the Mach-Zehnder interferometeris to put a sample in one of the two arms to measure the corresponding phase shift.This configuration is also widely used as a simple and low-cost technologyto measure the sample's density distribution, pressure, temperature, etc.This type of sensor should be modeled by ĝ_1 (Fig. <ref>(c)).Since these two models are physically different, they may lead todifferent outcomes in our problem; the MZI with vacuum in one inputport. That is, they could have different fundamental precision limitswith vacuum in one input port.We will rigorously analyze each model in the following. Remedy: Full quantum Fisher information matrix treatment— The MZI sensing with the ĝ_s-ĝ_d model in its full generality is a two-parameter estimation problemsince there are two unknown parameters, ϕ_s and ϕ_d, in the system(although usually only the phase difference ϕ_d is an interesting quantity to measure).Therefore, a two-by-two quantum Fisher information matrix (QFIM) is considered.The problem in Eq. (<ref>) is in fact due to the ignorance of the phase sum ϕ_s [In Ref. <cit.>, the QFIM of the system considered was calculated.However, they reduce it to the single-parameter estimation(i.e. drop off the terms for ϕ_s) which loosesthe tightness of the bound. Note that this problem does not appearin Eq. (<ref>) since with T=1/2, the non-diagonal term ofthe QFIM goes to zero and thus the problem reduces to two independentsingle-parameter estimations. Nevertheless, in Ref. <cit.>,they also consider the non-vacuum input case wherethe bound may have some looseness. ]. In multi-parameter estimation, the QCRB is given byΣ≥ℱ_Q^-1/m, where Σ is the covariance matrix of the estimatorincluding both ϕ_s and ϕ_d, m is the number of trials,and ℱ_Q is the two-by-two QFIM:ℱ_Q = [ [ F_dd F_sd; F_ds F_ss ]] , where s and d correspond to ϕ_s and ϕ_d.The first diagonal element of ℱ_Q^-1in Eq. (<ref>)corresponds to the estimation limit of ϕ_d, which is explicitly given byF_ss/F_ss F_dd - F_sd F_ds .For an arbitrary mixed quantum state, the QFIM is in general not easy to calculate.However, the optimal input state that maximizes ℱ_Qis always given by a pure-state input.This is the consequence of the convexity of the QFIM:for ρ̂_ϕ = p σ̂_ϕ + (1-p) τ̂_ϕ,ℱ_Q(ρ̂_ϕ) ≤ p ℱ_Q(σ̂_ϕ)+ (1-p) ℱ_Q(τ̂_ϕ), holds. This can be proved by using the monotonicity of the QFIM underthe completely positive trace preserving (CPTP) map <cit.>and extending the proof of the convexity for the QFI <cit.>.(See Supplementary Material 2.)The statement basically says that a statistical mixture of the input stateswill never increase the QFIM and thus implies that the QFIM is maximizedwith a pure state input.The optimal pure state for the QFIM is also optimal forthe multi-parameter QCRB (<ref>)since the QFIM is a positive matrixand for positive matrices A and B,B^-1≥ A^-1 holds if and only if A ≥ B.That is, a statistical mixture of the input stateswill never increase the sensitivity for both single- andmultiple-parameter estimation.Therefore, by considering a pure input state |χ⟩, the elements of the QFIM are given byF_ij = 4 ( ⟨ĝ_i ĝ_j ⟩- ⟨ĝ_i ⟩⟨ĝ_j ⟩), where i,j takes s and d.These are explicitly given byF_dd={1-(1-2T)^2}n̅_χ + (1-2T)^2 V_χ,F_ss= V_χ,F_ds= F_sd = -(1-2T) V_χ, where note that F_dd corresponds to Eq. (<ref>).Inserting these into (<ref>), we getΔ^2 ϕ_d ≥1/4T(1-T)mn̅_χ, where the minimum of the right hand side is obtained with T=1/2 as 1/(m n̅_χ), which is the SNL, as it should be.That is, no matter how highly nonclassical the input state ρ̂_ in is, and no matter what POVM you deploy, the SNL cannot be surpassesfor ĝ_d so long as the other input to the interferometeris the vacuum state. Thus, this result establishes the no-go theorem in a most general form,which includes the beam splitter transmissivity as a free parameter. Phase shift in one arm (ĝ_1)— The ĝ_1 model is a single-parameter estimation problem andthus (<ref>) is directly applied to the QCRB, whichsuggests the sub-SNL sensitivity with high V_χ, that is,input states with high photon number fluctuation such as squeezed vacuum.Then as mentioned at the introduction, the QFI-only approach may havethe pitfall that the optimal POVM attaining the QCRB could contain huge amountof hidden resources as pointed out by Jarzyna and Demkowicz-Dobrzański<cit.>.In other words one can fool oneself into thinking,via the QFI-only approach, that there is somequantum metrological advantage, where none actually exists.There are two remedies.The first, and the one we recommend, is that if authors wish to claima quantum metrological advantage from a QFI-only calculation,they then must provide a detection scheme that actually hitsthe related QCRB, so all resources hidden in the associated POVM may bethen laid bare for all to see. (We should note that in Ref. <cit.>, the authorswere careful to back up the QFI calculation by providinga detection scheme — the parity operator — that actually hits the QCRB.)The second remedy, besides producing the POVM that hits the limit,is to rule out any external resource that might give some phase information tothe measurement device. Such a “rule-out" protocolwas introduced by Jarzyna and Demkowicz-Dobrzański <cit.>.They resolved this issue by introducing the idea of a phase-averaged input state, wherethe two-mode input state from the two input ports isaveraged by a common phase shift, which preserves the relative phasebetween two modes, but does not allow any phase information to be brought in from the outside of the interferometer, e.g.from the measurement devices themselves (a similar discussion appearedin the context of superselection rule <cit.>).Therefore, the QFI of the phase-averaged input gives the proper phase-sensing limit without any external phase reference.A simple way to understand the phase averaging is to think of itas a type of phase randomization akin to preparing a thermal state.A thermal state can be used in an MZI for SNL interferometry, eventhough it contains no coherence, because each photon— as in Dirac's dictum —only ever interferes with itself.In this way the advantage of any hidden resource in the POVM is mitigated.Here we apply these two remedies separately.First, we employ the phase-averaging approach to eliminate any hidden resourcein the POVM.For the two input states, we consider a vacuum andan arbitrary quantum state with the density matrix ofρ̂_ in = ∑_n,m=0^∞ c_nm |n ⟩⟨ m|, where |n⟩ is the n-photon number state.Then the phase-averaged input is given by Ψ_avg=∫dθ/2πV̂^A_θV̂^B_θ( ρ̂_ in^A ⊗ |0 ⟩⟨ 0|^B )V̂^A†_θV̂^B†_θ =∑_n,m=0^∞∫dθ/2π e^iθ (n-m) c_nm |n⟩⟨ m|^A⊗ |0⟩⟨ 0|^B =∑_n=0^∞ p_n |n⟩⟨ n|^A ⊗ |0⟩⟨ 0|^B , where V̂^A_θ = e^iθâ^†â,V̂^B_θ = e^iθb̂^†b̂, andp_n = c_nn is a real positive number satisfying∑_n p_n =1. The state after the first beamsplitter of the MZI and the phase shiftingis given byΨ_avg^ϕ=Û_ϕ^(1)ABB̂^AB_T Ψ_avgB̂_T^† ABÛ_ϕ^(1)†AB =∑_n=0^∞p_n |ψ_n(ϕ)⟩⟨ψ_n(ϕ)|_AB , where|ψ_n(ϕ)⟩ _AB=∑_j=0^ne^-ijϕ√(nj) ×√(T)^j√(1-T)^n-j|j⟩ _A⊗|n-j⟩_B.By using the convexity of the QFI and noticing that|ψ_n(ϕ)⟩ and |ψ_n'(ϕ)⟩are orthogonal for nn', we haveF^(1)_Q(Ψ_avg^ϕ) = ∑_n=0^∞ p_nF^(1)_Q(|ψ_n(ϕ)⟩) , where F_Q^(1)(|ψ_n(ϕ)⟩) = 4(⟨ĝ_1^2⟩ -⟨ĝ_1⟩^2)= 4n̅T(1-T).(See Supplementary Material 3 for the detailed derivation.)The maximum in Eq. (<ref>) is attained at T=1/2,and is equal to n̅, as it should be.Consequently, the QFI for Ψ^ϕ_ ave is given as F_Q(Ψ_avg^ϕ) =∑_n=0^∞ n p_n = n̅ ,where n̅ is the average photon number of ρ̂_ in,and thus we find that the phase sensitivity is lower bounded asΔ^2 ϕ_A≥1/m n̅.That is, if the optimal POVM is not allowed to have external phase information,the estimation precision is limited by the shot-noise limit. There is, however, one question remaining: if one is allowed to usesome additional resource at the measurement, is it possible to surpass the SNLwith respect to the total number of resources used at the input andthe detection process? As our last result, we prove that the answer is affirmativeby showing a concrete measurement scheme.Figure <ref> illustrates the concrete input stateand the measurement.The input state is a single-mode squeezed vacuum, which is generatedfrom vacuum by applying the squeezing operation Ŝ(r), wherer is the squeezing parameter. The measurement is a time-reversed process, that is, itconsists of the complex-conjugate beam splitter B̂_T^†,anti-squeezing operation Ŝ^†(r), and photon detectors thatdiscriminate zero and non-zero photons (so-called on-off detectors).For simplicity of the analysis, we consider only two outcomes:zero photons in both detectors or other events,{|0⟩⟨0| ⊗ |0⟩⟨0|,I - |0⟩⟨0| ⊗ |0⟩⟨0|}(photon number discrimination may further improve the performance butto simplify the discussion here, we leave it for a future work).Note that this mirror-image like detection strategy has been considered inthe context of state discrimination <cit.> and also the phase estimation via coherent-state input <cit.>.Since we need the phase information of the input state atthe anti-squeezing process, this is a phase-sensitive measurement,and so the POVM has access to external phase information and the phase of the squeezer Ŝ^†.The input average photon number is given by n̅= sinh^2 r.Since the measurement device also uses the same amount of squeezing,the average photon number of the all resources are counted asn̅_ tot = 2n̅ (to consider the fundamental limitation,here we assume a unit efficiency parametric downconverter wheresinh^2 r is directly a function of the pump energy used for the converter).The attainable precision limit (in the asymptotic limit, r→∞) is specified bycalculating its CFI<cit.>:F(ϕ_A)=E[ - d^2/d ϕ_A^2log p_ϕ_A], where p_ϕ_A = P(x|ϕ_A) is the conditional probability of obtainingthe measurement outcome x for given ϕ_A andx=0,1 represent the photon detection outcome|0⟩⟨0|⊗|0⟩⟨0| andI-|0⟩⟨0|⊗|0⟩⟨0|, respectively.F(ϕ_A) is calculated by the characteristic function approach(e.g. Ref. <cit.>), and the derived analytical expressionof F(ϕ_A) is complicated (see Supplementary Material 4). Taking the limit of ϕ_A → 0, we getF(ϕ_A) = 2n̅_ tot T (1+T+n̅_ totT), where we remind the reader that n̅_ tot = 2n̅ is the total resourceused for the input state and the detection process.Thus we get the (classical) CRB around ϕ_A = 0 asΔ^2ϕ_A ≥1/2mn̅_ tot T (1+T+n̅_ totT), which surpasses the SNL of the total resource for any T0.This example shows how a QFI-only calculation could contain hiddenresources in the unknown optimal POVM, that are unfairly not counted. Here, by taking all resources into account, we conclude that it is possible to beat the SNLfor the ĝ_1 estimation if one uses an additional energy and phase resourcesat the detection.Conclusions.—In this paper, we revisit the ultimate limit of the MZI sensing precisionwhen an input into one port is vacuum.We show a full statement of the problem with a rigorous proof:the statement depends on your choice of the phase shift unitary operator— in other words, on the physical setup of your sensing application. First, if both two arms of the MZI have different unknown phase shiftsin your application (e.g. gravitational wave detection) andinput vacuum in one port, then no matter what you put in the other port,and no matter what your detection scheme you deploy, you can neverdo better than the SNL in phase sensitivity. The statement holds even if the first beamsplitter of the MZI isnon-50:50.The proof is based on the fact that it is intrinsically a two-parameter estimation problemalthough the phase sum ϕ_s is often treated as a “global phase” and ignored in real experiments.Intuitively, we see that the state after the phase shift is e^i ĝ_s ϕ_s e^i ĝ_d ϕ_d |Ψ⟩_AB(where |Ψ⟩_AB is a probe state before the phase shift) which clearly contains ϕ_s.This implies if one knows nothing about ϕ_s, the effective state should be randomized over ϕ_swhich might give a different conclusion than that from the analysis considering only ϕ_d as a single-parameter estimation.This is why we need to take into account both ϕ_s and ϕ_d even for deriving the precision bound of only ϕ_d.This type of sensing includes the gravitational wave detection<cit.>, long-baseline interferometry <cit.>,and differential interference contrast microscopy <cit.>, for example.In these applications, if one input is vacuum, our result rules outthe possibility of doing something “quantum” at the detector(such as putting in a squeezer or doing photon addition or subtraction)to beat the SNL.Second, if only one of the MZI arms has an unknown phase shiftin your application (sensing a sample placed at one arm of the MZI),the ultimate precision limit depends on the detector restriction.If you do not allow the detector to use any external phase reference andpower resource, then the precision is limited by the SNL.However, if you allow the detector to use such resources,you can beat the SNL in terms ofthe total resource used at the input and detector.The explicit sensing scheme which uses squeezers for both input and detectoris given.This type of sensing includes simple MZI devices measuring sample's density,pressure, temperature, etc, and also LIDAR-type sensing <cit.>.In these applications, only if you put nonclassical light into at least one input port,is there a hope to beat the SNL by doing something quantumat the detector, even if the other port is vacuum. § ACKNOWLEDGEMENTS MT would like to acknowledge suport from the Open Partnership Joint Projects of JSPS Bilateral Joint Research Projects and the ImPACT Program of Council for Science, Technology and Innovation, Japan. KPS, and JPD would like to acknowledge support from theAir Force Office of Scientific Research, the Army Research Office, the National ScienceFoundation, and the Northrop Grumman Corporation. CY would like to acknowledge supportfrom an Economic Development Assistantship from the the Louisiana State University System Board of Regents.apsrev4-1 § SUPPLEMENTAL MATERIAL 1: QUANTUM FISHER INFORMATIONFOR THE MACH-ZEHNDER INTERFEROMETER PHASE SENSING WITH A VACUUM INPUT Here we derive Eqs. (<ref>), (<ref>), and(<ref>)–(<ref>) in the main text.Consider |χ⟩⊗ |0⟩ as an inputto the MZ interferometer. For the calculation, it is useful to expand|χ⟩ in a coherent state basis:|χ⟩ = ∫ d^2 αf(α) |α⟩ , where |α⟩ is a coherent state with complex quadratureamplitude α.Then the average photon number and the variance of the state aregiven byn̅_χ=⟨χ|n̂|χ⟩ =∫ d^2 α ∫ d^2 βf^*(α) f(β)⟨α| n̂ |β⟩ =∫ d^2 α ∫ d^2 βf^*(α) f(β)α^* β⟨α|β⟩ =∫ d^2 α ∫ d^2 βf^*(α) f(β)α^* β ×exp[ -1/2( |α|^2 + |β|^2 - 2 α^* β) ] , and V_χ=⟨χ|n̂^2|χ⟩ - n̅_χ^2=∫ d^2 α ∫ d^2 βf^*(α) f(β){ (α^* β)^2 + α^* β} ×exp[ -1/2( |α|^2 + |β|^2 - 2 α^* β) ]- n̅_χ^2, where we use the fact thatn̂^2 = â^†2â^2 + â^†â.The state after the beam splitter with transmittance Tis given by|Φ⟩_AB =∫ d^2 αf(α)|√(T)α⟩_A|√(R)α⟩_B , where R=1-T.§.§ QFI with ĝ_1 (Eq. (<ref>))The quantum Fisher information (QFI) is calculated from F_Q(|χ⟩, ĝ_1, T) = 4 (⟨Φ| ĝ_1^2 |Φ⟩- ⟨Φ| ĝ_1 |Φ⟩^2) . We have ⟨Φ| ĝ_1^2 |Φ⟩=∫ d^2 α ∫ d^2 βf^*(α) f(β)⟨√(T)α|_A ⟨√(R)α|_B (â^†2â^2 + â^†â) | √(T)β⟩_A | √(R)β⟩_B=∫ d^2 α ∫ d^2 βf^*(α) f(β){ (T α^* β)^2 + T α^* β}⟨√(T)α|. √(T)β⟩⟨√(R)α|. √(R)β⟩ =T^2 ⟨χ|n̂^2|χ⟩+ T(1-T) ⟨χ|n̂|χ⟩ , and⟨Φ| ĝ_1 |Φ⟩^2 =( ∫ d^2 α ∫ d^2 βf^*(α) f(β)⟨√(T)α|_A ⟨√(R)α|_B (â^†â) | √(T)β⟩_A | √(R)β⟩_B )^2 =T^2 ⟨χ|n̂|χ⟩^2.In total, we haveF_Q(|χ⟩,ĝ_1,T) = 4 (⟨Φ| ĝ_1^2 |Φ⟩- ⟨Φ| ĝ_1 |Φ⟩^2) =4 { T^2 V_χ + T(1-T) n̅_χ}. For T=1/2, it is V_χ + n̅_χ and thuswe get Eq. (<ref>).§.§ QFIM for ĝ_d and ĝ_s [Eq. (<ref>), (<ref>)–(<ref>)]For pure states, the elements of the QFIM are given byF_ij = 4 ( ⟨ĝ_i ĝ_j ⟩- ⟨ĝ_i ⟩⟨ĝ_j ⟩), where i,j takes s and d.Recall thatĝ_d = (â^†â - b̂^†b̂)/2 andĝ_s = (â^†â + b̂^†b̂)/2.Then we have 4 ⟨Φ| ĝ_d^2 |Φ⟩=∫ d^2 α ∫ d^2 βf^*(α) f(β) ×⟨√(T)α|_A ⟨√(R)α|_B (â^†2â^2 + â^†â + b̂^†2b̂^2 + b̂^†b̂- 2 â^†âb̂^†b̂) | √(T)β⟩_A | √(R)β⟩_B=∫ d^2 α ∫ d^2 βf^*(α) f(β) ×{ (T α^* β)^2 + T α^* β+ (R α^* β)^2 + R α^* β- 2 RT (α^* β)^2}⟨√(T)α|. √(T)β⟩⟨√(R)α|. √(R)β⟩ =∫ d^2 α ∫ d^2 βf^*(α) f(β){α^* β + (T-R)^2 (α^* β)^2 }exp[ -1/2( |α|^2 + |β|^2 - 2 α^* β) ] =⟨χ| n̂ |χ⟩+ (1-2T)^2 ( ⟨χ| n̂^2 |χ⟩- ⟨χ| n̂ |χ⟩). Similarly, we have4 ⟨Φ| ĝ_s^2 |Φ⟩=⟨χ| n̂^2 |χ⟩ , 4 ⟨Φ| ĝ_d ĝ_s |Φ⟩=4 ⟨Φ| ĝ_s ĝ_d |Φ⟩ =-(1-2T) ⟨χ| n̂^2 |χ⟩ . Also2 ⟨Φ| ĝ_d |Φ⟩=∫ d^2 α ∫ d^2 βf^*(α) f(β)⟨√(T)α|_A⟨√(R)α|_B( â^†â - b̂^†b̂)|√(T)β⟩_A |√(R)β⟩_B =∫ d^2 α ∫ d^2 βf^*(α) f(β)( Tα^* β - Rα^* β)⟨√(T)α| .√(T)β⟩⟨√(R)α| .√(R)β⟩ =(1-2T)∫ d^2 α ∫ d^2 βf^*(α) f(β)α^* βexp[ -1/2( |α|^2 + |β|^2 - 2 α^* β) ] =(1-2T) ⟨χ| n̂ |χ⟩ ,and similarly,2 ⟨Φ| ĝ_s |Φ⟩ =⟨χ| n̂ |χ⟩ . By using the above results, we haveF_dd= F_Q(|χ⟩,ĝ_d,T) =⟨χ| n̂ |χ⟩+ (1-2T)^2 ( ⟨χ| n̂^2 |χ⟩- ⟨χ| n̂ |χ⟩) - (1-2T)^2 ⟨χ| n̂ |χ⟩^2 ={ 1- (1-2T)^2 }n̅_χ+ (1-2T)^2 V_χ , F_ss=⟨χ| n̂^2 |χ⟩- ⟨χ| n̂ |χ⟩^2 =V_χ , F_ds= F_sd = -(1-2T) ⟨χ| n̂^2 |χ⟩-(1-2T) ⟨χ| n̂ |χ⟩^2 =-(1-2T) V_χ .§ SUPPLEMENTAL MATERIAL 2: CONVEXITY OF QUANTUM FISHERINFORMATION MATRIX Here we prove the convexity of the quantum Fisher informationmatrix (QFIM):ℱ_Q(ρ̂_φ) ≤ p ℱ_Q(σ̂_φ)+ (1-p) ℱ_Q(τ̂_φ), for ρ̂_φ = p σ̂_φ + (1-p) τ̂_φ. Here ρ̂_φ, σ̂_φ, and τ̂_φare (maybe mixed) quantum states whereφ = {φ_1 , … , φ_M }is a set of M unknown parameters. To begin with, we briefly review the definition and the structure of the QFIMthat we will use in the proof. Detailed review on the QFI and QFIMcan be found for example in Ref. <cit.>.The QFIM for ρ̂_φ is given by an M × M matrixℱ_Q(ρ̂_φ) = [F_ij(ρ̂_φ)]_ij(i,j = 1, … , M) where each entry is defined asF_ij(ρ̂_φ) = 1/2 Tr[ ρ̂_φL̂_i L̂_j +ρ̂_φL̂_j L̂_i ] , and L̂_i, called the symmetrized logarithmic derivative,is a Hermitian operator satisfying∂/∂φ_iρ̂_φ = 1/2( L̂_i ρ̂_φ + ρ̂_φL̂_i ) . Let ρ̂_φ = ∑_k λ_k |λ_k⟩⟨λ_k|be the spectral decomposition of ρ̂_φ.Then we can explicitly describe L̂_i asL̂_i = 2 ∑_k,l⟨λ_k| ρ̂_φ^(i) |λ_l⟩/λ_k + λ_l |λ_k⟩⟨λ_l| , where ρ̂_φ^(i) = ∂ρ̂_φ /∂φ_i. Combining it with Eq. (<ref>), the QFIM is expressed asF_ij(ρ̂_φ) = 2 ∑_k,l⟨λ_k| ρ̂_φ^(i) |λ_l⟩⟨λ_l| ρ̂_φ^(j) |λ_k⟩/λ_k + λ_l . We also use an important property of the QFIM:monotonicity under completely positive trace preserving (CPTP) mapℒ <cit.>, ℱ_Q (ρ̂_φ)≥ℱ_Q (ℒ(ρ̂_φ)). The proof of the convexity of the QFIM is basically given by extendingthe proof for the QFI (i.e. single-parameter case) in Ref. <cit.>.Consider the bipartite state ρ̃_φ^AB= p |e_0 ⟩⟨ e_0|^A ⊗σ̂_φ^B+ (1-p) |e_1 ⟩⟨ e_1|^A ⊗τ̂_φ^B,where |e_k⟩ is an orthonormal basis in A.Note that Tr_A [ρ̃_φ^AB] = ρ̂_φ^B.Then we haveℱ_Q(ρ̃_φ^AB) =p ℱ_Q (σ̂_φ^B)+ (1-p) ℱ_Q (τ̂_φ^B) . This is justified by the following observation.Since |e_k⟩ is independent of the unknown parameters φ_i,ρ̃_φ^(i) =p |e_0 ⟩⟨ e_0| ⊗σ̂_φ^(i)+ (1-p) |e_1 ⟩⟨ e_1| ⊗τ̂_φ^(i),for any i.Also the spectral decomposition of ρ̃_φis described asp |e_0 ⟩⟨ e_0| ⊗∑_i λ_i^σ|λ_i^σ⟩⟨λ_i^σ| +(1-p) |e_1 ⟩⟨ e_1| ⊗∑_i λ_i^τ|λ_i^τ⟩⟨λ_i^τ|,where ∑_i λ_i^σ|λ_i^σ⟩⟨λ_i^σ| and∑_i λ_i^τ|λ_i^τ⟩⟨λ_i^τ| arethe spectral decompositions of σ̂_φ andτ̂_φ, respectively.Plugging them into the expression of QFI in Eq. (<ref>),we getF_ij(ρ̃_φ^AB) =p F_ij(σ̂_φ^B) + (1-p) F_ij(τ̂_φ^B). Since this holds for all i and j, we get Eq. (<ref>).By using Eq. (<ref>),the monotonicity (<ref>), andthe fact that partial trace is a CPTP map, we haveℱ_Q(ρ̂_φ^B)≤ ℱ_Q(ρ̂_φ^AB) =p ℱ_Q (σ̂_φ^B)+ (1-p) ℱ_Q (τ̂_φ^B) , which completes the proof of the convexity of the QFIM. § SUPPLEMENTAL MATERIAL 3: QUANTUM FISHER INFORMATIONFOR Ĝ_1 WITH PHASE RANDOMIZINGHere we calculate the QFI for |n⟩⊗ |0⟩ withthe generator ĝ_1 = â^†â andsee that it coincides with that of ĝ_d.The state past the beam splitter and the phase-shift transformationis given by|ψ_n(ϕ)⟩ _AB=∑_j=0^ne^-ijϕnj^1/2 ×T^j/2(1-T)^(n-j)/2|j⟩ _A⊗|n-j⟩_B.For |ψ_n(ϕ)⟩, we find ⟨â^†â⟩=∑_j=0^njnjT^j(1-T)^n-j=nT, ⟨b̂^†b̂⟩= n(1-T), ⟨â^†2â^2⟩ = ∑_j=0^nj(j-1)njT^j(1-T)^n-j = n(n-1)T^2 ,and the QFI evaluated as 4(⟨ĝ_1^2⟩ -⟨ĝ_1⟩^2) is found to beF_Q^(1)=4(⟨â^†2â^2⟩+⟨â^†â⟩ -⟨â^†â⟩^2) =4{ n(n-1)T^2 + nT - n^2 T^2 } =4nT(1-T).The maximum is attained at T=1/2 and is equal to n.§ SUPPLEMENTAL MATERIAL 4: DERIVATION OF F(Φ_A) The calculation of Fisher information can be performed bythe characteristic function approach.For the details of the characteristic function formalism in quantum optics,see Ref. <cit.> for example.Here we follow the definition and the methodology developedin Ref. <cit.>.Then the covariance matrix of the two-mode vacuum is given byγ_ in = I(4), where I(4) is the four-by-four identity matrix.The beam splitter unitary transformation is represented bythe symplectic transformation:S_ BS = [ [√(T) 0√(1-T) 0; 0√(T) 0√(1-T); -√(1-T) 0√(T) 0; 0 -√(1-T) 0√(T) ]] Similarly, the unknown phase shift is given byS_ PS = [ [cosϕ_Asinϕ_A 0 0; -sinϕ_Acosϕ_A 0 0; 0 0 1 0; 0 0 0 1 ]], and the squeezing in the first arm is given byS_ SQ(r) = [ [ e^-r000;0e^r00;0010;0001 ]] , where e^-r = √(n̅+1) - √(n̅)and e^r = √(n̅+1) + √(n̅) (remember n̅ = sinh^2 r). Then the covariance matrix of the state beforethe photo detectors is calculated to be γ_ out=S_SQ(-r) S_ BS^T S_ PS S_ BS S_SQ(r)γ_ in × S_SQ^T(r) S_ BS^T S_ PS^T S_ BS S_SQ^T (-r) where the superscript T denotes the matrix transpose. The probability of having no-clicks at both detector(i.e. the projection onto |0⟩⟨0|⊗|0⟩⟨0|)is given by <cit.>, P_00 = 4/√( (γ_ out + I(4))) .Then the Fisher information for ϕ_A is calculated byF(ϕ_1) = 1/P_00(d P_00/d ϕ_A)^2+ 1/1-P_00(d (1-P_00)/d ϕ_A)^2 . The calculation is performed by Mathematica.Since the expression of F(ϕ_A) is quite complicated,we consider the limit of small ϕ_A.Then we getlim_ϕ_A → 0 F =4 n̅ T (1+ T + 2 n̅T) . Replacing n̅ with n̅_ tot = 2n̅, we getlim_ϕ_A → 0 F =2 n̅_ tot T (1+ T + n̅_ tot T) , which implies that in the limit of small phase shifts,the Fisher information of our protocol can surpass the SNL in terms ofthe total resource for any T0, and particularly for T=1/2. | http://arxiv.org/abs/1705.09506v1 | {
"authors": [
"Masahiro Takeoka",
"Kaushik P. Seshadreesan",
"Chenglong You",
"Shuro Izumi",
"Jonathan P. Dowling"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170526100016",
"title": "Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum"
} |
Edge waves and localisation in lattices containing tilted resonators [=======================================================================§ INTRODUCTION Recently high temperature superconductivitywith T_c in the range of 43-123 K has been reported followingdifferent growth procedures in a potassium doped aromatichydrocarbon p-Terphenyl <cit.>.The superconducting crystalline phase is expected to be K_3C_18 H_14.Superconducting pairing with a large 15 meV gap opening at about 60Kwas confirmed on a K-doped surface of a p-Terphenyl single crystal <cit.>.Para-Terphenyl, a linear molecule made of a chain of 3 benzene rings,and its derivatives are aromatic biological molecules present in edible mushrooms <cit.>.Pharmaceutical research is in progress for their use as immunosuppressive, anti-inflammatory and anti-tumor agents, moreover it has technological applications as laser dye, sunscreen lotion and in photon detectors. If these results will be confirmed, K_x p-Terphenyl provides today the recordfor the highest critical temperature in carbon based materials and larger than in many cuprates oxides and in many iron based superconductors.The search for macroscopic quantum coherence at high temperature in organics and organometallics has been a long standing search for the holy grail of room temperature superconductors.Superconductivity in graphite intercalation compounds has been an active topic for several decades, but T_c rangesonly up to 11.5 K for CaC_6 <cit.>.The highest critical temperature T_c=38 K in doped fullerides A_3C_60 A=K,Rb,Cs has been found in Cs_3C_60 with A15 structure by application of 7 Kbar hydrostatic pressure <cit.>. This system provides a complex phase of condensed matter where structural polymorphism controls both magnetic and superconducting properties <cit.> and shows a fluctuating microscopic heterogeneity made of the coexistence of both localized Jahn Teller active and itinerant electrons <cit.>. Recently the material research for high temperature superconductors was oriented toward metal-intercalated aromatic hydrocarbons by the discovery of superconductivity with T_c=18Kby doping potassium into picene (C_22H_14) <cit.>. Picene is a hydrocarbon molecule made of five benzene rings condensed in an armchair manner.Looking for high T_c in alkali-metals or alkali-earth-metals dopedpolycyclic-aromatic-hydrocarbons (PAHs), the previous record T_c=33 K was held by in K_x 1,2:8,9-dibenzopentacene (C_30H_18) <cit.>.The recent indications for T_c=123K in p-Terphenyl open a new road map for the searchof higher critical temperatures in the large family of doped metal-organic compounds, with the hope to overcome the highest superconducting critical temperature known so far, T_c=203K in pressurized H_3S <cit.>. There is today high interest on understanding both the quantum mechanismbeyond the emergence of T_c=123K in p-Terphenyl and the relation betweenthe nanostructure and quantum functionality in organics. This is needed to develop novel quantum plastics materials, taking advantage that conducting polymers can be prepared in a large variety of polymericheterostructures at atomic limit <cit.> and possess the combination of easy processability,light weight, and durability.Using density functional theory and Eliashberg's theory of superconductivityin the single band approximation the superconducting critical temperature has been predictedto be in the range 3<T_c<8 K in K_3 picene <cit.> and around 6.2 Kin K_2C_6H_6 moreover, it was arguedthat all hydrocarbons should show T_c in a similar temperature range of 3<T_c<7K <cit.>. A hot topic today is the search for superconductivityin the extreme high pressure phases of benzene <cit.>. Therefore unconventional pairing mechanisms were invoked for high T_c in doped p-Terphenyl:a) the Bose-Einstein Condensation (BEC) of preformed bipolarons <cit.>;b) the Resonating Valence Bond (RVB) theory in a scenario with two coexisting doped Mott insulators <cit.>c) the so called s_+/- pairing mechanism mediated by a repulsive interaction in a scenario with two stronglycorrelated bands forming two similar Fermi surfaces connected by a nesting vector <cit.>.Here we propose the mechanism for T_c amplification driven by Fanoresonance <cit.> in a multigap superconductorbetween different gaps occurring where the chemical potential is tuned at a electronic topological Lifshitz transition <cit.>.In this regime the configuration interaction between the pairing scattering channels in the (n-1)-th bands with high Fermi energy E_F(n-1) in the BCS approximation and the n-th pairing scattering channel in the new appearing n-th small Fermi surface with a low Fermi energy E_Fn) can give a resonance in the superconducting gaps, called shape resonance<cit.> likein nuclear physics and molecular physics <cit.> or Feshbach resonance as in the jargon of ultracold gases <cit.>. The Fano resonance give high T_c domes around Lifshitz transitions in the energy range of E_Fn of the order of the energy of the pairing cut-off and the n-th condensate is in the BCS-BEC crossover <cit.>.The configuration interaction between pairing channels is determined by the symmetry and interference between the wavefunctions of electron pairs at the coexisting multiple Fermi surfaces thereforethe nanoscale material architecture plays a central role. In ref <cit.> particular material architectures made by heterostructures at atomic limit which could give high-T_c superconductors have been disclosed. Panel (A) in Fig. <ref> shows one of these heterostructures at atomic limit: a superlattice of stripes.In this case the material is made of nanoscale modules, metallic stripes of atomic thickness, as for example metallic graphene or phosphorene nanoribbonsassembled in a superlattice of stripes separated by potential barriers fromneighbor stripes in the same plane or in the neighbor plane. The size of the nanoscale units or modules and the superlattice period λ_pis required to be in the nanoscale. In fact the Fano resonance to get the highest T_c need to tune the wavelength λ_Fn of electrons at the Fermi level in the n-th band in the range of λ_p. The crystalline structure of undoped p-Terphenyl <cit.>with a monoclinic space group P21/a is shown in Panel (B) of Fig. <ref> where gray dots are carbon atoms and red dots are hydrogen atoms, can be described as a packing of parallel p-Terphenyl nanoribbons or stripes of about 1.4 nm width running in theb-axis direction indicated by the black arrows. The figure shows the similarity of the arrangements of the p-Terphenyl nanoscale stripes with the schematic drawing of the superlattice of stripesshown in panel (A) of Fig. <ref> from ref <cit.>.The projections in the ac, bc and ba crystal planes are shown in Fig. <ref>.Two parallel stripes are clearly seen in the bc projection.The ba projection in Fig. <ref> shows a variable torsion angle (up to 14 degrees) of the middle ring relative with the two lateral ones. While the H - H bond is 0.32 nm between the lateral rings of the neighbor molecules very short H - H intermolecular bonds (dashed lines connecting neighbor molecules in the stripe in the b direction) are established for the central benzene ring, as short as 0.22 nm, like the shortest values observed in the high pressure phases of benzene <cit.>. The stripes are not connected with the neighbor stripes also in the a-axis direction indicating the quasi one dimensional electronic structure of the stripes.The electronic structure of p-Terphenyl <cit.> confirms the presence of one dimensional electronic states in the conduction band. A portion of the electronic structure of the conduction band <cit.> is shown in the upper part of Fig. <ref>. The band structure, upper panel of Fig. <ref>, displays in a clear way a strong anisotropic character in the wave-vector space. Indeed, in the irreducible representation of the band structure reported in the upper partof Fig. <ref> <cit.> shows two directions in the Brillouin zone for which the band dispersions are very narrow. A narrow bandwidth of the order of hundred meV, indicates that large potential barriers have to be penetrated by the electrons to tunnel from one p-Terphenyl stripe to its neighbor stripe. Interestingly and relevantly for this work, all the bands of the band-bundle due to the different overlapping orbitals of the p-Terphenyl, have this property and are very narrow along these two independent directions, indicating that the potential barrier formed by the crystal structure and by the characteristics of the electronic bonds is felt by all the conduction and valence electrons. In the b-axis direction in the real space, orthogonal to the upper surface of the Brillouin zone the band dispersion is much broader, of the order of 700 meV, pointing toward a quasi-free conduction of the electrons with a moderate effective mass along the b direction. The energy scale of the pairing,145 meV, is taken from the relevant phonons detected at 1171 cm-1, due to a mode of the C-C bond, and/or the mode at 1222 cm-1 of the C-H mode, detected by Raman spectroscopy in metallic K_3 p-terphenyl <cit.>.We have simulated the band dispersion and the DOS with a periodic potential barrier with 1.4 nmperiodicity which reproduces the narrow band dispersion in the transversal direction of thestripes shown in the lower panel of Fig. <ref>. We have obtained an electronic structurewhich grabs the key feature of the evolution of the electronic structure where the chemical potential is tuned at a band edge in a superlattice of stripes as shown in the upper panel of Fig. <ref> showing the drawing of the patent for material design of heterostructures at atomic limit <cit.>. Once doped by K, the chemical potential will be raised in the conduction band and depending of the doping (and/or misfit strain, pressure, orientational disorder, magnetic field ...) at the n-th band edge giving rise to a complex network of Fermi surfaces, with electron and hole-like small pockets of Fermi surfaces and Fermi arcs. The key point of this work is to predict the superconducting properties of the K-doped p-Terphenyl with numerical calculations through a simplified model of its nanoscale structure getting the key electronic structure near a band edge. The bands and wave-functions are created by using a proper superlattice of stripes, which allow the solution of the Bogoliubov gap equations <cit.>without standard BCS approximations. giving high-T_c superconducting state with multigaps and multi-condensates at different BCS/BEC pairing regimes. The pairing is thought to be driven by an attractive interaction within each Fermi surface,but the non-diagonal interaction between condensates can be either repulsive or attractive.The multicomponent character of the pairing and the geometry of the system will determine shape resonances in the superconducting gaps and T_c, with peculiar features predictedfor the isotope effect and the gap to T_c ratios, which can be tested in future experiments, together with the expectedanisotropic transport of electrons in doped (or out of equilibrium) single crystals. The presence of high temperature superconducting domes where the chemical potentialcrosses the Lifshitz transitions has now well confirmed by experiments in iron based superconductors. Here we present the possible scenario of high temperature superconductivity in p-Terphenyl where the chemical potential is driven by potassium doping at a Lifshitz transition.Our model predict that at the Lifshitz transition the Fermi surface is made of multiple components: circular Fermi surface pockets and Fermi arcs as shown in Fig.4. The superconducting properties have been calculated using the Bianconi-Perali-Valletta (BPV) theory which has been used to predict the mechanism driving the emergence of high T_c incuprates <cit.>,where the 1996 proposed scenario of the coexistenceof Fermi arcs and Fermi pockets was confirmed in 2009 <cit.>. It has been able to predict the anomalous deviation of the isotopecoefficient from the BCS predicted value <cit.>. The BPV theory has been applied to diborides <cit.>,and iron based superconductors <cit.> where it has beenconfirmed by recent experiments <cit.>.In these experiments it has been reported compelling evidence thatthe high T_c domes occur in the proximityof the Lifshitz transitions for the appearing of a new Fermi surface pocket. Recently similar scenarios have been proposed for pressurized sulfur hydrides <cit.> and in superconducting nanofilms <cit.>Let us consider a system made of multiple bands with index n. The energy separation between the chemical potentialand the bottom of the n-th band defines the Fermi energy of the n-th band.This formulation was proposed for systems with a band crossing the chemical potential having a steep free electron like dispersion in the x direction and a flat band-like dispersion in the y direction. The wavefunctions of electrons at the Fermi level are calculated using a lattice with quasi-one dimensional latticepotential modulation where the chemical potential is tuned near a Lifshitz transition, like in magnesium diborides, A15, cuprates, iron based superconductors and we propose here for doped p-terphenyl.The superconducting critical temperature T_c in the BPV theory is obtained by numerical solution of the gaps equation <cit.> considering the simplest case of a two dimensional system <cit.> of stripes, but the extension to three dimensional system <cit.> is straightforward.The T_c is determined by solving the following selfconsistent system of equations,Δ_n,k_y= -1/2N∑_n',𝐤'V_𝐤,𝐤'^n,n'tanh(E_n,k'_y+ϵ_k_x-μ/2T_c)/E_n,k'_y+ϵ_k_x-μΔ_n',k'_y,where the E_n,k_y+ϵ_k_x is the energy dispersion and μ the chemical potential. We consider a superconductor with multiple gaps Δ_n,k_y in multiple bands n with flat band-like dispersion in the y direction and steep free-electron-likedispersion in the x direction for a simple model of a two dimensional metal with a one-dimensionalsuperlattice modulation in the y-direction. The self consistent equation for the gaps at T=0 where each gap depends on the other gaps is given by Δ_n,k_y=-1/2N∑_n',k'_y,k'_xV_𝐤,𝐤'^n,n'Δ_n',k'_y/√((E_n',k'_y+ϵ_k'_x-μ)^2+Δ_n',k'_y^2), where N is the total number of wave-vectors in the discrete summation and V^n,n'_𝐤,𝐤' is the effective pairing interactiontaken in the separable and energy cutoff approximation, V_𝐤,𝐤'^n,n'= V_𝐤,𝐤'^n,n'×θ(ω_0-|E_n,k_y+ϵ_k_x-μ|)θ(ω_0-|E_n',k'_y+ϵ_k'_x-μ|). Here we take account of the interference effects between thesingle-particle wave functions of the pairing electrons in the different bands, where n and n' are the band indexes, k_y(k_y') is thesuperlattice wave-vector and k_x(k_x') is the component of the wave-vector in the free-electron-like direction of the initial (final) state in the pairing process.V_𝐤,𝐤'^n,n' = - λ_n,n'/N_0S××∫_Sψ_n',-k'_y(y)ψ_n,-k_y(y) ψ_n,k_y(y)ψ_n',k'_y(y)dxdy, N_0 is the DOS at the Fermi energy E_F without the lattice modulation, λ_n,n' isthe dimensionless coupling parameter, S=L_xL_y is the surface of the plane and ψ_n,k_y(y) are the eigenfunctions in the 1D superlattice.We have used weak coupling intraband coupling constants 0.14for the (n-1)-th bands, 0.2 for the appearing n-th band and a small 0.1 interbandattractive or repulsive exchangeinterband pairing constant and the distribution of calculated intraband pairing terms(n,n) (n-1,n-1) together with the(n,n-1) interband terms is shown in Fig. <ref>.The system of equations for the gaps need to besolved iteratively. The anisotropic gaps depend onthe band index and on the superlattice wave-vector k_y. According with Leggett<cit.>the ground-state BCS wave function corresponds to an ensemble of overlapping Cooper pairs at weak coupling (BCS regime) and evolves tomolecular (non-overlapping) local pairs with bosonic character in the BEC regime. This approach remains valid also if a particular band is in the BCS-BEC crossover and beyond Migdal approximation because all other bands are in the BCS regime and in the Migdal approximation. In this crossover regime the chemical potential μresults strongly renormalized with respect to the Fermi energy E_Fn of the non interacting system. In the case of a Lifshitz transition, as described in this paper, nearly all electrons in the new appearing Fermi surface condense forming a condensate having a BCS-BEC crossover character. Therefore at any chosen value of the charge density ρ for a number of the occupied bands N_b, the chemical potential in the superconducting phase has to be renormalized by the gap opening solving the following equation: ρ =1/L_xL_y∑_n^N_b∑_k_x,k_y[1- E_n,k_y+ϵ_k_x-μ/√((E_n,k_y+ϵ_k_x-μ)^2+Δ_n,k_y^2)].Three distinct regimes of multi-gap superconductivity are obtained as a function of the chemical potential tuned around the 2D-1D Lifshitz transition, as reported in Fig. <ref>. At the n-th band bottom, when the new Fermi surface pocket starts to appear, there is a coexistence of a BCS-like condensate of the pairs of the (n-1)-th band together with a BEC-like condensate of the pairs of the n-th band, which because of the very low density condense completely in a bosonic liquid. In this regime the critical temperature is extremely low and small variations of the parameters can lead to large variations in the gaps and in the T_c, determining a large peaked value of the isotope coefficient. The peak in the isotope coefficient reported in panel (B) is a finger print of our BPV theory and it signals the strong interplay between the Lifshitz transition at the band bottom and the onset of shape resonant superconductivity. Increasing the chemical potential trough the second Lifshitz transition, the resonant regime of maximum T_c and gaps is obtained. This resonant regime is characterized by an interesting coexistence of BCS-like pair condensate of the (n-1)-th band and BCS-BEC crossover-like pair condensate of the n-th band. In the resonance regime the two gaps differ by a sizable 2.5 factor, the isotope effect gets its smallest values and T_c can reach the high value of 123 K with coupling strengths in the different channels not exceeding 0.3, values still typical of metals, as Nb. The large energy scale of the phonon (145 meV) determines not only a large prefactor for the critical temperature, but also induces a large width of the resonant regimes, making relatively simple to find the right doping levels giving the highest critical temperatures. Finally, for larger chemical potentials, a third regime of conventional two-band superconductivity is reached, with coexistence of two-particles condensates having both BCS-like character, confirmed also by values of the isotope coefficient approaching 0.5 and small values of the gaps typical of weakly-coupled superconductors. The predicted zero temperature superconducting gaps to T_c ratios, panel (A), and the Isotope coefficient as function on the critical temperature, panel (B) are reported in Fig. <ref> so that the present theoretical work can be confirmed by direct experiments. This theory does not include superconducting fluctuations.However it should be noticed that in a multigap / multiband system another fundamental phenomenon helps in stabilizing high temperature superconductivity: the screening of the superconducting fluctuations.In fact the multi-band BCS-BEC crossover in a two-bandsuperconductor (one condensate in the BCS regime, the other in the BCS-BEC regime) can determine the optimal condition to allow the screening of the detrimental superconducting fluctuations due to the large stiffness of the BCS-like condensate in the deep band. Finally an arrested or frustrated phase separation is expected to occur at a topological Lifshitz transition <cit.> as it was observed in cuprates, and diborides <cit.>, diborides <cit.>. This work has been supported by superstripes-onlus. Discussions with Gianni Profeta, Augusto Marcelli, Nicola Poccia, Andrea Ienco, Corrado Di Nicola, Fabio Marchetti, Claudio Pettinari are acknowledged. 0aGao Y., Wang R.-S., Wu X.-L., Cheng J., Deng T.-G., Yan X.-W. Huang, Z.-B Acta Physica Sinica652016077402. bWang R.-S., Gao Y., Huang Z.-B. Chen X.-J.Arxiv:1703.058042017cWang R.-S., Gao Y., Huang Z.-B.Chen X.-J.Arxiv:1703.066412017 dLi H., Zhou X., Parham S., Nummy T., Griffith J., Gordon K., Chronister E. L. Dessau D. S.Arxiv:1704.042302017eLiu J.-K., Hu L., Dong Z.-J. Hu Q. Chemistry & Biodiversity12004 601-605.fWeller T. E. et al.Nature Phys.1200539-41ganin1Ganin A. Y., Takabayashi Y., Khimyak Y. Z., et al. Nature Materials72008367-371. ganin2Ganin A. Y., Takabayashi Y., Jeglic P. et al. Nature4662010221-225.zadikZadik R. H., Takabayashi Y., Klupp G., et al.Science Advances1 2015e1500059. mitsuMitsuhashi R. et al. 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Tosatti E. arxiv:1705.050662017 cuprates1 Bianconi A.Solid State Communications891994933.cuprates0Bianconi A.US Patent 6,265,0192001priority date Dec 7, 1993.LifshitzLifshitz I.M.Sov. Phys. JETP1119601130.Lifshitz1Volovik G.E. Low Temperature Physics43201747-55.shapeBianconi A.J. Phys.: Conf. Ser.4492013012002. shape2 Bianconi A.Journal of Superconductivity182005625. vittorini Vittorini-Orgeas A.Bianconi A.Supercond. Nov. Magn.222009215.fes10Chin C., Grimm R., Julienne P.Tiesinga E.Rev. Mod. Phys.8220101225. perali-bcs-bec Perali A., Pieri P.Strinati G.C.Phys. Rev. Lett.932004100404. Guidini-bcs-becGuidini A. Perali A.Supercond. Sci. Technol.272014124002.riceRice A., Tham F. Chronister E. Journal of Chemical Crystallography43201314-25. lechLechner R. E., Toudic B. Cailleau H.J. Phys. C: Solid State Phys.171984405-420.puscPuschnig P., Heimel G., Weinmeier K., Resel R. Ambrosch-Draxl C. 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Technol. 25 2012 124002 isotope4Innocenti D., Bianconi A.Supercond. Nov. Magn.2620131319-1324. mgb0Bianconi A., Di Castro D., Agrestini S., Campi G., Saini N. L., Saccone A., De Negri S. Giovannini M. Journal of Physics: Condensed Matter1320017383. mgb3 Bussmann-Holder A. Bianconi A.Phys. Rev. B 672003132509.mgb7 Innocenti D. et al. Phys. Rev. B822010184528. iron1Caivano, R. et al. Supercond. Sci. Technol.222009014004. iron2 Innocenti D., Valletta A. Bianconi A.Supercond. Nov. Magn.2420111137. iron3 Bianconi A. Nature Physics 92013536.iron4Kordyuk A.A. et al.Supercond. Nov. Magn.2620132837. iron5Kordyuk A.A. Low Temperature Physics 412015319-341. iron6C. Liu et al.Physical Review B 842011 020509. iron7 Ideta S. et al.Phys. Rev. Lett.1102013107007. iron10Charnukha A. et al. Scientific Reports5201510392. annette2Bussmann-Holder A. et al.Supercond. Nov. Magn.302017151-156doriaDoria M. M., Cariglia M. Perali A. Physical Review B 942016224513. legLeggett A. J.Modern Trends in the Theory of Condensed Matter,Lecture Notes in Physics115Pekalski A. Przystawa R.Springer-Verlag, Berlin1980 kugel1Kugel K.I., Rakhmanov A.L., Sboychakov A.O., Poccia N.Bianconi, A.Phys. Rev. B782008165124. kugel2Bianconi A., Poccia N., Sboychakov A.O., Rakhmanov A.L. Kugel K.I. Supercond. Sci. Technol.282015024005. campi1Campi et al.Nature5252015359-362. saini1Saini, N. L. et al.Eur. Phys. J. B36200375-80. campi2Campi, G. et al. Eur. Phys. J. B52200615-21. | http://arxiv.org/abs/1705.09690v1 | {
"authors": [
"Maria Vittoria Mazziotti",
"Antonio Valletta",
"Gaetano Campi",
"Davide Innocenti",
"Andrea Perali",
"Antonio Bianconi"
],
"categories": [
"cond-mat.supr-con"
],
"primary_category": "cond-mat.supr-con",
"published": "20170526192849",
"title": "Possible Fano resonance for high Tc multi-gap superconductivity in p-Terphenyl doped by K at the Lifshitz transition"
} |
Skip-Gram with Negative Sampling is a Weighted Logistic PCA Shintaro Koshida1, Yuzuru Yoshii2,3,[8], Yukiyasu Kobayashi4, Takeo Minezaki2, Keigo Enya5, Masahiro Suganuma4, Hiroyuki Tomita2, Tsutomu Aoki6, and Bruce A. Peterson7=========================================================================================================================================================================== We show that the skip-gram formulation oftrained with negative sampling is equivalent to a weighted logistic PCA. This connection allows us to better understand the objective, compare it to other word embedding methods, and extend it to higher dimensional models. § BACKGROUND <cit.> introduced the skip-gram formulation for neural word embeddings, wherein one tries to predict the context of a given word. Their negative-sampling algorithm improved the computational feasibility of training the embeddings. Due to their state-of-the-art performance on a number of tasks, there has been much research aimed at better understanding it. <cit.> showed that skip-gram with negative-sampling algorithm (SGNS) maximizes a different likelihood than the skip-gram formulation poses and further showed how it is implicitly related to pointwise mutual information <cit.>. We show that SGNS is a weighted logistic PCA, which is a special case of exponential family PCA for the binomial likelihood.<cit.> showed that the skip-gram formulation can be viewed as exponential family PCA with a multinomial likelihood, but they did not make the connection between the negative-sampling algorithm and the binomial likelihood.<cit.> showed that SGNS is an explicit matrix factorization related to representation learning, but the matrix factorization objective they found was complicated and they did not find the connection to the binomial distribution or exponential family PCA. § WEIGHTED LOGISTIC PCA Exponential family principal component analysis is an extension of principal component analysis (PCA) to data coming from exponential family distributions. Letting Y = [y_ij] be a data matrix, it assumes that y_ij, i = 1, …, n, j = 1, …, d, are generated from an exponential family distribution with corresponding natural parameters θ_ij. Exponential family PCA decomposes Θ = [θ_ij] = AB^T, where A∈ℝ^n × f, B∈ℝ^d × f, and f < min(n, d). This implies that θ_ij = a_i^T b_j, where a_i ∈ℝ^f is the ith row of A and b_j ∈ℝ^f is the jth row of B. When the exponential family distribution is Gaussian, this reduces to standard PCA. When it is Bernoulli (y_ij∈{0, 1}, Pr(y_ij = 1) = p_ij), this is typically called logistic PCA and log likelihood being maximized is∑_i,j y_ijθ_ij - log( 1 + exp(θ_ij)),where θ_ij = log( p_ij/1 - p_ij) is the log odds and is approximated by the lower dimensional a_i^T b_j.Just as in logistic regression, when there is more than one independent and identically distributed trial for a given (i, j) combination, the distribution becomes binomial. If there are y_ij successes out of n_ij opportunities, then the log likelihood is∑_i,j n_ij( p̂_ijθ_ij - log( 1 + exp(θ_ij)) ),where p̂_ij = y_ij/n_ij is the proportion of successes. This can be viewed as a weighted logistic PCA with responses p̂_ij and weights n_ij.§ SKIP-GRAM WITH NEGATIVE SAMPLING SGNS compares the observed word-context pairs with randomly-generated non-observed pairs and maximizes the probability of the actual word-context pairs, while minimizing the probability of the negative pairs.Let n_w, c be the number of time word w is in the context of word c, n_w and n_c be the number of times word w and context c appears, |D| be the number of word-context pairs in the corpus, P_D(w) = n_w/|D|, P_D(c) = n_c/|D|[In <cit.> define P_D(c) ∝ n_c^0.75, but without loss of generality, we use the simpler definition in this paper.], and P_D(w, c) = n_w, c/|D| be the distributions of the words, contexts, and word-context pairs, respectively, and k be the number of negative samples. Letting σ(x) = 1/1 + e^-x, <cit.> showed that SGNS maximizes∑_w ∑_c n_w, c( logσ(v_w^T v_c) + k E_c^'∼ P_D[logσ(- v_w^T v_c^')] ),where v_w and v_c are the f-dimensional vectors for word w and context c, respectively.The SGNS objective can be rewrittenℓ=∑_w ∑_c n_w, c( logσ(v_w^T v_c) + k E_c^'∼ P_D[logσ(- v_w^T v_c^')] )=∑_w{[ ∑_c n_w, clogσ(v_w^T v_c) ] +[ ∑_c n_w, c k E_c^'∼ P_D[logσ(- v_w^T v_c^')]] } =∑_w {[ ∑_c n_w, clogσ(v_w^T v_c) ] + [ k n_w E_c^'∼ P_D[logσ(- v_w^T v_c^')] ] } =∑_w {[ ∑_c n_w, clogσ(v_w^T v_c) ] + [ k n_w∑_c^' P_D(c^') logσ(- v_w^T v_c^') ] } =∑_w ∑_c { n_w, clog( exp(v_w^T v_c)/1 + exp(v_w^T v_c)) + k n_w P_D(c) log( 1/1 + exp(v_w^T v_c)) } =∑_w ∑_c { n_w, c (v_w^T v_c) - (n_w, c + k n_w P_D(c)) log( 1 + exp(v_w^T v_c) ) } =∑_w ∑_c (n_w, c + k n_w P_D(c)) ( n_w, c/n_w, c + k n_w P_D(c) (v_w^T v_c) - log( 1 + exp(v_w^T v_c) ) ). Define the proportionx_w, c = n_w, c/n_w, c + k n_w P_D(c) = P_D(w, c)/P_D(w, c) + k P_D(w) P_D(c).Then SGNS maximizes∑_w ∑_c (n_w, c + k n_w P_D(c)) ( x_w, c (v_w^T v_c) - log(1 + exp(v_w^T v_c)) ),which is logistic PCA with responses x_w, c and weights (n_w, c + k n_w P_D(c)).Multiplying by the constant 1 / |D|, the objective becomes∑_w ∑_c (P_D(w, c) + k P_D(w) P_D(c)) ( x_w, c (v_w^T v_c) - log(1 + exp(v_w^T v_c)) ),which gives the weights a slightly easier interpretation.§ IMPLICATIONS*InterpretationInterpreting the objective, weights will be larger for word-context pairs with higher number of occurrences, as well as for word and contexts with higher numbers of marginal occurrences. The response x_w, c is 0 for all non-observed pairs and will be closer to 1 if the number of word-context pair occurrences is large compared to the marginal word and context occurrences. The number of negative samples per word, k, has the effect of regularizing the proportions down from 1. Larger k's will also diminish the effect of the word-context pairs in the weights. *Comparison to Other ResultsWe can easily derive the main result from <cit.>, the implicit factorization of the pointwise mutual information (PMI), under this interpretation. For each combination of w and c, there are n_w,c positive examples and k n_w P_D(c) negative examples. The maximum likelihood estimate of the probability is x_w, c. The log odds of x_w, c is log( n_w, c |D|/n_w n_c) - log k = PMI(w, c) - log k, which is the same result as in <cit.>. *Comparison to Other MethodsWeighted logistic PCA has been used in collaborative filtering of implicit feedback data by Spotify <cit.>, where it was referred to as logistic matrix factorization. <cit.> was a modification of a previous method which performed matrix factorization with a weighted least squares objective <cit.>. <cit.> reported that weighted logistic PCA had similar accuracy to <cit.>'s weighted least squares method, but could achieve it with a smaller latent dimension.With that in mind, we can consider an alternative weighted least squares version of SGNS (SGNS-LS), ∑_w ∑_c (P_D(w, c) + k P_D(w) P_D(c)) ( x_w, c - v_w^T v_c )^2.Possible advantages include improved computational efficiency and a further comparison with GloVe <cit.>, which also uses a weighted least squares objective. Ignoring the word and context bias terms, GloVe's objective is∑_w ∑_c f(n_w, c) ( log n_w, c - v_w^T v_c )^2,where f(n_w, c) is a weighting function, which equals 0 when n_w, c is 0, effectively removing the non-observed word-context pairs.Comparing the two objectives, they both have weights increasing as a function of n_w,c, but SGNS-LS's weights are dependent on the number of marginal occurrences of the words and contexts. Both methods transform the number of word-context occurrences, SGNS-LS converting it to a proportion and GloVe taking the log. We believe the weighting scheme for SGNS-LS has a conceptual advantage over that of GloVe. For example, let n_i, j = n_k, l = 1 with n_i ≫ n_k and n_j ≫ n_l. GloVe treats them both the same, but SGNS-LS will have x_i, j < x_k, l and will give more weight to x_i, j because n_i, j being small is much more unlikely due to random chance than n_k, l being small. *TrainingThe connection of SGNS to weighted logistic PCA allows us to conceive of other methods to train the word and context vectors. For example, once the sparse word-context matrix has been created, one can either use the MapReduce framework of <cit.> or GloVe's approach: sample elements of the matrix and perform stochastic gradient descent with AdaGrad (and similarly for SGNS-LS, with different gradients). GloVe only samples non-zero elements of the matrix, whereas SGNS(-LS) must sample all elements, because the non-occurrence is important for SGNS. *ExtensionFinally, with this connection to logistic PCA, SGNS can be extended to include other factors in a higher order tensor factorization, analogous to the extension for skip-gram described in <cit.>. Of particular interest is training document vectors along with the word and context vectors. chicago | http://arxiv.org/abs/1705.09755v1 | {
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The Locus of the apices]The Locus of the apices of projectile trajectories under constant drag Departamento de Ciencias Básicas,Universidad Autónoma Metropolitana at Azcapotzalco,Av. San Pablo 180, México 02200 D.F., Mexico.We present an analytical solution for the projectile coplanar motion under constant drag parametrisedby the velocity angle. We found the locus formed by theapices of the projectile trajectories. The range and time offlight are obtained numerically and we find that the optimal launching angle is smaller than in the free dragcase. This is a good example of problems with constant dissipation of energy that includes curvature,and it is proper for intermediate courses of mechanics.[ H. Hernández-Saldaña December 30, 2023 ========================§ INTRODUCTIONProjectile trajectory under constantdrag has deserved a lot of attention in literature, not only as it appears as a common problem in undergraduate physicsand can be given recent exact analytical results and new analysis<cit.>.The power-law velocity dependent dragf⃗ = -∑_n mg b_n v^n v⃗v,with v=||v⃗||, is a series approximation for the real complex problem. The linear, n=1, and quadratic, n=2, cases are of much used, not only for the analysis of the motion ofa particle in midair butas well to model other energy dissipation process. In the quantum scales, n=1 is a usual model for theenergy losses<cit.>. Notwithstanding the usefulness of linear approximation, and that allows analytical solutions for the projectile motion, equation (<ref>) have another case: n=0, the constant drag case.It is not trivial if, as it is usual, the projectile motion is coplanar, i.e., the vector v⃗/v changes with the orientation of the orbit.[In one dimension this term only changes the sign of the force in order to keep it in opposition to the movement.]This case has deserved few attention, the only reported work we found are <cit.>.There is not evidence that exists a regime where the drag could be considered constant, however the problem studied here is important for the following reasons: i)A series expansion for a retarding force has to have a no null zeroth term, take for instance,theintegrable Legendre casesf (v) = 1n(a+b v^n),where the constant a ≠ 0 appears<cit.>. ii) The motion of an object in a non-newtonian fluid with yield stress could be constant, see forinstance <cit.>, i.e., the problem of a particle launched in oil or liquid chocolate contains this constant term. Even more, spheres into loose granular media are another example of an object moving in afluid with presence of yield stress<cit.>.iii) As an undergraduate problem, aconstant retarding force could be considered as a point rocket with the thrust pointed against the motion. iv) As a simple example of friction that depends on the curvature.In the present paper we analyse such a case, obtaining both the explicit solutions of the problem in the next section andthe description of the locus which give title to this work. We discuss the range and the time of flight are given in section <ref>. Conclusions are presented in section <ref>. § THE PROJECTILE PROBLEM WITH CONSTANT DRAG The constant drag problem is governed by the following equations in rectangular coordinatesm d^2/dt^2r⃗ = -mg-mgb_0 v⃗/v.Notice that the friction is constantin the direction of motion, i.e., it changes with velocity. We choose the drag force in units of weight mg in order to compare with linear and quadratic drag results. In order to be clear on what kind of differential equations we are dealing with, we explicitly rewritethe above equations in cartesian coordinates[m ẍ=-mgb_0 ẋ√(ẋ^2+ẏ^2),;m ÿ= -mg -mg b_0 ẏ√(ẋ^2+ẏ^2). ] Here, we use a dot for a time derivative. The above equations are coupled and non-linear. However an analytical solution parametrised withthe velocity angle can be obtained. Someother results require of standard numerical methods<cit.>. The solutions presented here for x and y do not requiere of any further numerical integration. § EXPLICIT SOLUTION PARAMETRISED BY Θ. In order to obtain a solution of the problem (<ref>) we first change the equations fornormal, n, and tangent, t, coordinates to themotion, hence, the corresponding force components areF_t = -mg sinθ -mg b_0,andF_n = -mg cosθ.If the mass is constant, we obtain m v̇ = -mg sinθ -mg - mg b_0andm v^2/ρ = -mg cosθ.where ρ = -d s/ d θ and, s is the arc length. The last equationcan be written as v θ̇ = -g cosθ,with the help of the chain rule: ρ = -d s/ d θ = - (d s/d t) (d t/d θ ). Equation (<ref>) for the tangent acceleration can be modified with the same rule and using (<ref>) the result is d v/d θ = v (tanθ + b_0 θ).For the initial conditions v(t=0) = v_0 and θ(t=0) = θ_0, we solve this first order differential equation obtaining v(θ) = v_0 cosθ_0 /cosθ(Δ/Δ_0),withΔ≡ (θ + tanθ)^b_0,and Δ_0 ≡Δ(θ_0).The solution for time is[ t(θ)=-1g∫_θ_0^θ v(θ) θ d θ; = -v_0 cosθ_0 g Δ_0 ((b_0- sinθ) Δ(b_0^2-1) η- (b_0- sinθ_0) Δ_0(b_0^2-1)η_0), ]beingη = (cosθ/2 -sinθ/2 ) (cosθ/2 + sinθ/2 ),and η_0 ≡η(θ_0).Using a similar procedure we obtain [ x(θ)= -1/g∫_θ_0^θ v(θ)^2 d θ; = -1 g (v_0 cosθ_0Δ_0)^2[ -(-2 b_0 + sinθ) Δ^2(2 b_0-1)(2b_0+1) η+(-2 b_0 + sinθ_0) Δ_0^2(2 b_0-1)(2 b_0+1)η_0], ]and[y(θ) = -1/g∫_θ_0^θ v(θ)^2 tanθ d θ; = -1 g (v_0 cosθ_0Δ_0) ^2[^2 θ(-3+cos 2 θ +4 b_0 sinθ) Δ^28 (b_0^2-1);-^2 θ_0 (-3+cos 2 θ_0 +4 b_0 sinθ_0 ) Δ_0^28 (b_0^2-1)]. ]So,(<ref>),(<ref>) and (<ref>) are, formally, the solutions tothe problem (<ref>). Unfortunately,explicit inversion of t(θ) is hard (if not imposible).Notwithstanding, these solutions are analytical and no additional integration is requiered. In search of an explicit time dependent solution homotopy analysis method could offer aguide as it was the case of quadratic drag <cit.>. In order to establish that previous expressions are as useful as the time parametrisation we shall use them to plot theusual graph of x(t) and y(t) as well asthe iconicy(x)(see <ref>).For comparison, we rewrite the free drag solutions as function of θ, the results t(θ) = -v_0 cosθ_0/g( tanθ - tanθ_0 ),x(θ) = -(v_0 cosθ_0)^2/g(tanθ - tanθ_0 ),andy(θ) = -(v_0 cosθ_0)^2/2 g(^2 θ - ^2 θ_0 ),are obtained bysolving (<ref>) and (<ref>) for b_0 = 0. It is an exercise to check that the previousexpression are the familiar solutions of parabolic motion.First we explain the solutions in angle parametrisation. To this endwe draw equation(<ref>) in figure <ref>(a), i.e.,time as function of θ for b_0 = 0.25 (black line), b_0 = 0.5 (red dashed line) and b_0 = 0.75 (blue dotted line) and the free drag case in blue dashed line, from (<ref>). The launching angle was set to θ_0 = π/4 here, other selection shall shift the graphs (not shown). The parameter θ go, asimptotically, to -π/2, since the reference frame change the orientation after the orbit reach its apex as it appears in figure <ref>.In figure <ref>b) solutions (<ref>) for x(θ) are presented in the same order as before (graphs diverging to∞ as θ→ -π/2) .The solutions (<ref>) fory(θ) are those that diverge to -∞ as θ→ -π/2. A close up of them (not shown) could show the angle wherey=0. The numerical solutions to this condition shall be discussed below. Again we draw in blue-dashed lines the drag free solutions from (<ref>) and (<ref>). In figure (<ref>)c) time solutions are presented forx(t) (upper graphs) and y(t) (lower graphs). Using (<ref>), (<ref>)and a simple computational program we can write the x(t(θ)) andx(t(θ)) data and plot it.We do that and we present the results for the same drag values and color code. We consider only the range of θ in order to show the y=0 condition. The y results show the larger the drag the shorter the maxima. The maxima are reached at a shorter times as the drag increases, as well. Finally, we present the iconic y(x) for projectile motion infigure (<ref>)d). As expected, the larger the b_0 value the shorter the path. Certainly, at first sight the paths are similar to those obtained with a linear drag, but a comparison require to compare energy losses, not similar values of b_0 and b_1<cit.>. §.§ The locus of the apices The solution in terms of the angle could be hard to handle but gives a straightforwardfor a particular locus: the locus formed by all the apices for initial launching angle θ_0. The cases for no drag<cit.> and linear drag has been studied previously<cit.>.The apex for each orbit is obtained by setting θ = 0 for x and y in (<ref>) and (<ref>) as can be seen at figure (<ref>).After rearranging factors in these equations and using (cosθ_0/2 -sinθ_0/2 ) (cosθ_0/2 + sinθ_0/2 )= cosθ_0[ Since the launching angle remain in the first quadrant], thelocus is writtenas x(θ_0) =-1/g(4 b_0^2-1)(v_0 cosθ_0 /Δ_0)^2 [ 2 b_0+(-2 b_0 + sinθ_0) Δ_0^2/cosθ_0 ]and [ y(θ_0)= 18 g (b_0^2-1)(dv_0 cosθ_0/Δ_0)^2 [ 2+ ^2 θ_0 (-3+cos 2 θ_0 +4 b_0 sinθ_0 ) Δ_0^2 ]. ] In figure (<ref>) we show the locus for parameters with values v_0 = 50 m/s, b_0= 0.15 and g = 9.81 m/s^2.The drag-free solutionx_m = ρsinθ_0 cosθ_0,y_m = ρ/2sin^2 θ_0,is shown for comparison. We add three orbits, those corresponding to launching angles θ_0 = 30^∘,θ_0 = 45^∘ and θ_0 = 60^∘ as is usual in the textbooks. §.§ Some important quantities in projectile motion: The range and the flight timeUnfortunately not all the important quantities are of mathematical significance. Meanwhile the apex has a mathematical meaning, other locus are important for practical reason. Such is the case of therange and its maximum.Their value are determined by our choose of the origin and the chord generated.The selection of the origin is determined in an arbitrary way and hence the length of the chord.Hence, it is not surprising that we need tosolve numerically (<ref>) for y = 0. This condition is translated from (<ref>) to solve the equation^2 θ(-3+cos 2 θ +4 b_0 sinθ) Δ^2= ^2 θ_0 (-3+cos 2 θ_0 +4 b_0 sinθ_0 ) Δ_0^2If we callp(θ)= ^2 θ(-3+cos 2 θ +4 b_0 sinθ) Δ^2, we are looking for solutions such that p(θ) = p(θ_0). Forsymmetrical functions the solutions is clear, but this is not the caseas can be seen in figure <ref>a).There we plot p for the indicated values of b_0 and the free-drag case ^2 θ (-4 + 2 cos^2 θ )=sec^2 θ_0 (-4 + 2 cos^2 θ_0 ) with the solution θ^* = ±θ_0 (in blue dashed line).In this figure the drag values considered are b_0=0.05 in black line, 0.15 in red dashed lineand 0.25 in dotted blue line. We add the extreme case of b_0 = 0.75 in order to show how asymmetric the curve p(θ) can be[ Calculations forvaluesofb_0 larger than 0.25 require of a better selection of the initial condition as we can be seen for b_0 = 0.75 in figure <ref>(a)]. The color code remains in the rest of the graphs.In figure<ref>b) we show the solution obtained viaNewton-Raphson for the equation p(θ ^*)-p(θ_0)=0and the corresponding case for the range x_max=x(θ ^*) as function of the launching angle in figure <ref>c). In the last figure the maximum rangeoccurred at θ_0 ≈ 0.7697,0.7226 and 0.6912 for theindicated values of b_0.All these values are smaller than θ_0 = π/4 ≈ 0.7854, the corresponding value for the drag free case (shown in blue broken line).For completeness, we present the time of flight as function of the launching anglein figure <ref>(d). Such a timeincreases with the angle. Notice that the drag free case and the solution for b_0 = 0.05 are so close that they appear superimposed.§ CONCLUSIONS We discussed the motion of a projectile under the influence of constant gravitational pull and constant drag. Such a case could be considered as the yieldstress in a non-newtonian fluid and as an example of a simple situation where the retarding force depends on the velocity direction. The two coupled non-linear differential equations in rectangular coordinates can be exactly solved bya change to normal and tangent coordinates. The solutions, (<ref>) and (<ref>), are parametrised as functions of the velocity angle. That allow us to obtain the locus of the apices in an explicit way. Other locus or quantities requiere of numerical calculation as the range and flight time presented in the previous section. This problem serves as a good example for introduce undergraduate students to problems with curvature and retarding forces,beyond the problem of an inclined plane with constant friction. § ACKNOWLEGMENTWe thank to AL Salas-Brito for valuable comments. § REFERENCES The text booksThe text book for university and college physicsThe intermediate mechanics cases. oogoogle See for instance . Knuth1 R.M. Corless, G.H. Gonnet, G.H. Hare, D.E.G. Jeffrey,and D.E. Knuth,“On the Lambert W function”, Adv. in Comp. Mathematics5, 329-359 (1996). Scott T.C. Scott, A.Lüchow, D. Bressanini, and J.D. Morgan III, “The Nodalsurfaces of Helium atom eigenfunctions”, Phys. Rev. A,75. 060101-060104 (2007). Wang R.D.H. Warburton and J. Wang, “Analysis of asymptotic motion with air resistance using theLambert W function”, Am. J. Phys.72, 1404-1407 (2004). PackelandYuen E. Packel and D. Yuen, “Projectile motion with resistance andthe Lambert function”, Coll. Math. J. 35(5), 337-350 (2004).SalasJ.L.Fernández-Chapou, A.L. Salas-Brito, and C.A. Vargas, “An elliptic property of parabolic trajectories”, Am. J. Phys.72,1109-1109 (2004).MacMillan W.D. MacMillan,Theoretical Mechanics: Static and the Dynamics of a Particle(McGraw-Hill, New York and London, 1927). Reprinted in (Dover, New York, 1958), pp. 249-254. Thomas G.B. Thomas, M.B. Weir, J. Hass, F.R. Giordano.Calculus. 11th Ed. (Addisson-Wesley, 2004). Steward2005a S.M. Steward, “Alittleintroductory and intermediate physics with theLambert function”, Proc. of the 16th BiennialCongress of the Australian Institute of Physics. M. Colla ed. Vol 2. pp 194-197. Australian Institute of Physics. Parville, VIC (2005). Stewart2006 S.M. Stewart, “Characteristics of the trajectory of a projectile in a linear resisting medium and the Lambert W function”, Australian Inst. of Physics. 17th. National Congress 2006. Paper No. WC0035.(2006). yabusita K. Yabusita, M. Yamashita and K. Tsuboi, “An analytic solution fo projectile motion with the quadratic resistence law using the homotopy analysis method." J. Phys. A: Math Theor. 40 8403-16 (2007) Razhavy Datta For CancunPartisanship or corporatism: a comparison ofmathematical models and actual data in Mexican elections.How we vote and what influence it deserved a lot of attention fromphysicists and mathematicians during the last two decades. Researches andmodels for "power laws" in electoralresults are common in literature.However, and unfortunately, politician are involved within, so, deviations from our beloved models appear.In this work we discuss how corporatism, more than partisanship, plays a role on elections in the Mexican case. We look for the some evidence of such a behaviour in Indian and Argentinian cases. | http://arxiv.org/abs/1705.10597v1 | {
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thmTheorem[section] acknowledgement[theorem]Acknowledgement algorithm[theorem]Algorithm axiom[theorem]Axiom case[theorem]Case claim[theorem]Claim conclusion[theorem]Conclusion condition[theorem]Condition conjecture[theorem]Conjecture criterion[theorem]Criterion example[theorem]Example exercise[theorem]Exercise notation[theorem]Notation problem[theorem]Problem remark[theorem]Remark solution[theorem]Solution summary[theorem]Summary | http://arxiv.org/abs/1705.09350v1 | {
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Direct Multitype Cardiac Indices Estimation via Joint Representation and Regression Learning Wufeng Xue, Ali Islam, Mousumi Bhaduri, and Shuo Li* Copyright (c) 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]. W. Xue, A. Islam, M. Bhaduri and S. Li are with the Department of Medical Imaging, Western University, London, ON N6A 3K7, Canada. W. Xue and S. Li are also with the Digital Imaging Group of London, London, ON N6A 3K7, Canada. * Corresponding author. (E-mail: [email protected]) =======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================Cardiac indices estimation is of great importance during identification and diagnosis of cardiac disease in clinical routine. However, estimation of multitype cardiac indices with consistently reliable and high accuracy is still a great challenge due to the high variability of cardiac structures and complexity of temporal dynamics in cardiac MR sequences.While efforts have been devoted into cardiac volumes estimation through feature engineering followed by a independent regression model, these methods suffer from the vulnerable feature representation and incompatible regression model.In this paper, we propose a semi-automated method for multitype cardiac indices estimation. After manual labelling of two landmarks for ROI cropping, an integrated deep neural network Indices-Net is designed to jointly learn the representation and regression models. It comprises two tightly-coupled networks: a deep convolution autoencoder (DCAE) for cardiac image representation, and a multiple output convolution neural network (CNN) for indices regression. Joint learning of the two networks effectively enhances the expressiveness of image representation with respect to cardiac indices, and the compatibility between image representation and indices regression, thus leading to accurate and reliable estimations for all the cardiac indices.When applied with five-fold cross validation on MR images of 145 subjects, Indices-Net achieves consistently low estimation error for LV wall thicknesses (1.44±0.71mm) and areas of cavity and myocardium (204±133mm^2). It outperforms, with significant error reductions, segmentation method (55.1% and 17.4%) and two-phase direct volume-only methods (12.7% and 14.6%) for wall thicknesses and areas, respectively. These advantages endow the proposed method a great potential in clinical cardiac function assessment. multitype cardiac indices, direct estimation, joint learning, deep convolution autoencoder, cardiac MR. § INTRODUCTION Cardiac disease is one of the leading cause of morbidity and mortality around the world. Accurate estimation of cardiac indices from cardiac MR images plays a critical role in early diagnosis and identification of cardiac disease. Cardiac indices are quantitative anatomical or functional information (wall thickness, cavity area, myocardium area, and ejection fraction (EF), etc.) of the heart, which help distinguish between pathology and health <cit.>. Two categories of solutions exist for cardiac indices estimation: traditional methods <cit.> and direct methods <cit.>. Traditional methods rely on the premise of cardiac segmentation, from which cardiac indices are then manually measured. However, obtaining good and robust segmentation is still a great challenge due to the diverse structure and complicate temporal dynamics of cardiac sequences, therefore resulting requirements of strong prior information and user interaction <cit.>. To circumvent these limitations, direct methods without segmentation have grown in popularity in cardiac volumes estimation, and obtained effective performance benefiting from machine learning algorithms <cit.>. Despite their effective performance for volume estimation, existing direct methods can not obtain satisfactory results for multitype cardiac indices estimation. 1) They are not designed for multitype cardiac indices estimation. Existing direct methods focus on estimation of volumes of ventricles and atriums only, whereas cardiac indices are far more than these volumes <cit.>. For other indices, such as wall thickness and myocardium area, more challenges compared to volume estimation arise (see Section <ref>). 2) They do not learn image representation and regression models jointly and cannot make them adapt to and benefit from each other maximally. Existing direct methods follow a two-phase framework, where image representation is usually based on task-unaware hand-crafted features and the regression model is then learned separately, therefore obtain only inferior performance. To provide an accurate and reliable solution for multitype cardiac indices estimation, we propose a semi-automated method based on an integrated deep neural network Indices-Net. It comprises two tightly-coupled networks: a deep convolution autoencoder (DCAE) for cardiac image representation, and a multiple output convolution neural network (CNN) for indices regression. When DCAE and CNN are learned jointly with proper initialization, Indices-Net can 1) remarkably enhance the expressiveness of image representation and the compatibility between image representation and indices regression models; and 2) simultaneously estimate with high accuracy two types of cardiac indices, i.e., linear indices (6 regional myocardium wall thicknesses), and planar indices (areas of LV myocardium and cavity). Experiments on 145 subjects show that Indices-Net achieves the lowest estimation error for linear indices (1.44±0.71mm) and planar indices (204±133mm^2). Fig. <ref> demonstrates how the proposed method differs from segmentation-based methods and existing two-phase volume-only direct methods. §.§ Multitype cardiac indicesTwo types of cardiac indices <cit.> that describe anatomical information are to be estimated: linear indices (i.e, regional wall thickness of LV myocardium) and planar indices (i.e, areas of LV cavity and myocardium), as demonstrated in the right of Fig. <ref>. These indices are closely related to regional and global cardiac function assessment, respectively. Detailed definitions and clinical roles of more cardiac indices can be found in Indices of cardiac function of <cit.>. Despite their clinical significance <cit.>, multitype cardiac indices estimation has never been explored in existing direct methods <cit.>. These methods only focus on volume estimation, which can be simplified as integration of cavity area along the long axis and is less difficult to estimate due to the high contrast of the boundary between LV cavity and myocardium, and the high density of the cavity area in cardiac MR images. More challenges arise during estimation of the above mentioned multitype cardiac indices. 1) Linear indices differ from planar ones in their relation with 2D spatial image structures, therefore more relevant and robust image representation is required to estimate both of them. 2) As for the specific indices, regional wall thicknesses and myocardium area are susceptible to the complicated dynamic deformation of myocardium during the cardiac cycle, and the invisible epicardium boundary near the lateral free wall. Regional wall thicknesses are also subject to the orientation of myocardium segments in different regions. The representation and regression models should be capable of tolerating the dynamic deformation, the imperceptible boundary and the orientation variation. §.§ Existing two-phase direct methods Two-phase framework employed in existing direct methods <cit.> is inadequate to achieve accurate estimation for multitype cardiac indices, for the reason that image representation and indices regression are separately handled, and no feedback connection exists between them during optimization. In the work of <cit.>, LV cavity area estimation was conducted through feeding directly to the neural network the proposed image statistics, which were based on the Bhattacharyya coefficient between image distributions. Avoiding the requirement of segmentation, this method still needs user interaction to indicate two closed curves within and enclosing the cavity. The proposed statistical features were further employed in the detection of regional LV abnormalities <cit.>, where manual segmentation of a reference frame was required. In the work of <cit.>, a Bayesian framework was build for bi-ventricular volume estimation based on multiple appearance features such as blob, homogeneity, and edge. Besides its intensive computation burden, an over simple linear correlation between areas of the two ventricular was taken into account in the prior model. Another direct estimation of bi-ventricular volumes was proposed in <cit.> with low level image features, i.e, Gabor features, HOG, and intensity, as input and random forest (RF) <cit.> for feature selection and regression. These handcrafted features were further replaced with a more effective image representation that learned from a multiscale convolutional deep belief network (MCDBN) <cit.>, leading to improved correlation between the estimated volumes and their ground truth. Supervised descriptor learning (SDL) was proposed in the work of four chamber volumes estimation <cit.>, which still employed a separate adaptive K-clustering random forest (AKRF) regression <cit.>. The compatibility between the descriptor and the regression model still can not be enhanced in the two-phase framework. §.§ Deep neural networkDeep neural networks have demonstrated great power in a broad range of visual applications, as well as medical image analysis <cit.> for the capability of learning effective hierarchy representations in an end-to-end fashion <cit.>. Image representation obtained in such a way is endowed with more expressiveness with respect to the manifold structures of the image space and the target space. This property makes deep network quite suitable for the problem of cardiac indices estimation. However, in the area of cardiac image, deep networks are mostly deployed in segmentation with dense supervision of manually segmented results. Deep belief network, stacked autoencoder, and convolution neural network have been employed in cardiac segmentation with optional refinement by traditional models <cit.>. Fully convolution network (FCN) was applied to cardiac segmentation <cit.> due to its success in semantic segmentation of natural image <cit.>. Recurrent FCN was later proposed to leverage inter-slice spatial dependencies in cardiac segmentation <cit.>. Only one work for cardiac volume estimation <cit.> was proposed leveraging the hierarchy representation of deep neural network. In this work, a volume estimation CNN network took as input end-systole and end-diastole cardiac images (chosen by thresholding) of all slice positions, and output only two volume estimations for frames of end-systole and end-diastole. This makes the learning procedure more data-demanding and the network incapable of giving frame-wise prediction for detailed and reliable cardiac function assessment. Our proposed method is capable of estimating multitype cardiac indices for all frames throughout the whole cardiac cycle. The remainder of this paper is organized as follows. Section II describes in detail the proposed Indices-Net, including the joint learning procedure, the representation network DCAE and the regression network CNN. Section III gives detailed descriptions of dataset, configuration, evaluation, and experiments. The results are reported and analyzed in Section IV. Conclusions and discussions are given in Section V. § METHODOLOGY We propose an integrated deep network Indices-Net (Fig. <ref>) to jointly learn the image representation and regression for multitype cardiac indices estimation. A deep convolution autoencoder DCAE (19 layers) is designed to extract common expressive representation for all the indices, leveraging the capability of extracting discriminative feature and reconstructing the image in a generative manner. A shallow network CNN (3 layers) is tightly coupled to DCAE, to further extract index-specific feature and predict the corresponding cardiac index. Joint learning of DCAE and CNN remarkably enhances the expressiveness of image representation and the compatibility between image representation and the regression, therefore leads to highly reliable and accurate estimations. The learning procedure and details of the two networks are given below. §.§ Joint learning of representation and regression Joint learning intends to improve the expressiveness of image representation with supervision of cardiac indices, and alleviates the demanding of complicated regression model. In our work, joint learning of DCAE and CNN is implemented by iterated forward/backward propagation of the integrated whole network. During forward propagation (solid arrows in Fig. <ref>), DCAE extracts common image representations from cardiac MR images, and then CNN disentangles from this representation index-specific features for each index, from which the indices are predicted. During backward propagation (dashed arrows in Fig. <ref>), the estimation error of cardiac indices is then back-propagated to update the parameters in CNN, and further backward through all layers of DCAE, to update the common representation for cardiac images and embed indices information into this representation. Iteration of this alternative forward/backward propagation constantly enhances the expressiveness of the obtained image representation with respect to cardiac indices, as well as its compatibility with the regression network. Once the learning procedure is finished, one-pass forward propagation is sufficient to achieve estimations of cardiac indices for novel images. §.§ Cardiac image representation learning with DCAE In this work, the representation network is designed as a customized deep convolution autoencoder (DCAE), with input dimension and filter numbers for each layer accommodated to the single-channel gray cardiac images. DCAE is composed of two subparts: convolution layers which are capable of extracting latent discriminative structural features from the input cardiac images; and deconvolution layers which are capable of reconstructing the output from these latent features in a generative manner. With both of them, DCAE is capable of building a cascade mapping of cardiac image (input layer)→ latent representation (fc6)→ indices-relevant structure by the bottom-up discriminative encoder (convolution layers) and the top-down generative decoder (deconvolution layers) together. In such a way, expressive representations with respect to the cardiac indices can be obtained. Convolution autoencoder (CAE) was firstly introduced in <cit.> by two convolution layers with their weight matrices being transposed of each other. A stack of CAE was trained to initialize a CNN-based classifier. The convolution layer with transposed weight matrix is later replaced by a deconvolution layer whose weights are learnable <cit.>. The deconvolution layer was proposed in <cit.> to build low and mid-level image representation in a generative manner and is closely related to convolutional sparse coding <cit.>. The learned filters act as the basis in sparse coding and are capable of capturing rich structures of different types.The architecture of our DCAE is shown in Fig. <ref>(a). The subpart of convolution has 9 convolution layers, with batch normalization layer and ReLU layer following each of them. Batch normalization layer <cit.> helps reduce internal covariance shift in very deep network and accelerate the network convergence. Max pooling is performed every two convolution layers. The subpart of deconvolution is a mirrored version of the convolution subpart, with convolution layer and pooling layer replaced by deconvolution layer and unpooling layer. A fully connected layer in between connects the two parts, transforming the features obtained from each part. In DCAE, two types of layers are important: deconvolution layer and unpooling layer. §.§.§ DeconvolutionDenoted by z_l∈ R^n_l× n_l× c_l the output of layer l with c_l channels, each associated with kernel h_l,k∈ R^m× m× c_l-1, k=1,...c_l, the convolution layer can be described as: z_l,k = f(z_l-1∗ h_l,k), k=1,...c_l.f denotes the element-wise nonlinear transformation of ReLU and batch normalization.Define ℛ_m(x) a operation which extracts all the patches of size m× m in x along the two spatial dimensions and rearrange them into columns of a matrix, the convolution layer with all output channels can be reformulated as:𝐳_l=f(𝐡_l^T𝐳_l-1)with 𝐳_l-1=ℛ_m(z_l-1), 𝐡_l=ℛ_m(h_l), 𝐳_l=ℛ_1(z_l) 𝐳_l-1∈ R^m^2c_l-1× n_l-1^2 is the matrix form of the input, and 𝐡_l∈ R^m^2c_l-1× c_l maps multiple inputs within a receptive field to one single output in 𝐳_l∈ R^c_l× n_l^2. The deconvolution layer, on the contrary, reverses the convolution operation by associating a single activation in its input with multiple activations in the output: 𝐳_l=f(𝐡_l^T𝐳_l-1)with 𝐳_l-1=ℛ_1(z_l-1), 𝐡_l=ℛ_1(h_l), 𝐳_l=ℛ_m(z_l)where 𝐳_l-1∈ R^c_l-1× n_l-1^2, 𝐡_l∈ R^c_l-1× m^2c_l, and 𝐳_l∈ R^m^2c_l× n_l^2. The output of a deconvolution layer can then be obtained by z_l=ℛ_m^-1(𝐳_l). Note that the difference of Eq. <ref> and <ref> lies in the patch size of the operation ℛ_m. The kernel of a deconvolution layer is updated in the same way as in a convolution layer independently.§.§.§ UnpoolingAnother important layer in DCAE is the unpooling layer, which reverses the corresponding pooling layer with respect to the switches of the max pooling operation. For a max pooling operation P_m with kernel size m× m, the output is:[p_l,k,s_l,k] = P_m(z_l,k)where the pooled maps p_l,k store the values and switches s_l,k record the locations. The unpooling operation U_m takes elements in p_l,k and places them in ẑ_l,k at the locations specified by s_l,k. ẑ_l,k=U_m(p_l,k,s_l,k)Unpooling layer is particularly useful to reconstruct the structure of input image by tracing back to image space the locations of strong activations. Details of deconvolution layer and unpooling layer can be found in <cit.>.Initialization with pre-trained DCAETo alleviate the training procedure of the whole network and equip DCAE with expressiveness of cardiac image structure, DCAE is initialized with parameters pre-trained from a cardiac image autoencoder Image-DCAE. Unsupervised pre-training has been proved to behave as a form of regularization towards the parameter space and support better generalization <cit.>. In our network, Image-DCAE is constructed by adding a deconvolution layer with one output channel on top of the DCAE to reconstruct the input image, as shown in Fig. <ref>(b). After being trained with cardiac MR images (no label is required here), Image-DCAE is capable of extracting different abstract levels of structures in cardiac images. Fig. <ref> shows the feature maps (a-f) of a cardiac image (g) and its reconstructed result (h) generated by Image-DCAE. For each considered layer, we show three representative feature maps. These feature maps favor some specific structures in the cardiac image that are responsive to cardiac indices considered in this work, such as LV cavity, background, and the myocardium. As these feature maps are forward-propagated to higher deconvolution layers, finer details of the cardiac structure can be revealed. With these parameters capturing cardiac structure as initialization, it becomes more efficient to obtain indices relevant information during the iterated joint learning procedure. Benefits of this initialization can also be found in Section <ref>.§.§ Multitype cardiac indices estimation with CNN To estimate multitype cardiac indices with image representation from DCAE, we design a simple CNN with index-wise connection on top of DCAE, as shown in Fig. <ref>. We find that there is no need for a deeper or more complex regression network, since the representation network DCAE has extracted expressive information from cardiac image. With the outputs of DCAE as common representations, the first two layers in CNN aim at disentangling index-specific features from them, resulting two feature maps for each index, as shown in Fig. <ref>. The third layer estimates each cardiac index from the corresponding feature maps with a simple linear model. Because common representations obtained by DCAE are employed here, the correlations among these indices are automatically embedded through overlapping of the extracted index-specific feature maps. Fig. <ref> demonstrates for 5 cardiac images (within color rectangles) the index-specific features obtained with layer conv-reg2. In each row of the rectangle show two features maps for one cardiac index indicated by the diagrams in the leftmost column. As shown in the figure, Indices-Net can capture the most responsive features (the bright/dark regions) from cardiac images to estimate each index. The overlapping of feature maps for different indices alleviate the estimation of some challenging indices, such as WT3 and WT4, by leveraging the correlation between neighbouring indices.§ EXPERIMENTS §.§ Dataset To evaluate the performance of our method, a dataset of 2D short-axis cine MR images with labelled cardiac indices is used, which includes 2900 images from 145 subjects. These subjects are collected from 3 hospitals affiliated with two health care centers (London Healthcare Center and St. Joseph’s HealthCare) using scanners of 2 vendors (GE and Siemens). The subjects age from 16 yrs to 97 yrs, with average of 58.9 yrs. The pixel spacings of the MR images range from 0.6836 mm/pixel to 2.0833 mm/pixel, with mode of 1.5625 mm/pixel. Diverse pathologies are in presence including regional wall motion abnormalities, myocardial hypertrophy, mildly enlarged LV, atrial septal defect, LV dysfunction, etc. Each subject contains 20 frames throughout a cardiac cycle. In each frame, LV is divided into equal thirds (basal, mid-cavity, and apical) perpendicular to the long axis of the heart following the standard AHA prescription <cit.> and a representative mid-cavity slice is selected for validation of cardiac indices estimation.Several preprocessing steps are applied prior to ground truth calculation. 1) Landmark labelling. Two landmarks, i.e, junctions of the right ventricular wall with the left ventricular, are manually labelled. 2) Rotation. Each cardiac image is rotated until the line between the two landmarks is vertical. 3) ROI cropping. The ROI image is cropped as a squared region centered at the mid-perpendicular of this line and with size twice the distance between the two landmarks. 4) Resizing. All the cropped images are resized to the dimension of 80× 80.After the preprocessing, all the cardiac images are manually contoured to obtained the epicardial and endocardial borders, which are double-checked by two experienced cardiac radiologists (A. Islam and M. Bhaduri). The ground truth values of LV cavity area and myocardium area can be easily obtained by counting the pixel numbers in the segmented cavity and myocardium. The linear-type regional wall thicknesses are obtained as follows. First, myocardial thicknesses are automatically acquired from the two borders in 60 measurements using the 2D centerline method <cit.>. Then the myocardium is divided into 6 segments (as shown in Fig.4 of <cit.>), with 10 measurements per segment. Finally, these measurements are averaged per segment as the ground truth of regional wall thicknesses. Papillary muscles and trabeculations are excluded in the myocardium. The obtained two types of cardiac indices are normalized according to the image dimension (80) and area (6400), respectively. During evaluation, the obtained results are converted to physical thickness (in mm) and area (in mm^2) by reversing the resizing procedure and multiplying the pixel spacing for each subject. §.§ ConfigurationsIn our experiments, 5-fold cross validation is employed for performance evaluation and comparison. The dataset is divided into 5 groups, each containing 29 subjects. Four groups are employed to train the prediction model, and the last group is used for test. This procedure is repeated five times until the indices of all subjects are obtained. The network is implemented by Caffe <cit.> with SGD solver. The configuration of the whole network Indices-Net is shown in Table. <ref>. Learning rate and weight decay are set to (0.0001, 0.005) for Image-DCAE and (0.05, 0.02) for Indices-Net. In both procedures, ‘inv’ learning policy is employed with gamma and power being (0.001, 2) and momentum 0.9. §.§ Performance evaluationWe first evaluate the estimation accuracy for the two types of cardiac indices with two criteria: correlation coefficient (ρ) and mean absolute error (MAE) between the estimated results and the ground truth. Denoteŷ_s,f^ind and y_s,f^ind the estimated and ground truth cardiac index of the sth subject and the fth frame, where ind∈{WT1,...WT6, A1, A2}, 1≤ s≤ 145, 1≤ f≤ 20. The two criteria are calculated as follows:MAE^ind = 1/145× 20∑_s=1^145∑_f=1^20|y_s,f^ind-ŷ_s,f^ind| ρ^ind=2∑_s=1^145∑_f=1^20(y_s,f^ind-y_m^ind)(ŷ_s,f^ind-ŷ_m^ind)/∑_s=1^145∑_f=1^20((y_s,f^ind-y_m^ind)^2+(ŷ_s,f^ind-ŷ_m^ind)^2)where y_m^ind and ŷ_m^ind are the mean value of the ground truth and estimated indices ind.To further evaluate the effectiveness of the estimated anatomical indices in cardiac function assessment, two cardiac functional indices are computed: Ejection Fraction, which quantifies the quantity of blood pumped out of the heart in each beat as percentage and indicates the global cardiac function; and Wall Thickening, which quantifies the change of myocardial wall thickness during systole as percentage and reflects regional cardiac function. For the sth subject, the two functional indices are computed as: Ejection Fraction_s =y_s,ED^A1-y_s,ES^A1/y_s,ED^A1100% Wall Thickening_s=y_s,ES^WT-y_s,ED^WT/y_s,ED^WT100%where ED and ES indicate end-diastole and end-systole frames, and the superscript WT indicates the mean value of WT1∼WT6. Correlation coefficients and MAE are computed for functional indices. §.§ ExperimentsExtensive experiments are conducted to validate the effectiveness of our Indices-DCAE from the following aspects.Firstly, performance of Indices-DCAE for multitype cardiac indices estimation and its effectiveness for cardiac function assessment are examined with our dataset following the five-fold cross validation protocol. Secondly, advantages of Indices-DCAE over existing segmentation-based and direct methods are extensively analyzed following the same five-fold cross validation protocol[The implementation of <cit.> is from the original author, while the rest <cit.> are implemented by ourselves strictly following the original papers.]. Statistical significance of the better performance of Indices-DCAE is examined by one-tailed F-test with significance level of 1%. The test statistic is variance ratio F=σ_0^2/σ^2, where σ_0^2 and σ^2 are variances of the estimation errors (which is essentially the mean squared error given the fact that the mean value of estimation error is near zero) for Indices-Net and one competitor to compare with. A test result of H=1 indicates that Indices-Net achieves significantly lower estimation error variance than its competitor. Thirdly, benefit of joint learning is evidenced. Joint learning is capable of enhancing the expressiveness of the representation model and thus make it more compatible with the regression model. To demonstrate this, the expressiveness of two representations is compared in terms of cardiac indices estimation: 1) F_Indices, which is obtained from Indices-Net, as jointly learned feature; 2) and F_Image, which is obtained from Image-DCAE, as non-jointly learned feature. Both of them are computed from the 16 feature maps of the last deconvolution layer in DCAE by averaging values of all non-overlapping 5× 5 blocks in these feature maps, as the way in GIST descriptor <cit.>, resulting a feature vector of length 4096 for each image. Once the two representation are available, random forest (RF) models with the same configuration (ntree=1000, mtry=200) are applied to them for cardiac indices estimation following the five-fold cross validation protocol.Fourthly, benefit of our initialization strategy is evidenced by comparing the performance of Indices-DCAE with and without initialization from pre-trained DCAE.Finally, two other deep networks for cardiac indices estimation are examined. 1) deep CNN, which contains only the convolution part of our DCAE followed by a additional fully connected layer for indices estimation. 2) FCN, which contains the first 10 convolution layers of FCN32s <cit.>, followed by convolution layers of 1024@5× 5, 1024@1× 1 and 8@1× 1 for indices estimation. Three models are examined for cardiac indices estimation: 1) deep CNN 1, which is trained from scratch; 2) deep CNN 2, which is finetuned from our pre-trained DCAE; 3) FCN, which is finetuned from PASCAL VOC-trained FCN-32s model (https://github.com/shelhamer/fcn.berkeleyvision.org).§ RESULTS AND ANALYSISIn this section, we demonstrate the results of the above mentioned experiments to validate the effectiveness of Indices-Net in the task of multitype cardiac indices estimation, as well as its advantages over existing segmentation-based, two-phase direct methods, and other deep architectures. §.§ Estimation accuracy and effectiveness for cardiac function assessmentThe proposed Indices-Net achieves accurate estimation of all the cardiac indices, as shown in the last row of Table. <ref>. Indices-Net estimates wall thicknesses with average MAE of 1.44±0.71mm, which is less than the mode of pixel spacing (1.5625mm/pixel) of the MR images in our dataset, and achieves average correlation of 0.758 with the ground truth. Among the six linear indices, the lateral wall thicknesses (WT3 and WT4) are more difficult to estimate due to the nearly invisible border between the lateral free wall of myocardium and the surroundings, and the presence of papillary muscle. Even so, Indices-Net is capable of obtaining accurate estimation with error of about one pixel, by leveraging the correlation between neighbouring indices with the index-wise connected CNN (see Fig. <ref>). For the two planar indices, Indices-Net estimates both with low MAE (185±162mm^2 and 223±193mm^2) and high correlation (0.953 and 0.853). The effectiveness of Indices-Net for cardiac function assessment is also demonstrated in the last row of Table. <ref>. For all the 145 subjects, Indices-Net achieves estimation error of 6.22±5.01% and correlation of 0.856 for ejection fraction, and (18.6±15.8%, 0.610) for myocardium wall thickening. This the first time that automatic cardiac wall thickening estimation is studied. §.§ Performance comparisonIndices-Net reveals great advantages for cardiac indices estimation and cardiac function assessment, when being compared with segmentation-based and existing two-phase direct methods (Tables. <ref> and <ref>).The average MAE reductions of Indices-Net over Max Flow <cit.> are 55.1% for the linear indices and 17.4% for the planar indices, even though Max Flow achieved high dice metric (0.913) for LV cavity segmentation. When the epicardium border is involved, Max Flow fails to deliver accurate estimation, as shown by the results of wall thicknesses and myocardium area. Our further analysis reveals that the dependency on manual segmentation of the first frame makes the estimation error of Max Flow increase as the estimated frame becomes far from the first frame within the cardiac cycle (Fig. <ref>). This makes it incapable of LV function assessment such as wall thickening analysis. Indices-Net outperforms the best of existing direct methods with clear MAE reductions (12.7%, 14.6%) and correlation improvements (0.123, 0.079) for the linear and planar indices. This evidences that the two-step framework in these methods is not adequate to achieve accurate estimation for multitype cardiac indices. Figs. <ref> and <ref> reveal with more details that Indices-Net can deliver more robust and accurate estimation than its competitors. The Bland-Altman plots show that Max Flow is prone to overestimate cardiac indices, while MCDBN+RF overestimates small indices and underestimates large indices. On the contrary, Indices-Net is capable of estimating them with consistently low error. The bar plots of frame-wise estimation error also demonstrate that Indices-Net estimates cardiac indices with consistently low error for all frames of one cardiac cycle. When applied to cardiac function assessment, Indices-Net performs best considering both the estimation error and correlation for LV ejection fraction and wall thickening (Table. <ref>).Table <ref> demonstrates the results of left-tailed F-test for Indices-Net and other competitors for all the 8 indices. The test result H, p-value, variance ratio and its confidence interval are demonstrated. The variance ratio and its confidence level show to which extent the proposed method differs from the competitors. Except the two cases where variance of estimation error of Max Flow for A1 and that of MCDBN+RF for WT5 are very close to those of Indices-Net, for all the rest cases, Indices-Net significantly outperforms these competitors. §.§ Benefit of joint learning frameworkFrom the results (Rows F_Indices+RF and F_Image+RF) shown in Table. <ref>, it can be drawn that F_Indices achieves average MAE of 1.46mm and 207mm^2 for the linear and planar indices, versus 1.82mm and 272mm^2 obtained by F_Image. This evidences that joint learning makes F_Indices more expressive with respect to cardiac indices, therefore leads to lower estimation error and better correlation.§.§ Benefit of initialization with pre-trained DCAE The loss curves (Fig. <ref>) of the proposed Indices-Net with and without pre-training of the DCAE network clearly show that pre-trained DCAE helps the network train faster and converge to lower estimation error for the training procedure, and generalize better to the test dataset. Comparing the performance of Indices-Net and Indices-Net (scratch) in Table. <ref>, we can draw that without pre-training, the deep network fails to deliver accurate prediction for both types of cardiac indices. With the pre-trained DCAE as initialization, our Indices-Net is capable of disentangling index-relevant features and delivering accurate estimation for multitype cardiac indices. §.§ Performance comparison with other deep networksTable <ref> demonstrates that our Indices-Net achieves much lower MAE than the single deep CNN and the FCN networks in estimation of the two types of cardiac indices. The superior performance of Indices-Net is mainly contributed by the DCAE for representation and the index-wise connection of CNN for regression. Both the discriminative encoder and the generative decoder in DCAE builds a cascade mapping of cardiac image (input layer)→ latent representation (fc6)→ indices-relevant structures. Then the index-wise connection of CNN effectively disentangles the most relevant index-specific features from these structures, while keeping inter-indices correlation. § CONCLUSION AND DISCUSSIONA deep integrated network Indices-Net was proposed to estimate frame-wise multitype cardiac indices simultaneously and achieved highly reliable and accurate estimation for all the cardiac indices when validated on a dataset of 145 subjects. Jointly learning of the two tightly-coupled networks DCAE and CNN enhanced the expressiveness of image representation and the compatibility between the indices regression and image representation. It is the first time that multitype cardiac indices estimation is investigated and the first time that joint learning of image representation and indices regression is deployed in cardiac indices estimation. The success of the proposed method for mid-cavity slice paved a great way to the true 3D (multi-slice) estimation of cardiac indices, which is usually used in clinical application. To achieve 3D estimation of multitype cardiac indices, Indices-Net can be directly trained with multi-slice cardiac MR images as in existing CNN-based 3D volume estimation <cit.>, or adapted with the recurrent neural network to model the dependencies of neighbouring slices in the latent space.IEEEtran | http://arxiv.org/abs/1705.09307v1 | {
"authors": [
"Wufeng Xue",
"Ali Islam",
"Mousumi Bhaduri",
"Shuo Li"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170525180141",
"title": "Direct Multitype Cardiac Indices Estimation via Joint Representation and Regression Learning"
} |
empty Multidesigns for a graph pair of order 6 Yizhe GaoDan Roberts[[email protected]]Department of MathematicsIllinois Wesleyan University Bloomington, IL 61701Given two graphs G and H, a (G,H)-multidecomposition of K_n is a partition of the edges of K_n into copies of G and H such that at least one copy of each is used.We give necessary and sufficient conditions for the existence of (C_6,C_6)-multidecomposition of K_n where C_6 denotes a cycle of length 6 and C_6 denotes the complement of C_6.We also characterize the cardinalities of leaves and paddings of maximum (C_6,C_6)-multipackings and minimum (C_6,C_6)-multicoverings, repsectively. § INTRODUCTIONLet G and H be graphs. Denote the vertex set of G by V(G) and the edge set of G by E(G).A G-decomposition of H is a partition of E(H) into a set of edge-disjoint subgraphs of H each of which are isomorphic to G.Graph decompositions have been extensively studied.This is particularly true for the case where H≅ K_n, see <cit.> for a recent survey.As an extension of a graph decomposition we can permit more than one graph, up to isomorphism, to appear in the partition.A (G,H)-multidecomposition of K_n is a partition of E(K_n) into a set of edge-disjoint subgraphs each of which is isomorphic to either G or H, and at least one copy of G and one copy of H are elements of the partition.When a (G,H)-multidecomposition of K_n does not exist, we would like to know how “close” we can get.More specifically, define a (G,H)-multipacking of K_n to be a collection of edge-disjoint subgraphs of K_n each of which is isomorphic to either G or H such that at least one copy of each is present.The set of edges in K_n that are not used as copies of either G or H in the (G,H)-multipacking is called the leave of the (G,H)-multipacking.Similarly, define a (G,H)-multicovering of K_n to be a partition of the multiset of edges formed by E(K_n) where some edges may be repeated into edge-disjoint copies of G and H such that at least one copy of each is present.The multiset of repeated edges is called the padding.A (G,H)-multipacking is called maximum if its leave is of minimum cardinality, and a (G,H)-multicovering is called minimum if its padding is of minimum cardinality.A natural way to form a pair of graphs is to use a graph and its complement.To this end, we have the following definition which first appeared in <cit.>.Let G and H be edge-disjoint, non-isomorphic, spanning subgraphs of K_n each with no isolated vertices.We call (G,H) a graph pair of order n if E(G) ∪ E(H) = E(K_n).For example, the only graph pair of order 4 is (C_4, E_2), where E_2 denotes the graph consisting of two disjoint edges.Furthermore, there are exactly 5 graph pairs of order 5.In this paper we are interested in the graph pair formed by a 6-cycle, denoted C_6, and the complement of a 6-cycle, denoted C_6.Necessary and sufficient conditions for multidecompositions of complete graphs into all graph pairs of orders 4 and 5 were characterized in <cit.>.They also characterized the cardinalities of leaves and paddings of multipackings and multicoverings for the same graph pairs.We advance those results by solving the same problems for a graph pair of order 6, namely (C_6,C_6).We first address multidecompositions, then multipackings and multicoverings.Our main results are stated in the following three theorems.The complete graph K_n admits a (C_6,C_6)-multidecomposition of K_n if and only if n ≡ 0,13 with n≥ 6, except n∈{7,9,10}.For each n≡ 23 with n≥ 8, a maximum (C_6,C_6)-multipacking of K_n has a leave of cardinality 1.Furthermore, a maximum (C_6,C_6)-multipacking of K_7 has a leave of cardinality 6, and a maximum (C_6,C_6)-multipacking of either K_9 or K_10 has a leave of cardinality 3. For each n≡ 23 with n≥ 8, a minimum (C_6,C_6)-multicovering of K_n has a padding of cardinality 2.Furthermore, a minimum (C_6,C_6)-multicovering of K_7 has a padding of cardinality 6, and a minimum (C_6,C_6)-multicoveirng of either K_9 or K_10 has a padding of cardinality 2. Let G and H be vertex-disjoint graphs.The join of G and H, denoted G∨ H, is defined to be the graph with vertex set V(G)∪ V(H) and edge set E(G)∪ E(H)∪{{u,v} u∈ V(G),v∈ V(H)}.We use the shorthand notation ⋁_i=1^tG_i to denote G_1∨ G_2∨⋯∨ G_t, and when G_i≅ G for all 1≤ i≤ t we write ⋁_i=1^tG.For example, K_12≅⋁_i=1^4K_3.For notational convenience, let (a,b,c,d,e,f) denote the copy of C_6 with vertex set {a,b,c,d,e,f} and edge set {{a,b}, {b,c}, {c,d}, {d,e}, {e,f}, {a,f}}, as seen in Figure <ref>.Let [a,b,c;d,e,f] denote the copy of C_6 with vertex set {a,b,c,d,e,f} and edge set{{a,b},{b,c},{a,c},{d,e},{e,f},{d,f},{a,d},{b,e},{c,f}}. Next, we state some known results on graph decompositions that will help us prove our main result.Sotteau's theorem gives necessary and sufficient conditions for complete bipartite graphs (denoted by K_m,n when the partite sets have cardinalities m and n) to decompose into even cycles of fixed length.Here we state the result only for cycle length 6. A C_6-decomposition of K_m,n exists if and only if m ≥ 4, n≥ 4, m and n are both even, and 6 divides mn. Another celebrated result in the field of graph decompositions is that the necessary conditions for a C_k-decomposition of K_n are also sufficient.Here we state the result only for k=6. Let n be a positive integer.A C_6-decomposition of K_n exists if and only if n≡ 1, 912. The necessary and sufficient conditions for a C_6-decomposition of K_n are also known, and stated in the following theorem. Let n be a positive integer.A C_6-decomposition of K_n exists if and only if n≡ 19. § MULTIDECOMPOSITIONSWe first establish the necessary conditions for a (C_6,C_6)-multidecomposition of K_n.If a (C_6,C_6)-multidecomposition of K_n exists, then* n≥ 6, and* n ≡ 0,1 3.Assume that a (C_6,C_6)-multidecomposition of K_n exists.It is clear that condition (1) holds, with the exception of the trivial case where n=1.Considering that the edges of K_n are partitioned into subgraphs isomorphic to C_6 and C_6, we have that there exist positive integers x and y such that n 2=6x+9y.Hence, 3 divides n 2, which implies n≡ 0,13, and condition (2) follows. §.§ Small examples of multidecompositionsIn this section we present various non-existence and existence results for (C_6,C_6)-multidecompositions of small orders.The existence results will help with our general constructions. §.§.§ Non-existence resultsThe necessary conditions for the existence of a (C_6,C_6)-multidecomposition of K_n fail to be sufficient in exactly three cases, namely n=7,9,10.We will now establish the non-existence of (C_6,C_6)-multidecompositions of K_n for these cases. A (C_6,C_6)-multidecomposition of K_7 does not exist. Assume the existence of a (C_6,C_6)-multidecomposition of K_7, call it 𝒢.There must exist positive integers x and y such that 7 2=21=6x+9y.The only solution to this equation is (x,y)=(2,1); therefore, 𝒢 must contain exactly one copy of C_6.However, upon examing the degree of each vertex contained in the single copy of C_6 we see that there must exist a non-negative integer p such that 6=2p+3.This is a contradiction.Thus, a (C_6,C_6)-multidecomposition of K_7 cannot exist.A (C_6,C_6)-multidecomposition of K_9 does not exist. Assume the existence of a (C_6,C_6)-multidecomposition of K_9, call it 𝒢.There must exist positive integers x and y such that 9 2=36=6x+9y.The only solution to this equation is (x,y)=(3,2); therefore, 𝒢 must contain exactly two copies of C_6.Turning to the degrees of the vertices in K_9, we have that there must exist positive integers p and q such that 8=2p+3q.The only possibilities are (p,q)∈{(4,0),(1,2)}. Note that K_6 does not contain two edge-disjoint copies of C_6.Since 𝒢 contains exactly two copies of C_6, there must exist at least one vertex a∈ V(K_9) that is contained in exactly one copy of C_6. However, this contradicts the fact that vertex a must be contained in either 0 or 2 copies of C_6.Thus, a (C_6,C_6)-multidecomposition of K_9 cannot exist.A (C_6,C_6)-multidecomposition of K_10 does not exist. Assume the existence of a (C_6,C_6)-multidecomposition of K_10, call it 𝒢.There must exist positive integers x and y such that 10 2=45=6x+9y.Thus, (x,y)∈{(6,1),(3,3)}; therefore, 𝒢 must contain at least one copy of C_6.However, if 𝒢 consists of exactly one copy of C_6, then the vertices of K_10 which are not included in this copy would have odd degrees remaining after the removal of the copy of C_6.Thus, the case where (x,y)=(6,1) is impossible.Upon examining the degree of each vertex in K_10, we see that there must exist positive integers p and q such that 9=2p+3q.The only solutions to this equation are (p,q)∈{(3,1),(0,3)}.From the above argument, we know that 𝒢 contains exactly 3 copies of C_6, say A,B, and C.Let X=V(A)∩ V(B).It must be the case that |X|≥ 2 since K_10 has 10 vertices.It also must be the case that |X|≤ 5 since K_6 does not contain two copies of C_6.If |X|∈{2,3}, then V(C)∩ (V(A)△ V(B))≠∅, where △ denotes the symmetric difference.This implies that there exists a vertex in V(K_n) that is contained in exactly 2 copies of C_6 in 𝒢, which is a contradiction.Observe that any set consisting of either 4 or 5 vertices in C_6 must induce at least 3 or 6 edges, respectively.Furthermore, X⊆ V(C) due to the degree constraints put in place by the existence of 𝒢.If |X|=4 or |X|=5, then X must induce at least 9 or at least 18 edges, respectively.This is a contradiction in either case.Thus, no (C_6,C_6)-multidecomposition of K_10 exists.§.§.§ Existence resultsWe now present some multidecompositions of small orders that will be useful for our general recursive constructions.K_13 admits a (C_6,_6)-multidecomposition.Let V(K_13)={1,2,…, 13}.The following is a (C_6,_6)-multidecomposition of K_13. { [1,2, 3;7,9,8], [1,4,5;9,12,10], [3,4,6;7,11,10], [2,5,6;8,12,11]} ∪{(13, 1,6,8,5,11), (13,2,4,7,6,12), (13,3,5,9,4,10), (13,7,12,3,9,6), (13,8,10,2,7,5), (13,9,11,1,8,4), (1,10,3,11,2,12)}K_15 admits a (C_6,_6) -multidecomposition.Let V(K_15)={1,2,…, 15}.The following is a (C_6,_6)-multidecomposition of K_15. { [1,5, 10;6,8,12], [4,8,13;9,11,15], [7,11,1;12,14,3], [10,14,4;15,2,6],[13,2,7;3,5,9]} ∪{(1,12, 11, 13, 5, 15), (4, 15, 14, 1, 8, 3), (7, 3, 2, 4, 11, 6), (10, 6, 5, 7, 14, 9),(13, 9, 8, 10, 2, 12), (1, 2, 11, 3, 6, 13), (4, 5, 14, 6, 9, 1), (7, 8, 2, 9, 12, 4),(10, 11, 5, 12, 15, 7), (13, 14, 8, 15, 3, 10)}K_19 admits a (C_6,C_6)-multidecomposition.Let V(K_19)={1,2,…, 19}.The following is a (C_6,C_6)-multidecomposition of K_19. { [2,11, 14; 17, 4, 18], [3, 12, 15; 18, 5, 19], [4, 13, 16; 19, 6, 11], [5, 14, 17; 11, 7, 12],[6,15, 18; 12, 8, 13], [7, 16, 19; 13, 9, 14], [8, 17, 11; 14, 10, 15], [9, 18, 12; 15, 2, 16],[10, 19, 13; 16, 3, 17]} ∪{(2,12, 14, 3, 11, 1), (3, 13, 15, 4, 12, 1), (4, 14, 16, 5, 13, 1), (5, 15, 17, 6, 14, 1), (6, 16, 18, 7, 15, 1), (7, 17, 19, 8, 16, 1), (8, 18, 11, 9, 17, 1), (9, 19, 12, 10, 18, 1),(10, 11, 13, 2, 19, 1), (2, 3, 10, 4, 9, 5), (2, 6, 8, 7, 3, 4), (2, 7, 4, 5, 3, 8),(2, 10, 8, 4, 6, 9), (3, 6, 10, 5, 7, 9), (5, 6, 7, 10, 9, 8)}§.§ General constructions for multidecompositionsIf n ≡ 0 6 with n≥ 6, then K_n admits a (C_6,C_6)-multidecomposition.Let n=6x for some integer x≥ 1.Note that K_6x≅⋁_i=1^xK_6.On each copy of K_6 place a (C_6,C_6)-multidecomposition of K_6.The remaining edges form edge-disjoint copies of K_6,6, which admits a C_6-decomposition by Theorem <ref>.Thus, we obtain the desired (C_6,C_6)-multidecomposition of K_n.If n ≡ 1 6 with n≥ 13, then K_n admits a (C_6,C_6)-multidecomposition.Let n=6x+1 for some integer x≥ 2.The proof breaks into two cases.Case 1: x=2k for some integer k≥ 1.Notice that K_12k+1≅ K_1∨(⋁_i=1^kK_12).Each of the k copies of K_13 formed by K_1∨ K_12 admit a (C_6,C_6)-multidecomposition by Example <ref>.The remaining edges form edge-disjoint copies of K_12,12, which admits a C_6-decomposition by Theorem <ref>.Thus, we obtain the desired (C_6,C_6)-multidecomposition of K_n.Case 2: x=2k+1 for some integer k≥ 2.Notice that K_12k+7≅ K_1∨ K_6∨(⋁_i=1^kK_12).The single copy of K_19 formed by K_1∨ K_6∨ K_12 admits a (C_6,C_6)-multidecomposition by Example <ref>.The remaining k-1 copies of K_13 formed by K_1∨ K_12 each admit a (C_6,C_6)-multidecomposition by Example <ref>.The remaining edges form edge-disjoint copies of either K_6,12 or K_12,12.Both of these graphs admit C_6-decompositions by Theorem <ref>.Thus, we obtain the desired (C_6,C_6)-multidecomposition of K_n.If n ≡ 3 6 with n≥ 15, then K_n admits a (C_6,C_6) -multidecomposition.Let n=6x+3 for some integer x≥ 2.The proof breaks into two cases.Case 1: x=2k for some integer k≥ 1. Notice that K_12k+3≅ K_1∨ K_14∨(⋁_i=1^k-1K_12).The remainder of the proof is similar to the proof of Case 1 of Lemma <ref> where the ingredients required are C_6-decompositions of K_12,12, and K_12,14, as well as (C_6,C_6)-multidecompositions of K_13 and K_15.Case 2: x=2k+1 for some integer k≥ 1.Notice that K_12k+9≅ K_1∨ K_8∨(⋁_i=1^kK_12).The remainder of the proof is similar to the proof of Case 2 of Lemma <ref> where the ingredients required are C_6-decompositions of K_9 (which exists by Theorem <ref>), K_8,12, and K_12,12, as well as a (C_6,C_6)-multidecomposition of K_13.If n ≡ 4 6 with n≥ 16, then K_n admits a (C_6,C_6) -multidecomposition.Let n=6x+4 where x≥ 2 is an integer.Note that K_6x+4≅ K_10∨(⋁_i=1^x-1K_6).The remainder of the proof is similar to the proof of Case 2 of Lemma <ref> where the ingredients required are C_6-decompositions of K_6,6 and K_6,10, a C_6-decomposition of K_10 (which exists by Theorem <ref>), as well as a (C_6,_6)-multidecomposition of K_6. Combining Lemmas <ref>, <ref>, <ref>, and <ref>, we have proven Theorem <ref>. § MAXIMUM MULTIPACKINGSNow we turn our attention to (C_6,C_6)-multipackings in the cases where (C_6,C_6)- multidecompositions do not exist. §.§ Small examples of maximum multipackings A maximum (C_6,C_6)-multipacking of K_7 has a leave of cardinality 6.Note that the number of edges used in a (C_6,C_6)-multipacking of any graph must be a multiple of 3, since (6,9)=3. Since no (C_6,C_6)-multidecomposition of K_7 exists the next possibility is a leave of cardinality 3.However, the equation 18=6x+9y has no positive integer solutions.Thus, the minimum possible cardinality of a leave is 6.Let V(K_7) = {1,...,7}. The following is a (C_6,C_6)-multipacking of K_7, with leave {{1,7}, {2,7}, {3,7}, {4,7}, {5,7}, {6,7}}.{[1,3,5;4,6,2], (1,2,3,4,5,6)} A maximum (C_6,C_6)-multipacking of K_8 has a leave of cardinality 1.Let V(K_8) = {1,...,8}. The following is a (C_6,C_6)-multipacking of K_8, with leave {3,6}.{[2,5,7;4,1,8], (1,2,3,4,5,6), (1,3,5,8,6,7), (3,8,2,6,4,7)}A maximum (C_6,C_6)-multipacking of K_9 has a leave of cardinality 3.Let V(K_9) = {1,...,9}. The following is a (C_6,C_6)-multipacking of K_9, with leave {{2,4},{2,9},{4,9}}. {[1,2,3;6,5,4], [1,4,7;9,8,3], [2,6,8;7,9,5], (1,5,3,6,7,8)}A maximum (C_6,C_6)-multipacking of K_10 has a leave of cardinality 3.A (C_6,C_6)-multipacking of K_10 with a leave of cardinality 3 can be obtained by starting with a C_6-decomposition of K_10.Then remove three vertex-disjiont edges from one copy of C_6, forming a C_6.This gives us the desired (C_6,C_6)-multipacking of K_10 where the three removed edge form the leave. A maximum (C_6,C_6)-multipacking of K_11 has a leave of cardinality 1.Let V(K_11) = {1,...,11}. The following is a (C_6,C_6)-multipacking of K_11, with leave {1,2}.{[1,7, 10;9,6,3], [1,5,6;4,10,2], [2,5,7;11,8,4], [1,3,11;8,2,9]} ∪{ (3,4,9,10,6,8), (4,5,9,7,11,6), (3,5,11,10,8,7)} A maximum (C_6,C_6)-multipacking of K_17 has a leave of cardinality 1.Let V(K_17) = {1,...,17}. The following is a (C_6,C_6)-multipacking of K_17, with leave {1,10}.{[ 2,3,5;7,8,1], [3,6,4;9,8,10],[2,4,9;6,5,7]} ∪{ (2,12,5,10,11,14), (2,10,17,4,13,11), (4,7,13,14,5,15), (4,11,15,8,16,12), (1,15,14,16,5,17), (3,12,11,17,6,15), (1,2,16,7,14,4), (2,13,5,8,14,17), (7,15,10,13,9,17), (1,13,6,9,11,16), (1,9,12,7,3,11), (3,10,12,8,4,16), (3,13,16,6,12,14), (2,8,13,17,12,15), (6,10,16,15,9,14), (5,9,16,17,8,11), (1,3,17,15,13,12), (1,6,11,7,10,14)} §.§ General Constructions of maximum multipackingsIf n ≡ 2 6 with n≥ 14, then K_n admits a (C_6,C_6)-multipacking with leave cardinality 1. Let n=6x+2 for some integer x≥ 2.Notice that K_6x+2≅ K_2∨(⋁_i=1^xK_6).Let {u, v} = V(K_2).Each of the x copies of K_8 formed by K_2∨ K_6 admit a (C_6,C_6)-multipacking with leave cardinality 1 by Example <ref>. Note that we can always choose the leave edge to be {u,v} in each of these multipackings.The remaining edges form edge disjoint copies of K_6,6, each of which admits a C_6-decomposition by Theorem <ref>.Thus, we obtain the desired (C_6,C_6)-multipacking of K_n.If n ≡ 5 6 with n≥ 11, then K_n admits a (C_6,C_6)-multipacking with leave cardinality 1.Let n=6x+5 for some integer x≥ 1.Case 1: x=2k for some integer k≥ 1.Notice that K_12k+5≅ K_1∨ K_16∨(⋁_i=1^k-1K_12).Each of the k-1 copies of K_13 formed by K_1∨ K_12 admit a (C_6,C_6)-multidecomposition by Example <ref>.The copy of K_17 formed by K_1∨ K_16 admits a (C_6,C_6)-multipacking with leave of cardinality 1 by Example <ref>.The remaining edges form edge disjoint copies of K_12,12 or K_12,16, each of which admits a C_6-decomposition by Theorem <ref>.Thus, we obtain the desired (C_6,C_6)-multipacking of K_n.Case 2: x=2k+1 for some integer k≥ 1.Notice that K_12k+11≅ K_1∨ K_10∨(⋁_i=1^kK_12).On each of the k copies of K_13 formed by K_1∨ K_12 admit a (C_6,C_6)-multidecomposition by Example <ref>.The copy of K_11 formed by K_1∨ K_10 admits a (C_6,C_6)-multipacking with leave of cardinality 1 by Example <ref>.The remaining edges form edge disjoint copies of K_12,12 or K_10,12, each of which admits a C_6-decomposition by Theorem <ref>.Thus, we obtain the desired (C_6,C_6)-multipacking of K_n. Combining Lemmas <ref> and <ref>, we have proven Theorem <ref>.§ MINIMUM MULTICOVERINGSNow we turn our attention to minimum (C_6,C_6)-multicoverings in the cases where (C_6,C_6)- multidecompositions do not exist. §.§ Small examples of minimum multicoverings A minimum (C_6,C_6)-multicovering of K_7 has a padding of cardinality 6.We first rule out the possibility of a minimum (C_6,C_6)-multicovering of K_7 with a padding of cardinality 3.The only positive integer solution to the equation 24=6x+9y is (x,y)=(1,2).In such a covering there would be one vertex left out of one of the copies of C_6.It would be impossible to use all edges at this vertex with the remaining copies of C_6 and C_6.Thus, the best possible cardinality of a padding is 6.Let V(K_7) = {1,...,7}. The following is a minimum (C_6,C_6)-multicovering of K_7, with padding of {{1,2}, {1,5}, {1,6}, {3,6}, {4,5}, {5,6}}.{[1,2,3;6,5,4],(1,4,7,6,3,5), (1,6,2,4,5,7), (1,2,7,3,6,5)}A minimum (C_6,C_6)-multicovering of K_8 has a padding of cardinality 2.Let V(K_8) = {1,...,8}. The following is a minimum (C_6,C_6)-multicovering of K_8, with padding of {{1,8}, {3,5}}.{[1,2,8;4,3,5], [1,5,6;3,7,8], (1,7,2,6,4,8), (2,4,7,6,3,5)}A minimum (C_6,C_6)-multicovering of K_9 has a padding of cardinality 3. A (C_6,C_6)-multicovering of K_9 with a padding of cardinality 3 can be obtained by starting with a C_6-decomposition of K_9.One copy of C_6 can be transformed into a copy of C_6 by adding the edges in a 1-factor on the vertices in a copy of C_6.This gives us the desired (C_6,C_6)-multicovering of K_9 where the three added edges form the padding. A minimum (C_6,C_6)-multicovering of K_10 has a padding of cardinality 3. A (C_6,C_6)-multicovering of K_10 with a padding of cardinality 3 can be obtained by starting with a C_6-decomposition of K_10.One copy of C_6 can be transformed into two copies of C_6 by carefully adding three edges.This gives us the desired (C_6,C_6)-multicovering of K_10 where the three added edges form the padding. A minimum (C_6,C_6)-multicovering of K_11 has a padding of cardinality 2.Let V(K_11) = {1,...,11}. The following is a minimum (C_6,C_6)-multicovering of K_11, with padding of {{3,4}, {8,11}}.{[1,2, 11;6,5,7], [1,3,5;10,2,9], [4,6,10;7,9,8]} ∪{ (3,4,5,8,11,6), (1,8,2,7,3,9), (2,4,9,11,8,6), (1,4,3,11,10,7), (3,8,4,11,5,10) }A minimum (C_6,C_6)-multicovering of K_17 has a padding of cardinality 2.Let V(K_17) = {1,...,17}.Apply Theorem <ref> and let ℬ_1 be a C_6-decomposition on the copy of K_9 formed by the subgraph induced by the vertices {9,…,17}.Apply Theorem <ref> and let ℬ_2 be a C_6-decomposition of the copy of K_6,8 formed by the subgraph of K_17 with vertex bipartition (A,B) where A={1,…,8} and B={12,…,17}.The following is a minimum (C_6,C_6)-multicovering of K_17, with padding of {{3,5}, {7,8}}.ℬ_1 ∪ℬ_2∪{[1,2,3;6,5,4], [1,4,8;7,2,6]} ∪{ (1,5,7,8,3,9), (1,10,3,7,4,11), (2,8,7,11,6,9) } ∪{ (5,11,8,9,7,10), (3,5,9,4,10,6), (2,11,3,5,8,10) }§.§ General constructions of minimum multicoveringsIf n ≡ 2 6 with n≥ 8, then K_n admits a minimum (C_6,C_6)-multicovering with a padding of cardinality 2. Let n=6x+2 for some integer x≥ 1. Notice that K_6x+2≅ K_8∨(⋁_i=1^x-1K_6).Each of the x-1 copies of K_6 admit a (C_6,C_6)-multidecomposition by Lemma <ref>.The copy of K_8 admits a (C_6,C_6)-multicovering with a padding of cardinality 2 by Example <ref>.The remaining edges form edge disjoint copies of K_6,6 or K_6,8, each of which admit a C_6-decomposition by Theorem <ref>.Thus, we obtain the desired (C_6,C_6)-multicovering of K_n. If n ≡ 5 6 with n≥ 11, then K_n admits a minimum (C_6,C_6)- multicovering with a padding of cardinality 2. Let n=6x+5 for some integer x≥ 1.The proof breaks into two cases.Case 1: x=2k for some integer k≥ 1.Notice that K_12k+5≅ K_1∨ K_4∨(⋁_i=1^kK_12).One copy of K_17 is formed by K_1∨ K_4∨ K_12, and admits a (C_6,C_6)-multicovering with a padding of cardinality 2 by Example <ref>.The k-1 copies of K_13 formed by K_1∨ K_12 admit a (C_6,C_6)-multidecomposition by Example <ref>.The remaining edges form edge disjoint copies of K_12,12 or K_4,12, each of which admits a C_6-decomposition by Theorem <ref>.Thus, we obtain the desired (C_6,C_6)-multicovering of K_n.Case 2: x=2k+1 for some integer k≥ 1.Notice that K_12k+11≅ K_1∨ K_4∨ K_6∨(⋁_i=1^kK_12).One copy of K_11 is formed by K_1∨ K_4∨ K_6, and admits a (C_6,C_6)-multicovering with a padding of cardinality 2 by Example <ref>.The k copies of K_13 formed by K_1∨ K_12 admit a (C_6,C_6)-multidecomposition by Example <ref>.The remaining edges form edge disjoint copies of K_12,12, K_4,12, or K_6,12, each of which admits a C_6-decomposition by Theorem <ref>.Thus, we obtain the desired (C_6,C_6)-multicovering of K_n. Combining Lemmas <ref> and <ref>, we have proven Theorem <ref>.§ FINAL NOTESThe cardinalities of the leaves of maximum (C_6,C_6)-multipackings and paddings of minimum (C_6,C_6)-multicoverings of K_n have been characterized.It is still an open problem to characterize the structure of those leaves and paddings.We would like to thank Mark Liffiton and Wenting Zhao for finding (C_6,C_6)-multidecompositions of K_11 and K_17 using the MiniCard solver.MiniCard source code is available at <https://github.com/liffiton/minicard>. AbueidaDavenA. Abueida and M. Daven, Multidesigns for Graph-Pairs of Order 4 and 5, Graphs and Combinatorics (2003) 19, 433–447. AdamsetalP. Adams, D. Bryant, and M. Buchanan, A Survey on the Existence of G-Designs, J. Combin. Designs 16 (2008), 373–410. AlspachetalB. Alspach, H. Gavlas, M. Šajna, and H. Verrall, Cycle decompositions IV: Complete directed graphs and fixed length directed cycles, J. Combin. Theory A 103 (2003) 165–208. KangZhaoMaQ. Kang, H. Zhao, and C. Ma, Graph designs for nine graphs with six vertices and nine edges, Ars Combin. 88 (2008), 379–395. SotteauD. Sotteau, Decomposition of K_m,n (K^*_m,n) into Cycles (Circuits) of Length 2k, J. Combin. Theory Ser. B 30 1981, 75–81. | http://arxiv.org/abs/1705.09638v2 | {
"authors": [
"Yizhe Gao",
"Dan Roberts"
],
"categories": [
"math.CO",
"05C51, 05C70"
],
"primary_category": "math.CO",
"published": "20170526162148",
"title": "Multidesigns for a graph pair of order 6"
} |
Probabilistic Global Scale Estimation for MonoSLAMBased on Generic Object Detection Edgar Sucar, Jean-Bernard Hayet Centro de Investigación en Matemáticas - Universidad de Guanajuato Jalisco S/N, Col. Valenciana CP: 36023 Guanajuato, Gto, México {[email protected], [email protected]} December 30, 2023 ========================================================================================================================================================================================================================= This paper proposes a novel method to estimate the global scale of a 3D reconstructed model within a Kalman filtering-based monocular SLAM algorithm. Our Bayesian framework integrates height priors over the detected objects belonging to a set of broad predefined classes,based on recent advances in fast generic object detection. Each observation is produced on single frames, so that we do not need a data association process along video frames. This is because we associate the height priors with the image region sizes at image places where map features projections fall within the object detection regions. We present very promising results of this approach obtained on several experiments with different object classes. § INTRODUCTION Live monocular Simultaneous Localization and Mapping (SLAM) is a classical problem in computer vision, with many proposed solutions throughout the last decade <cit.>. It has numerous applications, from augmented reality to map building in robotics. However, when performing a 3D reconstruction with a calibrated, monocular system, it is well known that the output reconstruction is an estimate of the real structure up to an unknown scale factor. This may be a problem when the considered application involves measuring distances, inserting real-size virtual objects with a coherent scale, etc. Of course, in some specific contexts (in mobile robotics, for example), the unknown scale factor can be recovered easily by integrating information from other sensors <cit.>, from some assumptions on the environment or on the position of the camera <cit.>, or from the introduction of known objects in the scene <cit.>. Nevertheless, one can note that this intrinsic limit of monocular systems does not prevent animal visual systems to have a rather precise depth perception, even with one sensor only, and in very general situations. This perception of depth based only on monocular cues is rather well documented <cit.>. In computer vision, the idea of inferring depth from monocular cues has been exploited in previous works relating monocular perception of texture and depth through machine learning techniques <cit.>, and it has been used to initialize the unknown global scale in monocular SLAM systems <cit.>. In this work, the motivation is to follow this line of research from a different, much less explored perspective. We use region-based, semantic information instead of pixel-based information, and rely on the notion of familiar size monocular cues, well studied in psychology <cit.>. To achieve this goal, as illustrated in Figure <ref>, we take advantage of the recent availability of very efficient techniques to detect instances of classes of objects (“bottle”, “mug”, “book”, etc.), made possible by the advent of deep learning techniques, and use an out-of-the-box generic object detector to extract observations from images of the video sequence. By associating size priors to these object classes, we are finally able to infer the global scale of a monocular SLAM system, within a Bayesian framework. To the best of our knowledge, this is the first work that couples tightly the detected instances of an object class detector with a probabilistic monocular SLAM system to perform live global scale estimation based on semantic information given by the detector, on a single frame basis, and in a Bayesian scheme. We did not develop the detector by ourselves, but relied on recent proposals from the deep learning community. In essence, our approach is limited to detections from classes that the detector has learned. The structure of this document is the following. In Section <ref> we review related work on scale estimation for monocular SLAM. In Section <ref> we give an overview of our estimation method, and in Section <ref> we present the likelihood model to integrate the object detection observations in the scale estimation framework. Section <ref> gives an overview of the implementation, and in Section <ref> we present the experimental results and evaluations of these results with respect to simpler forms of estimating the unknown global scale. Finally, we summarize our work and discuss potential improvements in Section <ref>. § RELATED WORK Monocular algorithms are widely used for 3D scene reconstruction and camera tracking. The reason for this is that they are cost-effective (only an inexpensive camera is required) and current methods are robust. There are mainly two approaches for doing this: optimization-based methods such as PTAM <cit.> and filtering-based methods such as MonoSLAM <cit.>. We have considered here the filtering-based approach, MonoSLAM, as it is of our interest to incorporate in our method the uncertainty associated with 3D feature locations. In monocular 3D reconstruction, the scale of the map is inherently unknown. Methods such as MonoSLAM and PTAM use a special initialization procedure of the system to set up the scale of the map. MonoSLAM uses a known object for system initialization and for fixing the scale of the map, which introduces the inconvenience of needing a specific object each time it is executed. PTAM assumes an initial known translation, which does not give a very precise scale initialization. Both of these methods also have in common the problem of scale drift due to inaccuracies in 3D reconstruction and tracking. Two main techniques have been used for automatic scale estimation during 3D reconstruction. The first technique consists of using sensors that intrinsically obtain measurements that allow scale to be estimated, such as depth of features using Kinect <cit.> or translation measurements using IMU sensors <cit.>. However, as they incorporate additional sensors, these techniques defeat one of the main advantages of monocular algorithms which is the possibility of relying only on an inexpensive RGB camera. The second family of techniques consists of using known “natural” objects to estimate the scale of the map. The disadvantage of these techniques is the need to train an object recognition algorithm for specific objects and to have these objects introduced in the scenes in which the 3D reconstruction algorithm will be run. In <cit.>, real-time 3D object recognition is used in a live dense reconstruction method based on depth cameras, to get 3D maps with a high level of compression andcomplementary interactions between the recognitions, mapping and tracking processes. A major difference with our system is that it considers a database of fixed, specific objects, whereas our aim is to handle fixed categories of objects.The use of semantics through objects in the SLAM context is also present in works similar to ours, as <cit.>, where an object recognition system, driven by a bag-of-words algorithm, is integrated within an object-aware SLAM system to impose constraints on the optimization. Again, a database of pre-defined specific objects is used, while our approach relies on much broader object categories.An alternative approach has been developed with the idea of using more generic objects, namely faces, using the front cellphone camera <cit.>. This method requires a special routine to be done during the 3D reconstruction by the back cellphone camera, in which 6DoF face tracking with the front facing camera withknown scaleis usedto obtain the scalefor the 3D reconstruction. This method does not generalize easily to other object classes as it depends on the precise 6DoF tracking algorithms for faces.In <cit.>, the most similar work to this one, a generic detector is also used, in the context of urban scenes, and priors on the sizes of the detected objects. However, a first difference to our work is that the reconstruction is done in a bundle adjustment framework and, above all, no connection is done between the reconstructed map and the detected object, so that the scale inference is done by using data association through consecutive frames, which is done prior to running the method. Our method does not require data association, as map features and detection regions are associated naturally on single frames. The Bayesian framework of our method also has the advantage that it is easy to incorporate additional uncertainties other than 3D feature locations, such as the uncertainty in the fit of the object's bounding box. Our approach aims to be ubiquitous for 3D reconstruction. It avoids the use of external sensors and it is robust, as it is based on a probabilistic framework. It uses a generic object recognition algorithm which runs in real time. Deep neural networks have achieved good accuracy for generic object recognition <cit.>, and recently methods that run in real time have been developed <cit.>. By combining probabilistic height priors for recognized generic objects with 3D reconstruction uncertainties within a Bayesian framework, we are able to estimate the scale for monocular SLAM. § OVERVIEW OF OUR APPROACH A monocular SLAM, such as MonoSLAM <cit.>, allows to reconstruct a scene as a sparse cloud of 3D points 𝒩={𝐩_i}_1≤ i ≤ N, where 𝐩_i = (x_i,y_i,z_i)^T. It also allows to track the configuration of the camera along time 𝐱_v,k, where k is an index corresponding to time/camera frame and 𝐱_v is a representation of the 3D configuration of the camera, i.e., an element of the Special Euclidean group, SE(3). The vision algorithms reconstruct the scene up to an unknown scale factor, this means that the true 3D points 𝐪_i are related to 𝐩_i through 𝐪_i = d 𝐩_i + ν, where d is a global scale factor for the whole scene, a priori unknown, andν is the reconstruction error noise. Our aim is to estimate d using Bayesian inference, based on object detections given by a generic object detector. We take the approach proposed in <cit.> to separate the state vector (3D coordinates of the map points and camera pose) into: (i) a dimensionless part that can be maintained based on the perspective projections equations as in <cit.> and, (ii) a recursive estimation scheme that maintains an estimate of the global scale d as explained hereafter. Then, our idea is to use those 3D points reconstructed by the SLAM algorithm that project into an “object" region. In some way, the data association between the objects and the current reconstruction is done implicitly at these points. Let D_k be the number of objects detected by the detector, and 𝒟^k={S_l}_1≤ l ≤ D_k be the set of D_k detected object zones (bottles/chairs/screens...). Each S_l encodes a detection performed in the image by the detector, and is associated to a rectangular window in the video frame k, containing a detected object, and a class c_l for the detected object. We handle a subset of 3D points 𝒯_k⊂𝒩, a subset of T_kpoints reconstructed up to time k, characterized by the fact that their projection lies within an “object" region as specified by the detector at time k. Let 𝐩_i ∈𝒯_k be one of these points.We will note at time k: * π^k_i the 2D projection of 𝐩_i on image k. * H_l the function that associates one “object" region to its real height, which we do not know in general, but for which we will have an associated prior.* c_l the function that associates one “object" region to the index of the object class to which it belongs. We are given a prior distribution on the possible heights for each object class c, i.e., a distribution, p_c(H), that we build beforehand. For the moment, this distribution is set arbitrarily (see the Experimental Results section) but we plan to learn it by using large categories databases. Now, what we want to estimate is, at time k, the posterior probability for the coefficient d, i.e., the distribution conditioned on the cloud reconstructed up to time k and the different detection-observations gathered: p(d|𝒩,𝒟^1,…,𝒟^k). For the sake of clarity, we will suppose that D_k=1, so that there is only one object, with its detection S_1, and with its associated height prior p(H_1). Then, by using Bayes rule, we can write the global scale posterior into p(d|𝒩,𝒟^1,…,𝒟^k) ∝ p(𝒟^k|d,𝒩,𝒟^1,…,𝒟^k-1)p(d|𝒩,𝒟^1,𝒟^2,…,𝒟^k-1)∝ p(S_1|d,𝒩,𝒟^1,…,𝒟^k-1)p(d|𝒩,𝒟^1,𝒟^2,…,𝒟^k-1)∝p(d|𝒩,𝒟^1,…,𝒟^k-1) ∫_H_1 p(H_1,S_1|d,𝒩,𝒟^1,…,𝒟^k-1) dH_1 ∝p(d|𝒩,𝒟^1,…,𝒟^k-1) ∫_H_1 p(S_1|H_1,d,𝒩,𝒟^1,…,𝒟^k-1)p(H_1|d,𝒩,𝒟^1,…,𝒟^k-1)dH_1∝p(d|𝒩,𝒟^1,…,𝒟^k-1) ∫_H_1 p(S_1|H_1,d,𝒩) p_c_1(H_1)dH_1. This means that at each step we can update the posterior on d by multiplying it by ∫_H_1 p(S_1|H_1,d,𝒩) p_c_1(H_1)dH_1. We approximate this termwith the help of a histogram representation over H_1, {p_c_1(H_1,m)}, and the prior probability that a detection of object 1, of some specific class, has a real height H_1,m (where the heights have been discretized): ∑_m p(S_1|H_1,m,d,𝒩) p_c_1(H_1,m). In the more general case of D_k>1, and by assuming conditional independence between the different detections observed in frame k, one can show that a similar development leads to p(d|𝒩,𝒟^1,…,𝒟^k)= p(d|𝒩,𝒟^1,…,𝒟^k-1) ∏_l=1^D_k∫_H_lp(S_l|H_l,d,𝒩) p_c_l(H_l)dH_l. The likelihood term p(S_l|H_l,m,d,𝒩) is the probability that the detected object has the dimensions in pixels with which it was detected, given that the object has some real size H_l,m, that the scale is d, and that the cloud is 𝒩. Its calculation is explained in the next section. The evaluations over d are done for a discrete set of values, d_i, within some specified interval [d_min,d_max], to ease the computational burden. § LIKELIHOOD OF THE OBSERVATIONS We will now describe the calculation of the likelihood term p(S_l|H_l,m,d,𝒩), which is the probability that the detected object has the dimensions in pixels with which it was detected, given that the object has some real size H_l,m, that the scale is d, and that the 3D cloud is 𝒩. Thegeneral idea for doing this is back projecting the object extremities found in the image into the 3D map and estimating the object's height using a feature in 𝒩 that projects into the object and the scale d. The height estimate is then compared with H_l,m to obtainp(S_l|H_l,m,d,𝒩). Let us assume that in the worldframe, i.e., the frame in which the 3D points are reconstructed by MonoSLAM, we know the vertical direction, that is, the direction perpendicular to the ground plane. This vertical direction is set when initializing the MonoSLAM system with a square marker which is perpendicular to the ground plane. Let also 𝐑^W_k be the 3D location of the camera at time k in the world frame W. The vectors and points we will be dealing with hereafter will be expressed in the camera frame. To evaluate p(S_l|H_l,d,𝒩), let us consider a 3D point 𝐩_i from the reconstructed cloud 𝒩, such that its current projection π^k_i ∈ S_l^k.Let λ be the line inthe image such that it containsπ^k_i and such that the plane obtained by back projecting this line onto the 3D map is parallel to the vertical direction. Let us define the intersections of this line with the boundary of the detection S_l, π_t, π_d = λ∩∂ S_l^k, as depicted in Figure <ref> with red dots, while π^k_i is the black dot. These two points in the image correspond to the vertical extremities of the object.Now let us consider the line Λ in the 3D map such that it contains 𝐩_i and that it is parallel to the vertical direction, i.e., we will suppose that the detected object of interest is aligned with the vertical direction in the world frame. Let π̂_t and π̂_d be the 3D map rays obtained by back projecting the image points π_t and π_d, respectively. Now we define𝐩_t = π̂_t ∩Λ and𝐩_d = π̂_d ∩Λ. These two points correspond to the vertical extremities of the object in the 3D map, as seen in Figure <ref>. Note that the coordinates of 𝐫^W_k, together with the coordinates of the point 𝐩_i, are given in the dimensionless state vector. Then the estimated object height can be approximated as the Euclidean distance D(𝐩_t,𝐩_d). Given f, a symmetric density centered at zero (e.g., a Gaussian), we can evaluate p(S_l|H_l,d,𝒩) as p(S_l|H_l,d,𝒩) =f(|d D(𝐩_t,𝐩_d)-H_l|;0,σ). The dispersion parameter σ in f is important to define to give a proper weight to each observation in the estimation scheme. The density f is modeled as a Gaussian density since the only source of uncertainty considered at the moment is the 3D position of 𝐩_i, which is estimated through a Kalman filter. In particular, as the position of 𝐩_i is uncertain, and is estimated as in <cit.> in the dimensionless space, with an uncertainty P_p_ip_i, we can estimate the expected variance on D(𝐩_t,𝐩_d). Let us associate the same covariance matrix P_p_ip_i (calculated by the MonoSLAM algorithm) to each of the points 𝐩_t,𝐩_d. Then, the variance on D can be calculated by standard uncertainty propagation: σ^2_D =∂ D/∂𝐩_t P_p_ip_i(∂ D/∂𝐩_t)^T + ∂ D/∂𝐩_d P_p_ip_i(∂ D/∂𝐩_d)^T,= 2 ∂ D/∂𝐩_t P_p_ip_i(∂ D/∂𝐩_t)^T=2 J P_p_ip_iJ^T, where J is the gradient of the distance function with respect to 𝐩_t, i.e., J=1/D(𝐩_t-𝐩_d). Then, given that f(d; 0, σ) is the density of aGaussian random variable with variance σ^2, F(d) = f(|d D(𝐩_t,𝐩_d)-H_l|; 0, σ) is the density of aGaussian random variable with variance σ^2/ D(𝐩_t,𝐩_d)^2 and for that reason we use σ^2=σ^2_D D(𝐩_t,𝐩_d)^2 as the variance of f, so that f(|d D(𝐩_t,𝐩_d)-H_l|; 0, σ) has variance σ^2_D. The global state posterior is updated every time a new observation is obtained by multiplying it by the likelihood of the new observation: ∑_H_l,m p(S_l|H_l,m,d,𝒩) p(H_l,m). An observation is generated by each 3D feature whose projection lies inside an object detection region each time the object detection algorithm is ran. The dispersion of the likelihood of the new observation is calculated using the covariance of its respective 3D feature as described above. We expect the likelihood to have a larger dispersion the bigger the covariance of the 3D feature is. The scale parameter d is calculated as the mode of p(d|𝒩,𝒟^1,…,𝒟^k), the global state posterior (MAP). We can also calculate a local scale estimate specific to each global state posterior update, as the mode of L(d) = ∑_H_l,m p(S_l|H_l,m,d,𝒩) p(H_l,m), the likelihood of the new observation; this local scale estimate can be interpreted as an observation of the true scale. For instance, Figure <ref> (where the graphs correspond to experiment 1 described in Section <ref>) depicts in (b) the evolution along the video frames of the global scale posterior, with a clear tendency to reduce the scale estimate variance, in (a) the likelihoods of the new observations corresponding to each global state posterior update, and in (c) the evolution of the global scale parameter estimate along the the local estimates obtained after each update. The local scale parameter estimates allow us to observe the variability of the “observations” being made. § IMPLEMENTATION Our implementation uses the SceneLib2 library <cit.> which is a reimplementation of Davison et al.'s <cit.> original algorithm for Kalman-based Monocular SLAM. It also uses the YOLO v2 algorithm <cit.>, which runs on real time on an NVIDIA GPU, for generic object detection. YOLO v2 was trained on the 2007 and 2012 PASCAL Visual Object Classes Challenge datasets <cit.>. Both of these algorithms run on real time, and our method requires little computational expense, which guarantees a real time execution of the final algorithm. The object detection function is run every 10 frames. Each feature point inserted in the map and whose projection falls inside an object region is used to update the global scale posterior as described in Section <ref>.Figure <ref> shows an instance of the algorithm for an specific frame, showingthe object detection result, (a), and camera tracking with 3D map features, (b) and (c). § EXPERIMENTAL RESULTS We first describe the evaluation methodology and then present the quantitative results. §.§ Evaluation method The evaluation of the method is run in parallel to the algorithm by comparing distances computed based on the scale estimate with distances obtained by a Kinect. For this comparison, we use a marker (the same that is used to initialize the MonoSLAM system) with four unambiguous 3D map features. Let y_1^W, y_2^W, y_3^W, and y_4^W be the locations of these features as estimated by the MonoSLAM algorithm. Each time the global state posterior is updated, a new scale parameter is estimated and this error is computed for this scale parameter.Let D_R() be a function that measures the true distance from the camera to each feature obtained with a Kinect. For each feature, an absolute error is computed as:e_i(d) = |D_R(y_i^W)- d D(y_i^W, R_k^W)|,with R^W_k the 3D position of the camera at the current frame k. Given that there are four feature points, the total absolute error is calculated as:ϵ(d)=1/4∑_i=1^4 e_i(d). The relative errorfor each feature is defined as:δ_i(d) = 100 |D_R(y_i^W)- d D(y_i^W, R_k^W)|/D_R(y_i^W).The total relative error is computed as:Δ(d)=1/4∑_i=1^4 δ_i(d).§.§ Results To evaluate the proposed method we performed three experiments considering different types and number of objects in the scene. In each case a we ran MonoSLAM and the object detector over a video sequence, estimating the scale with our method. The absolute and relative errors are estimated according the procedure described above. We report the median and standard deviation of these errors for all the scale updates after the scale has converged, which is determined visually, and show the evolution of the relative error. To have an indication on the expected fluctuations, the relative errors are also calculated without using the scale estimate, that is, setting d=1 in equation <ref>, which maintains the initial scale that is set by the initialization of MonoSLAM with a marker of known dimensions. The purpose of this is determining the cause of fluctuations in the relative error of the scale.§.§.§ Experiment 1In this first experiment, 4 bottles of the same size are used. Our height prior assigns the true bottle height a probability of 1, zero elsewhere (i.e., the object size is known). The sequence that we used contains 950 frames, with a total of of 226 global scale posterior updates; the statistics are reported after update number 80. The median absolute error is 0.0191m with standard deviation of 0.0097m; the median relative error is 1.7197 % with a standard deviation of 0.8621 %. Figure <ref> (a) shows the evolution of the relative error of the scale, and (b) the evolution of the relative error with our estimated scale vs. the evolution of the relative error with a scale set to 1.§.§.§ Experiment 2In our second experiment, another object (a microwave oven) is used. The height prior assigns the true microwave height a probability of 1 (i.e., the oven size is known). The sequence that we used contains 1073 frames, with a total of of 92 global state posterior updates; the statistics are reported after update number 16. The median absolute error is 0.0243m with standard deviation of 0.0171m; the median relative error is 2.1455 % with standard deviation of 1.7411%. Figure <ref> (a) shows the evolution of the relative error of the scale, and (b) the evolution of the relative error with our estimated scale vs. the evolution of the relative error with a scale set to 1.§.§.§ Experiment 3In our third experiment, 4 bottles of different sizes are used. For each object (bottles), the height prior is defined as a uniform histogram assigning to each of the four different bottle sizes a probability of 1/4. The sequence that we used contains 1081 frames, with a total of of 403 global scale posterior updates; the statistics are reported after update number 215. The median absolute error is 0.0123m with a standard deviation of 0.0095m; the median relative error is 1.3390 % with standard deviation of 1.0266%.Figure <ref> (a) shows the evolution of the relative error of the scale, and (b) the evolution of the relative error with our estimated scale vs. the evolution of the relative error with a scale set to 1. §.§ Discussion We observe in the three experiments how the error decreases as more object observations are integrated in the scale estimate, obtaining in all cases a median relative error very close to 2%, after the scale estimate has converged. As expected, when there is a larger variance in the heights of the objects (Experiment 3), the error takes more updates to converge. However, even in this case, the median error is very low after convergence, with a median of1.3390 %.In the three experiments we can observe a bigger error in the first few updates of the scale. The reason for this is the large covariance of the 3D features used to estimate the scale, since the MonoSLAM algorithm is at its initial stage.It may appear that the scale estimate is not stable when observing the graphs of the evolution of the relative error. However, when analyzing Figure <ref> and <ref>, the scale appears rather stable. These fluctuations present in the relative error appear to be due uncertainties from the Kinect measurements and the MonoSLAM position estimates. This is verified by comparing the evolution of the relative error with our estimated scale vs. the evolution of the relative error with a fixed scale of 1 (keeping the initial scale from MonoSLAM, which is fixed by the features inserted on a marker with known dimensions), as can be seen in Figures <ref> (b), <ref> (b), and <ref> (b), where the fluctuations appear greatly correlated. § CONCLUSIONS AND FUTURE WORKWe have developed a novel method to estimate the global scale of a 3D reconstruction for monocular SLAM. Based on the recent advances in generic object recognition, we use a Bayesian framework to integrate height priors over observations of detected object sizes in single frames. The method does not require temporal data association as map features and detection regions are associated naturally on single frames. The proposed approach takes advantage of recently developed techniques for object recognition based on deep learning that run in real time.Experimental results considering different number and types of objects give evidence of the feasibility of our approach, obtaining median relative errors in average of 2 %. These preliminary experiments show that it is possible to perform a rather precise global scale estimation based on priors on semantic classes of objects.This precision could allow applications in Augmented Reality to insert objects in a scene with coherent size or scale drift correction for navigation of autonomous cars and drones.There are a few limitations to our approach that we are currently searching to improve. In particular, the map features projections may fall on the detection regions while the corresponding 3D ray does not intersect the 3D object. An outlier rejection scheme could be used to filter out these observations.Another option would be to use a semantic segmentation algorithm for a more precise outline of the object to avoid3D map features not in the object falling in the detection outline.We have used object classes with relatively small intra-class size variance; we need to test this approach on more complex priors.We intend to develop a methodology for constructing prior height distributions for different classes of objects using measuring sensors such as Kinect, sampling a wide range of specific objects for each given class. As the proposed method is mostly independent of the 3D reconstruction and tracking algorithm, it should allow us to migrate the approach tostate of the art monocular SLAM systems as a future work.This would allow us to test the method in public datasets such as the KITTI dataset <cit.> for a precise comparison with other methods for scale estimation. ieee | http://arxiv.org/abs/1705.09860v1 | {
"authors": [
"Edgar Sucar",
"Jean-Bernard Hayet"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170527201431",
"title": "Probabilistic Global Scale Estimation for MonoSLAM Based on Generic Object Detection"
} |
OmegaCAM discovers multiple sequences in the color-magnitude diagram of the Orion Nebula ClusterOmegaCAM discovers multiple sequences in the ONCG. Beccari et al. European Southern Observatory, Karl-Schwarzschild-Strasse 2, 85748 Garching bei München, Germany, [email protected] Astrofisico di Arcetri, L.go E. Fermi 5, 50125, Firenze, ItalyDipartimento di Fisica e Astronomia Galileo Galilei, Vicolo Osservatorio 3, I-35122, Padova, ItalyScience Support Office, Directorate of Science, European Space Research and Technology Centre (ESA/ESTEC), Keplerlaan 1, 2201 AZ Noordwijk, The NetherlandsEuropean Southern Observatory, Alonso de Córdova 3107, Casilla 19001, Santiago, ChileSchool of Physics, Astronomy and Mathematics, University of Hertfordshire, College Lane Campus, Hatfield, AL10 9AB, UKDepartamento de Astronomía, Universidad de Chile, Casilla 36-D, Correo Central, Santiago, ChileCSIC-INTA Centro de Astrobiologia Carretera Ajalvir km 4 28550 Madrid, Spain Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USAArmagh Observatory, College Hill, Armagh BT61 9DG, United KingdomAstrophysics Group, Keele University, Keele ST5 5BG, UKExcellence Cluster Universe, Garching bei München, Germany As part of the Accretion Discs in Hα with OmegaCAM (ADHOC) survey, weimaged in r, i and Hα a region of 12×8 square degrees around the Orion Nebula Cluster. Thanks to the high-quality photometry obtained, we discovered three well-separated pre-main sequences in the color-magnitude diagram. The populations are all concentrated towards the cluster's center. Although several explanations can be invoked to explainthese sequences we are left with two competitive, but intriguing, scenarios: a population of unresolved binaries with an exotic mass ratio distribution or three populations with different ages. Independent high-resolution spectroscopy supports the presence of discrete episodes of star formation, each separated by about a million years. The stars from the two putative youngest populations rotate faster than the older ones, in agreement with the evolution of stellar rotation observed in pre-main sequence stars younger than 4 Myr in several star forming regions. Whatever the final explanation, our results prompt for a revised look at the formation mode and early evolution of stars in clusters. A Tale of Three Cities G. Beccari1 M.G. Petr-Gotzens1 H.M.J. Boffin1M. Romaniello1,12D. Fedele2G. Carraro3G. De Marchi4W.-J. de Wit5J.E. Drew6 V.M. Kalari7C.F. Manara4 E.L. Martin8S. Mieske5N. Panagia9L. Testi1J.S. Vink10J.R. Walsh1N.J. Wright6,11December 30, 2023 ========================================================================================================================================================================================================================================= § INTRODUCTIONYoung stellar clusters are conspicuous components of our Galaxy.They are the best test beds of the stellar initial mass function because they are assumed to be entities of a common origin, i.e. born from the same molecular cloud material, and at the same time. Observations have shown, however, that the stars born in one and the same cluster are not as coeval as expected. In fact, stellar age spreads up to several million years have been postulated for several clusters from fitting isochronal ages to the position of cluster member stars in the Hertzsprung-Russell diagram <cit.>. But, if this observation is a consequence of a continuous star formation process lasting over several dynamical time scales, or rather is caused by different accretion histories within an otherwise co-eval population of pre-main sequence (PMS) stars, is widely debated <cit.>.The Orion Nebula Cluster (ONC) is the nearest <cit.>, populous young stellar cluster and hence the best laboratory to test the existence of stellar age spreads. <cit.> were the first to point out an apparent large age spread of ∼10 Myr based on the evidence of lithium depletion in some stellar cluster members. <cit.> interprets the luminosity spread of the ONC's PMS as an age spread of 0.3-0.4 dex with a mean age of 3-4 Myr depending on the models.In this letter we present a photometric study of a 12×8 area including the ONC. We report the detection for the first time of multiple sequences in the observed optical color-magnitude diagram of the ONC.§ OBSERVATIONS AND DATA REDUCTIONThe images used in this work were collected with the wide field optical camera OmegaCAM on the 2.6-m VLT Survey Telescope (VST) at Cerro Paranal in Chile. OmegaCAM consists of a mosaic of 32 CCDs and samples a 1 deg^2 Field of View (FoV) with a pixel sampling of 0.21 arcsec pixel^-1. Orion was sampled through the r,i broad band filters and the Hα narrow band as part of the Accretion Discs in Hα with OmegaCAM (ADHOC) survey (PI: Beccari).Each target region is sampled in groups of 3 overlapping fields, and in each group the fields are contiguous with a footprint close to 3×1 deg^2. For each position in sky we acquire 2 exposures of 25 sec in r and i and 3 images of 150 sec exposure with the Hα filter. The frames were collected between October and December 2015 and the median image quality is 091±016.The pattern to these observations is similar to that of the VPHAS+ survey, and the data have passed through the same pipeline <cit.>. The entire data-set was fully processed, from the bias, flat-field and linearity correction to the stellar photometry, at the Cambridge Astronomical Survey Unit (CASU). The magnitude for each star is extracted using aperture photometry adopting an algorithm based on IMCORE[Software publicly available from http://casu.ast.cam.ac.uk.] <cit.> and the nightly photometric calibrations are also performed.We downloaded the astrometrically and photometrically calibrated single bandcatalogs from the VST archive at CASU[http://casu.ast.cam.ac.uk/vstsp/].Stars lying in the overlap region between adjacent fields were used to adjust residual photometric offsets. The photometric calibration of the final band-merged catalog covering the entire area was checked against a catalog of stars from the AAVSO Photometric All Sky Survey, used as secondary standard catalog. The final catalog allows us to homogeneously sample the stellar populations in a region of 12×8 size in Orion down to r∼20.§ THREE DISTINCT PRE-MAIN SEQUENCES IN THE ONC We show in Fig. <ref>a the (r-i) vs. r color-magnitude diagram (CMD) of the stars located inside a radius of 15 from the center of the ONC.The population of PMS objects is well detectedand occupies the reddest side of the CMD above the population of back/fore-ground stars in the magnituderange 14<r<20 and (r-i)>1, which roughly corresponds to PMS stars of masses between 0.2 and 1 M_⊙ <cit.>.A remarkable feature is well visible on the CMD shown in Fig. <ref>a, i.e. the presence of at least two distinct and near-parallel sequences of PMS stars.Following <cit.>, in Fig. <ref>b we have applied to the CMD the “unsharp-masking” technique of <cit.> in order to enhance the high-frequency features in the diagram. The result shown in Fig. <ref>b further supports the detection of at least two distinct PMSs in the CMD diagram.We stress here that by inspecting the entire surveyed area we found that this feature is clearly detectable only in the ONC region.We used a simplified version of the method described in <cit.> to further investigate the existence of two or more distinct sequences in the CMD. In Fig. <ref>a we show a portion of the CMD zoomed on the population of the PMS. In order to increase the contrast of the ONC population against the back/fore-ground stars, we show only stars inside a radius of 05 from the center of the cluster. The black line shows the mean ridge line of the blue PMS. We then calculate the distance in r-i color of each star in a magnitude range 15.5<r<16.5 from the mean ridge line arbitrary chosen as reference line. We adopted this magnitude range to provide an adequate statistical balance between the number of stars in the PMS population andthe contamination from field stars. The distance of each staras a function of the r magnitude is shown on panel (b) of the same figure. Panel (c), the histogram of the distances in (r-i) color, clearly shows the presence of even three distinct populations of PMS stars well separated in color on the CMD. Indeed the Hartigans' dip test confirms that the distribution of the color distances is incompatible with a uni-modal's one. The number fraction of stars belonging to the two populations with the reddest colors (green and red in the figure)compared to the reference one (blue) is 0.5 and 0.15, respectively. These fractions hold (within the statistical uncertainties) even after accounting for contamination, which we estimated to be between 15-40% based on the CMD of a control field located a few degrees west of the ONC, and spectroscopic membership information from APOGEE spectra (see Sec. <ref>). Next, we investigate the spatialdistribution of stars belonging to the different sequences. In Fig. <ref> we show the surface densities of the three populations. These density plots are calculated using all the stars belonging to the three PMS populations selected in the color and magnitude ranges shown in the CMD of Fig. <ref>. We calculate the densities using the GATHER method from <cit.>. In the maps of Fig. <ref> we show the position of the stars with respect to the ONC nominal center (solid circles) together with the density contours, which were scaled to the maximum value of each population. It emerges that the density distributions of the three populations all peak around a common center. The blue population seems to be slightly more sparsely distribute with respect to the spatial distribution of the stars belonging to the green and red ones.We used the Minimal Spanning Tree <cit.> to assess if any difference in the spatial distribution of the populations is present and at which level of significance. The MST is the unique set of straight lines ("edges") connecting a given sample of points ("vertices"; in this case the star coordinates) without closed loops, such that the sum of the edge lengths is the minimum possible.Hence, the length of the MST is a measure of the compactness of a given sample of vertices <cit.>. The MST is a powerful algorithm to study population distributions since it can be used without assuming that the studied populations are distributed around the same center of gravity, which is mandatory when using the Kolmogorow-Smirnov (KS) test. The weakness of this method is that it must be assumed that the photometric completeness of the compared populations is the same. We have selected the populations in a common range of magnitudes that are affected by the same level of completeness. The fact that the three populations are affected by the same contamination from field stars, makes it not possible to use the MST when the number of genuine members is low, which is the case for the red population. For this reason we limit the use of the MST to the blue and green populations.We first estimate the MST of all the stars in the green population. Then we randomly extract 1,000 sets of stars belonging to the blue population equal in number to the number of stars in the green one. We compute the MST of the blue populations as the mean and the standard deviation of the distribution of 1,000 MSTs. Hence, we calculate[We remind here that Λ MST=1 means that the compared populations are equally distributed in space while a value greater than one indicates that the population at the numerator is more spatially extended then the one at the denominator.] Λ MST_blue= MST_blue/MST_green. We find that Λ MST_blue=1.16±0.03, which indicates that the blue sequence is slightly more sparsely distributed with respect to the green one with a 5σ significance. § THE UNRESOLVED BINARIES/MULTIPLES SCENARIO The appearance of multiple parallel sequences in the CMD as shown in Fig. <ref> could be caused by (1) a single coeval population consisting of single stars and unresolved binaries and higher order multiples; or populations (2) at different distances; (3) with different extinction A_V; or (4)with different ages. In this and the following sections we will discuss these four hypotheses.Any unresolved binary system would appear in a CMD as a single star with a flux equal to the sum of the fluxes of the two components. This effect produces a systematic over-luminosity of these objects and a shift in color which depends on the magnitudes (and hence mass) of the two components in each passband. When the mass ratio of two stars in the binary system is q=1 (equal mass binary)the unresolved binary system will appear 0.752 mag brighter than the individual component's brightness. The peak of the distribution of the green population is ∼0.75mag brighter than the blue one (Fig. <ref>a) and this distribution may therefore well reflect the presence of a population of unresolved ONC binaries. Under the assumption that the green and red sequences represent the binaries and higher-order multiples of the ONC members, we derive a multiplicity fraction of 39%. This number would support the unresolved binary hypothesis given that this fraction overall agrees with that seen for low-mass stars in other young clusters <cit.> and with the multiplicity fraction among field M-dwarfs <cit.>.Beside the total fraction, the companion mass ratio distribution plays an important role in shaping the CMD appearance. Determinations of the mass ratio distribution for ONC binaries are rare. Those studies that have derived mass ratios for close visual systems and spectroscopic systems find no indication for an equal mass preference <cit.>, i.e. for thosesystems that lead to the largest displacement in luminosity in the CMD. At most, the observed mass ratio distribution slightly increases from low q to higher q <cit.>, although this result is most likely affected by incompleteness and selection effects at low q. <cit.> performed a companion survey of 245 late-K to mid-M (K7-M6) dwarfs within 15 pc and found that the mass ratio distribution across the q=0.2-1.0 range is flat. This seems to be a general result. Numerical simulations of primordial binaries <cit.> produce an f(q) that is rather flat too. Moreover f(q) seems to be rather insensitive to dynamical disruptions and interaction processes within the cluster <cit.>.The marked CMD morphology and in particular the presence of the two gaps in the distribution of colors shown in Fig.<ref>c, allows us to investigate which combination of total binary frequency and companion mass ratio distribution could explain the observed CMD, and which can be excluded. For this, we performed Monte Carlo simulations and test a range of total binary fractions and mass ratio distributions f(q)[Note that when doing this, we only use the blue and green populations, and ignore for now the red population.]. We draw randomly a star from the blue sequence (with its r and i magnitudes) and, using the mass-luminosity relation from <cit.>, we determine its mass. We then draw a mass ratio from a given f(q) distribution, and add a companion with a mass q m_1. We then compute the color and magnitude of the resulting binary and can then see where the so obtained binary falls in the CMD. Finally, we compare the obtained histograms of colors with that observed in Fig.<ref>c. A range from 40% to 100% in binary fraction and different f(q) of the form uniformly flat, linearly increasing, quadraticallyincreasing, or step-like were explored. In Fig. <ref> we show a few representative results.It is clear that the canonical case of 40% binary fraction with a uniform mass-ratio distribution, as shown in the upper panels of Fig. <ref>, does not provide a satisfactory agreement with the observations. Increasing thefraction of binaries even worsens the comparison with observations, as does a lower total binary fraction. We conclude that a uniformly flat mass ratio distribution is not able to reproduce the observations, in particular it is not capable in reproducing the obvious, significant gaps in the CMD.When assuming other mass ratio distributions we can achieve reasonable fits (e.g. second to fourth row panels in Fig. <ref>). Here we considered as reasonable fit any solution with a reduced χ^2 < 2 which indicates that it can betrusted with 99.5% confidence. The solutions come with some caveats, though. A linearlyincreasing f(q) (second row panels of Fig. <ref>) appears only possible in combination with a totalbinary fraction of around 60%. However, such high overall binary frequencies among ONC low-mass stars are not observed. For visual binaries with separations betweena few tens and a few hundreds of AU (which would indeed appear unresolved in our OmegaCAM observations) various studies have consistently found that the ONC binary fraction is even slightly lower than in the field <cit.>. Also very close binaries with separations <10 AU, that have been traced by a multi-epoch spectroscopic study <cit.> do not show any excess in companions, but are again consistent with field star fractions. Hence, the overall binary fraction cannot noticeably exceed 40% unless these excess binaries are all in the narrow separation range of ∼10-40 AU whichseems very unlikely. On the other hand, if we enforce mass ratio distributions that are strongly skewed towards high mass ratios, such as a quadratically increasing or step-like function where the majority of systems would have q>0.6, then a simulated population with a total binary fraction as low as 35-40% can reproduce the observations (third and fourth row panels of Fig. <ref>). Actually, also higher total binary fractions, up to 65% would be feasible according to our simulation. In this case, however, the blue population would contain a large number of unresolved low q systems. Such specific mass ratio distributions are at odds with observations. We are thus led to the conclusion that in order for the binary hypothesis to be valid, one would need to postulate a very distinct mass-ratio distribution.In the same vein, the reddest sequence we find in the CMD would be even more difficult to explain, as it would require triple systems, where, the mass ratio distributions would have also to be quite tailored, i.e. the mass ratios of both secondaries and tertiaries would need to be strongly peaked to q > 0.8 and be exactly similar.Although formally we cannot exclude the presence of a population of binaries with an unusual f(q), the fact that no young binary population in any star forming region has shown indications of such an usual f(q), casts some doubts on binaries as the origin for the observed multiple sequences. On the other hand, if confirmed this fact would certainly challenge many of the studies published so far on the stellar population in the ONC whenever such stars were considered as single objects <cit.>.§ DIFFERENTIAL A_V AND DISTANCES SCENARIO Any effect of differential extinction can be rejected by the fact that the reddening vector as shown on the CMD of Fig. <ref>a runs parallel to the PMSs.Moreover, as shown in Fig. <ref> we have verified that the stars from the three sequences have approximately the same distribution of visual extinction A_V. To this aim we adopted the A_V provided by the spectroscopic analysis of <cit.>. An alternative explanation for the presence of the green and red sequences could be a PMS population located more than 100 and 200 pc in the foreground of the blue sequence, respectively. <cit.> suggested that the population of PMS around ι Ori belongs to the association NGC 1980 and represents a population of 4-5 Myr old stars located at ∼30 pc in the foreground of Orion A <cit.>. In their detailed spectroscopic analysis <cit.> rule out this hypothesis finding that the candidate foreground population is kinematically indistinguishable from the Orion A's one. They conclude that the old population studied by <cit.> witnesses the earliest (i.e. oldest) episode of star formation in the ONC. The three distinct PMS populations found in our CMD and their spatial distribution are not compatible with the presence of a old foreground population of PMS stars. <cit.> performed and extensive spectroscopic analysis of 691 “foreground” stars in the Orion A region and confirm that NGC 1980 is not a foreground population. Considering that the candidate foreground population studied by <cit.> is located more than 1 deg south with respect to the peaks of the density maps of the three populations shown in Fig. <ref> we think that it is unlikely to be related.§ THREE DISCRETE EPISODES OF STAR FORMATIONIn the previous sections we show that differential extinction or distance offsets are very unlikely to explain the discovered features in the CMD. The presence of unresolved binary and tertiary populations reproduce the CMD morphology only if the underlying mass ratio distribution is rather unusual (Fig. <ref>).We here explore the possibility that the age is the origin of the discreetness of the color distribution of the PMS in the ONC. We verified that the distance in magnitude between a 1 Myr and a 3 Myr PMS isochrone from <cit.> in the r, r-i CMDis indeed ∼0.75, i.e. equal to the the shift in luminosity due to unresolved binaries. This unfortunate fact is at the root of why it is hard to distinguish between the binary hypothesis and the multiple population scenarios[And we have checked that using different combinations of filters wouldn't help in this respect.].In order to assign ages to the stars belonging to the three distinct PMS populations, we use the measurements presented by <cit.>. D16 performed a spectroscopic study of the young stellar population of the Orion A molecular cloud with the APOGEE spectrograph. In their work they measured accurate stellar parameters (T_eff, logg, vsini) and extinctions and conclude that star formation in the ONC proceeded over an extended period of ∼3 Myr age.We have cross-correlated our stars with the catalog of stellar parameters (including age, temperature, extinction) published in table 4 of D16. We used these stellar parameters to study the properties of the three candidate populations of PMS stars as selected in Fig. <ref>. In order to have a sample which is as clean as possible from any contamination we removed stars from their study that did not clearly obey a PMS mass-luminosity relation and that, based on the effective temperature, mass and luminosity, cannot populate the PMS region in the HRD. We were left with 111, 63, and 24 stars for the three sequences (blue, green, and red), respectively.In Fig. <ref> we show the distributions of the ages derived byD16 of the stars in the three samples. These age histograms indicate that the three sequences that we discovered in the photometric study have distinct ages, with the bluer population being the oldest. We have fitted the distributions with Gaussians, using a χ^2-minimization technique, to derive the mean and standard deviation (σ). As these are distributions of the logarithm of the ages, we then estimated the corresponding distributions of the ages and computed the respective 1-σ and 5–95% intervals. Our results are shown in Table <ref>. This finding is consistent with the hypothesis that the formation of the population of PMS stars in the ONC that was thought to be the outcome of a single episode extended over 3 Myr is instead best described by 3 discrete and sequential episodes of star formation over the same time span.D16 provide also stellar rotational velocities. However, spectroscopy only provides v sin i, where i is the unknown inclination of the rotation axis on the plane of the sky. As we may assume that i is isotropically distributed in the sky, it is possible to deconvolve, using a Richardson-Lucy method <cit.>, the v sin i distributions to obtain the distributions of v. These are shown in Fig. <ref> and Table <ref>, where it is clear that not only do the three populations have different ages, but they have also different rotational velocities: the younger the population, the faster its members rotate. Obviously it can be argued that the age and v sin i estimations provided in D16 are incorrect were the redder stars entirely populated by unresolved binaries belonging to the main population.We investigated what it would need to obtain a broadening of the spectral line profile corresponding to v=25 km/s (the peak of the young/green population) starting from a binary system containing 2 stars having v=14 km/s (the peak of the old/blue population). It appears that this can only be explained if the binaries have orbital periods between ∼2 and ∼12 months, i.e. a very narrow range of periods. Indeed,if the orbital period is greater than ∼ 1 year then the maximum difference in velocities between the 2 stars is too small and cannot widen the line and make it appear as a fast rotating star. If, on the other hand, the orbital period is less than 2 months the difference in velocity is such that a double peaked line profile is detected and should have been well visible in the APOGEE spectra used by D16. The proponents of the binary hypothesis could also argue that the difference in rotational velocities would be due to tidal effects, as they would spin-up the stars. However, this can only work if the stars were very close to each other, i.e. with an orbital period of a few days or less <cit.>.Thus, whatever explanation one chooses, one needs to have a very specific orbital period distribution, unlike what is observed for such stars.Coupling this fact with the narrow distribution of f(q), makes the overall multiplicity scenario very unlikely.On the other hand, in their seminal work, <cit.> studied the evolution ofperiods and projected rotational velocities, v sin i, of youngPMS stars in the range K5-M2 in several star-forming regions, including the ONC. They find a decrease in meanv sin i as a function of age. They conclude that a significant fraction of all PMS stars must evolve at nearly constant angular velocity during the first 3–5 Myr.Several studies have been performed to look for a suitable explanation of this behavior which seems to be related to a process of disk locking that might regulate the PMS star angular momentum during the early evolution. In short, stars with accretion discs are found to be rotating at much slower rates, suggesting that significant angular momentum removal mechanisms must operate during the first few Myr of formation <cit.>.However, this scenario has not yet found a general consensus <cit.>. In the context of this paper, we notice that our result on stellar rotation is not unexpected and would support what was already shown by <cit.>. We will investigate the interesting scenario of a relation between stellar rotation and ongoing accretion from a circumstellar disk in a dedicated paper (Beccari et al. in preparation).§ CONCLUSION We have presented a wide-field optical survey of the stellar population in a region of 12×8 in Orion. Our CMD shows the presence of at least two parallel sequences among the PMS stars. The distribution of the r-i colors in the range of magnitude 15.5<r<16.5 reveals the presence of even three populations. We investigate the origin of these sequences that were newer observed before. We use detailed information (including age, A_v, rotational velocities) published in the spectroscopic workof D16 and look for comparative properties of the three populations. We can safely exclude that differential extinction or different projected distancescould be responsible for the feature revealed by our CMD. We are hence left with two competitive, but as intriguing, explanations:a population of unresolved binaries or three populations with different ages.We find that a flat mass ratio distribution is not able to reproduce the observed CMD, regardless the binary fraction. On the other hand, we can reproduce the observed color distribution shown in Fig <ref> by assuming a 35% to 65% of binaries and a f(q) strongly skewed towards high mass ratio where the majority of the binaries in the ONC have mass ratios q>0.6. This result, if confirmed, would provide the first and most solid constraint on the nature of compact binaries in a population of PMS objects. A dedicated spectroscopic monitoring campaign is urgently required to constrain the multiplicity fraction among these populations. This will certainly allow to unequivocally disentangle between the two scenarios. The fact that such a population of binaries populates the CMD of the ONC would inevitably challenge many of the previously published studies of stellar populations in the ONC.The comparison with the spectroscopic measurements published by D16 provide convincing observational evidences in support to the hypothesis that the ONC contains three populations of PMS stars, with different ages and rotational velocities. In particular we find that the younger the population the larger is the mean rotation velocity. This result seems not to be unexpected. <cit.> already reported a decrease of vsin i in the first 5 Myr among PMS stars in the same spectral range studied in this paper and in several star forming regions, including Orion. It has been speculated that the evolution of the angular momentum in solar-type PMS objects might be regulated by actively accreting circum-stellar disks trough a mechanism of disk-locking. This process wouldimpact the rotational properties of young stars and influence their rotational evolution <cit.>.In support of this hypothesis <cit.>, using Spitzer mid-IR data for about 900 starsin Orion in the mass range 0.1-3 M_⊙ find that slowly-rotating stars are indeed more likely to posses disks thanrapidly-rotating stars. There is no agreement in the literature on this matter and our data-set willbe used in a dedicated paper to further investigate any connection between stellar rotation and ongoing accretion. While the unresolved binary hypothesis cannot be ruled out, the evidence described so far seem to point towardthe first detection of distinct generations of PMS stars in the ONC. Interestingly, using Hubble Space Telescope (HST) observations of the ONC, <cit.> found that the youngest stars in the cluster are more clustered towards the center, while the oldest ones are distributed almost homogeneously in space. Their figure 12 strongly resemble the populations' distribution shown in our Fig. <ref>. Such scenario has further interesting implication.In the context of investigating the origin of 'blue hook' stars in the globular cluster ω Cen, <cit.>predicted that these evolved stars would originate from the evolution of a rapidly rotating second-generation PMS population whose accretion discs suffered an early disruption in the dense environment of the cluster's central regions. The result shown in this work might be pointing towards the first observational evidence that such a mechanism takes place in the early stage of a cluster's formation.Since the centers of the spatial distributions of the 3 populations are not significantly different, we speculate that star formation has been progressing along the line of sight and that the youngest population formed, on average, further away from us. This view seems consistent with the fact that the A_ V distributions for the stars in the different populations are not significantly different, but the youngest population showing a lack of very low A_ V sources. The clear decrease of number of stars belonging to the younger populations, with respect to oldest one, indicates that the overall cluster formation process is coming to an end, and that the major activity took place in the beginning when the cluster started to form. This is opposed to the model of accelerated star formation proposed by <cit.>. In fact, the 1σ age intervals from Table <ref> indicate that the typical age spreads of the three populations are∼0.5-0.8 Myrs, which is in excellent agreement with the characteristic dynamical time-scale of 7×10^5 yr for the ONC derived by <cit.>. Clearly, this prompts for a revised look at the formation time-scales of stars in clusters.Based on data collected through ESO programme 096.C-0730(A). 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J., et al. 2015, , 449, 2618 | http://arxiv.org/abs/1705.09496v1 | {
"authors": [
"G. Beccari",
"M. G. Petr-Gotzens",
"H. M. J. Boffin",
"M. Romaniello",
"D. Fedele",
"G. Carraro",
"G. De Marchi",
"W. -J. de Wit",
"J. E. Drew",
"V. M. Kalari",
"C. F. Manara",
"E. L. Martin",
"S. Mieske",
"N. Panagia",
"L. Testi",
"J. S. Vink",
"J. R. Walsh",
"N. J. Wright"
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"categories": [
"astro-ph.SR",
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"primary_category": "astro-ph.SR",
"published": "20170526092953",
"title": "A Tale of Three Cities: OmegaCAM discovers multiple sequences in the color-magnitude diagram of the Orion Nebula Cluster"
} |
[ [ Received ; accepted =======================[1]Laboratory for Analysis and Architecture of Systems (LAAS), French National Center for Scientific Research (CNRS), 7, avenue du Colonel Roche, Toulouse, 31000, France ([email protected]). The research was funded by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement 666981 TAMING). mmsxxxxxxxx–x We consider the problem of solving a linear system of equations which involves complex variables and their conjugates. We characterize when it reduces to a complex linear system, that is, a system involving only complex variables (and not their conjugates). In that case, we show how to construct the complex linear system. Interestingly, this provides a new insight on the relationship between real and complex linear systems. In particular, any real symmetric linear system of equations can be solved via a complex linear system of equations. Numerical illustrations are provided. The mathematics in this manuscript constitute an exciting interplay between Schur's complement, Cholesky's factorization, and Cauchy's interlace theorem.Linear algebra, Orthogonal projection, Schur's complement, Cholesky factorization.myheadings plain CÉDRIC JOSZReal and Complex Linear Systems § INTRODUCTIONLet ℕ, ℝ and ℂ respectively denote the set of natural, real and complex numbers, and let i denote the imaginary number satisfying i^2=-1. Let z, z, and z̅ respectively denote the real part, the imaginary part, and the complex conjugate of z ∈ℂ. Let ℳ_n(ℝ) and ℳ_n(ℂ) respectively denote the set of real-valued and complex-valued square matrices of size n ∈ℕ. Consider M,N ∈ℳ_n(ℂ) and p ∈ℂ^n and the systemMz + Nz̅ = p , z ∈ℂ^n.The set of solutions to (<ref>) is a real affine space, but not a complex affine space in general. For instance, if n=1, M=N=1, and p=0, then the system reads z + z̅ = 0, z∈ℂ.We raise the question of when system (<ref>) can be reduced to a complex linear system A z = b , z ∈ℂ^nwhere A ∈ℳ_n(ℂ) and b ∈ℂ^n, and if so, how A and b can be constructed. Naturally, the set of solutions to (<ref>) is a complex affine space. Thus the system reduction can take place if and only if the set of solutions of system (<ref>) is a complex affine space. The paper is organized as follows. Section <ref> contains the main results which are applied to real linear systems in Section <ref>. Section <ref> concludes our work. § REDUCTION TO COMPLEX LINEAR SYSTEM When system (<ref>) has no solution, it can be reduced to a complex linear system (<ref>) with A = 0 and b = p. When it has one solution or more, Proposition <ref> provides a constructive reduction. We thus introduce the notation ℛ(M,N) to denote the range of system (<ref>), that is to sayℛ(M,N) := { p ∈ℂ^n | ∃ z ∈ℂ^n : Mz + Nz̅ = p }.Let's equip ℂ^2n with Hermitian inner product ⟨·, ·⟩ defined by ⟨ u , v ⟩ = ∑_k=1^2n u_k v_k. Let ℛ and Ker respectively denote the range and Kernel of a linear application. The key ingredient to obtain the reduction is the direct sumℂ^2n = ℛ[ N; M ]⊕[ℛ[ N; M ]]^⊥and, in particular, the orthogonal projection P_⊥ onto the latter set in the direct sum.Let n ∈ℕ and M,N ∈ℳ_n(ℂ). The following are equivalent: * { z ∈ℂ^n | Mz + N z̅ = 0 } is a complex vector space.* The application defined from ℂ^n to ℂ^2n defined by z ⟼ P_⊥[z; z̅ ] is injective.* ∃ U,V ∈ℳ_n(ℂ): ∀ p ∈ℛ(M,N),[ { z ∈ℂ^n | Mz + N z̅ = p };=; { z ∈ℂ^n | (UM + VN)z = Up + Vp }. ] The implication (3 ⟹ 1) follows from taking p=0. As for (1 ⟹ 2), consider z ∈ℂ^n such that P_⊥[z; z̅ ] = 0.Thus [z; z̅ ]∈ℛ[ N; M ].As a consequence, there exists w ∈ℂ^n such that [z; z̅ ] = [ Nw; Mw ]The first line minus the conjugate of the second line yields Nw - M w = 0.Multiplying by the imaginary number, we get M(iw) + N(iw) = 0. Thus iw belongs to { z ∈ℂ^n | Mz + N z̅ = 0 }, which is a complex vector space thanks to Point 1. Thus i(iw) = w also belongs to it, so that Mw + Nw = 0. Together with (<ref>), this implies that Nw = Mw = 0. Going back to (<ref>), it holds that z = 0. We now prove (2 ⟹ 3). Let p ∈ℛ(M,N) and consider z∈ℂ^n. Point 2 implies that the following four equations are equivalent:Mz+Nz̅ = p [ Mz; Nz ] + [ N z̅;Mz̅ ] =[ p; p ] P_⊥[ Mz; Nz ] + P_⊥[ N z̅;Mz̅ ] =P_⊥[ p; p ] P_⊥[ Mz; Nz ] = P_⊥[ p; p ]since, by definition of P_⊥, we haveP_⊥[ N z̅;Mz̅ ] = 0.Let [ Û V̂ ] denote the matrix representation of P_⊥ in the canonical basis of ℂ^2n. The complex matrices Û and V̂ have 2n rows and n columns. Equation (<ref>) is thus equivalent to(ÛM + V̂N)z = Ûp + V̂p.The matrix ÛM+V̂N has 2n rows and n columns so the above system has 2n equations and n unknowns. We would like to obtain a system with n equations. First, note that it has at least one solution because p ∈ℛ(M,N). Second, a linear system of equations with more equations than unknowns either has no solutions, or any equation that is a linear combination of other equations can be removed without changing the set of solutions. Thus, consider a set I ⊂{ 1 ,, 2n } of n rows that generate the row space of ÛM+V̂N. Define U,V ∈ℳ_n(ℂ) such that[ ∀ i ∈ I, ∀ j ∈{1,,n}, U_ij := Û_ij, V_ij := V̂_ij. ]We stress that U and V do not depend on the vector p, but only on the matrices M and N. We now decude that (<ref>) is equivalent to (UM + VN)z = Up + Vp, which terminates the proof.Point 2 in Proposition <ref> provides a necessary and sufficient condition on M and N to determine whether system (<ref>) can be reduced to a system of the form (<ref>). Let's write the condition more explicitly. The projection P_⊥ is an endormorphism of ℂ^2n, so let's consider its block decomposition P_⊥ = [ P_11 P_12; P_21 P_22 ]where P_11, P_12, P_11, P_11 are endormorphisms of ℂ^n. For all z ∈ℂ^n, we then haveP_⊥[z; z̅ ] = [ P_11 P_12; P_21 P_22 ][ x+iy; x-iy ] = [P_11+P_12 i(P_11-P_12);P_21+P_22 i(P_21-P_22) ][ x; y ]where x :=z and y :=z. Point 2 is thus equivalent toKer[P_11+P_12 P_12 -P_11;P_21+P_22 P_22 -P_21;P_11+P_12P_11-P_12;P_21+P_22P_21-P_22 ] = {0}where the domain of the application in (<ref>) is ℂ^4n and its co-domain is ℂ^2n. We illustrate Proposition <ref> with the following example:M := [ 0-i 0 2-i5i; 03i 0 39i;-1 51-3i -3+3i1+7i; 0 -2i 0-i-i; 0 1-i 0 i -2-3i ], N := [-i5i-3+i3-3i 7+i; 0 -3 + 2i 0-2 -1+3i; 0-1 0 -1+2i 5; 0 1 0 i-i; 0 5 07+4i1+2i ], p := [1-i;3; -1+i;5+i;1 ].Using a reduced column echelon form and a Grahm-Schmidt orthogonalization, we compute the matrix associated to P_⊥ in the canonical basis of ℂ^2n. We then check whether the system can be reduced using (<ref>). In this example, reduction is possible. We then compute a set of 5 rows I = {3,4,5,6,7} that generate the row space ofP_⊥[ M; N ].We then deduceU :=[0.0000 + 0.0000i -0.1686 - 0.0351i0.6782 + 0.0000i0.0612 + 0.0175i -0.0133 - 0.0054i;0.0000 + 0.0000i0.1592 - 0.0009i0.0612 - 0.0175i0.9171 + 0.0000i0.0044 - 0.1532i;0.0000 + 0.0000i0.0309 - 0.0069i -0.0133 + 0.0054i0.0044 + 0.1532i0.0329 + 0.0000i;0.0000 + 0.0000i -0.0351 + 0.1686i0.0000 + 0.3218i0.0175 - 0.0612i -0.0054 + 0.0133i;0.0000 + 0.0000i0.2393 - 0.1929i -0.0152 + 0.1880i -0.0881 + 0.0651i0.0169 + 0.0212i ]andV:= [0.0000 - 0.3218i -0.0152 - 0.1880i0.0000 + 0.0000i -0.0128 - 0.1089i0.1766 + 0.0448i;0.0175 + 0.0612i -0.0881 - 0.0651i0.0000 + 0.0000i0.0510 + 0.0511i0.0000 + 0.0433i; -0.0054 - 0.0133i0.0169 - 0.0212i0.0000 + 0.0000i -0.0523 + 0.0314i -0.0300 - 0.0397i;0.6782 + 0.0000i -0.1880 + 0.0152i0.0000 + 0.0000i -0.1089 + 0.0128i0.0448 - 0.1766i; -0.1880 - 0.0152i0.4239 + 0.0000i0.0000 + 0.0000i0.1337 - 0.0184i0.2104 - 0.0583i ].Next, we compute the reduced complex linear system (<ref>) whereA:= [ -0.3563 + 0.0000i2.4429 + 0.0891i0.3563 - 1.0690i -0.2151 + 0.8851i0.5264 + 0.8788i; -0.1225 + 0.0349i0.6516 - 1.0495i0.0178 - 0.4024i0.7686 - 0.0611i -0.1645 + 2.0554i;0.0266 - 0.0107i -0.0681 - 0.0324i0.0057 + 0.0905i -0.0774 - 0.1673i -0.1734 - 0.0619i;0.0000 + 0.3563i0.0891 - 2.4429i -1.0690 - 0.3563i0.8851 + 0.2151i0.8788 - 0.5264i;0.0303 - 0.3759i0.5094 + 1.6202i1.0975 + 0.4669i0.0983 - 3.0761i -1.1428 + 0.6953i ]andb := [ -0.6283 - 0.6565i;5.0216 + 0.9709i; -0.1995 + 0.8183i; -0.6565 + 0.6283i;2.0157 - 1.0104i ].We can now compute the solutions to the reduced system:[ -27.4310 +50.9483i;-4.0647 + 5.7543i; 0.0000 + 0.0000i; 2.7694 - 1.2220i; 0.6875 + 0.9203i ] +ℂ[ 1-3i;0;1;0;0 ]We then arbitrarily chose one solution to the reduced system, and then check whether it satisfies the original system. If so, then the set of solutions to the original system and to the reduced system coincide. If not, then the original system has no solutions. In this example, the original system and the reduced system coincide. We now turn our attention to the special case when the system (<ref>) has a unique solution. When in addition M or N is invertible, there exists an analytic expression for the matrices U and V in Proposition <ref> that define the reduction to a complex linear system. Below, the matrix I denotes the identity of ℳ_n(ℂ). Let n ∈ℕ and M,N ∈ℳ_n(ℂ).[ ∀ p ∈ℂ^n, ∃ ! z ∈ℂ^n: Mz + N z̅ = p;⟺;[ M N; N M ] is invertible. ]Assume that either equivalent property holds in (<ref>). Then, for all p ∈ℂ^n,[{ z ∈ℂ^n | Mz + N z̅ = p }; =; { z ∈ℂ^n | (UM + VN)z = Up + Vp } ]where U := I and V:= -NM^-1 if M is invertible, and where U := N^-1 and V := -M^-1 if M and N are invertible. (⟹) Consider z,w ∈ℂ^n such that[ M N; N M ][ z; w ] = [ 0; 0 ].This implies after conjugation that[ M N; N M ][ w; z ] = [ 0; 0 ].Adding the two previous equations yields[ M N; N M ][ z+w; z+w ] = [ 0; 0 ].Thus z+w is a solution to system (<ref>) for p=0, whose unique solution is 0. Hence z+w = 0. As a result, Mz - Nz̅ = 0, that is to say M(iz) + N(iz) = 0. Again by unicity of the solution, i z = 0. To conclude, z = w = 0. (⟸) Existence of a solution: there exists z,w ∈ℂ^n such that[ M N; N M ][ z; w ] = [ p; p ]which implies after conjugation that[ M N; N M ][ w; z ] = [ p; p ]Since the system has a unique solution, it must be that z=w. Thus z is a solution to (<ref>).Unicity of the solution: consider z,w ∈ℂ^n such that Mz +Nz̅ = p and Mw +Nw = p. It must be that M(z-w) + N(z-w) = 0, thus[ M N; N M ][ z-w; z-w ] = [ 0; 0 ].Since the above block matrix is invertible, it follows that z = w. To find suitable choices for the matrices U and V, it suffices for them to satisfy that UN + V M = 0 and that the application from ℂ^n to itself defined byz ⟼ Uz + Vz̅is injective. Indeed, in that case, we have the following equivalences for z∈ℂ^n:Mz+Nz̅ = p [ Mz; Nz ] + [ N z̅;Mz̅ ] =[ p; p ] (UM + VN)z + (UN+VM)z̅ = Up + Vp (UM + VN)z = Up + Vp.Let's assume that M is invertible. The relationship UN + V M = 0 is easy to prove. As for the injectivity, consider z ∈ℂ^n such that z -NM^-1z̅ = 0. Then w := M^-1z satisfies Mw - Nw = 0, so that M(iw) + N(iw) = 0. By assumption, the unique solution is iw = 0, hence z =0. To treat the case where M and N are invertible, it suffices to multiply U and V by N^-1. § APPLICATION TO REAL LINEAR SYSTEM Consider m ∈ℕ, a matrix A ∈ℳ_m(ℝ) that is positive definite, a vector b ∈ℝ^m, and the linear systemA x = b , x ∈ℝ^m.Thanks to a Gauss pivot, we may assume that m=2n and view the system over the set of complex numbers. The first half of the variables can be viewed as real parts of complex variables, and the second half as their imaginary parts. To this end, define B,C,D ∈ℳ_n(ℝ) and z,p ∈ℂ^n such thatA =: [ B C; C^T D ] , z := [ x_1; ⋮; x_n ] + i [ x_n+1; ⋮;x_2n ] , p := [ b_1; ⋮; b_n ] + i [ b_n+1; ⋮;b_2n ],where (·)^T stands for transpose. System (<ref>) is equivalent toM z + N z̅ = p , z ∈ℂ^n,where M := (B+D-i(C-C^T))/2≻ 0 (positive definiteness will become clear in the following) and N := (B-D+i(C+C^T))/2. Indeed, system (<ref>) can be written as either of the three following equivalent systems: * 1/2[ B C; C^T D ][z+z̅; i(z̅-z) ] = b,* {[ B-iC/2z + B+iC/2z̅=p; C^T-iD/2z + C^T+iD/2z̅= p,;]. * B+D-i(C-C^T)/2 z + B-D+i(C+C^T)/2z̅ = p.Proposition <ref> informs us that system (<ref>) is in turn equivalent to(M - NM^-1N)z = p - NM^-1p.As a result, the real linear system (<ref>) is equivalent to the complex linear system (<ref>). To the best of our knowledge, this relationship between real and complex linear systems has not been presented in past literature. With regards to numerical aspects, an efficient method for solving the linear system (<ref>) is the Cholesky factorization <cit.> which requires roughly m^3/3 = 8n^2/3 flops. It searches for a lower triangular matrix L ∈ℝ^m × m such that A = LL^T. Next, it successively solves for Ly=b, y ∈ℝ^m and L^Tx = y, x ∈ℝ^m. Unfortunately, the matrix multiplication in the complex linear system (<ref>) makes it more expensive to solve. However, the spectra of the positive definite matrices M∈ℳ_n(ℂ) and M - NM^-1N∈ℳ_n(ℂ) are contractions of the spectrum of A ∈ℳ_2n(ℝ). They hence have a better conditioning number (ratio of maximum to minimum eigenvalue) than A. This is relevant since a Cholesky factorization of both these matrices is required to solve the complex linear system (<ref>). For work on the complex Cholesky factorization, see <cit.>. We now clarify our statement regarding spectrum contraction. Given a Hermitian matrix U ∈ℳ_n(ℂ), let λ_1(U), , λ_n(U) denote its eigenvalues in increasing order. Given λ∈ℝ and z,w ∈ℂ^m, it holds that[ M N; N M ][ z; w ] = λ[ z; w ]⟺[ B C; C^T D ][ (z+w); (z+w) ] = λ[ (z+w); (z+w) ]thereby equating the spectra of the two square matrices in the equation. Since M is an extraction of the left-hand matrix, Cauchy's interlace theorem <cit.> implies that λ_k (A) ⩽λ_k(M) ⩽λ_k+n (A) , k = 1,, n.In addition, the interlacing property of the eigenvalues of the Schur complement of a Hermitian matrix <cit.> implies that λ_k (A) ⩽λ_k(M-NM^-1N) ⩽λ_k+n (A) , k = 1,, n.See <cit.> for a nice reference on the Schur complement. We now provide a numerical experiment. Consider the real linear system (<ref>) with m := 6,A:= ( [2.0483 -0.30650.7403 -0.33380.94311.4834; -0.30651.2538 -1.11440.7319 -0.24120.1729;0.7403 -1.11441.8337 -0.6019 -0.05180.6788; -0.33380.7319 -0.60191.65250.43130.0371;0.9431 -0.2412 -0.05180.43131.48930.3625;1.48340.17290.67880.03710.36251.5775 ])and b := ( [ -1.7746; -1.3900; -1.9215; -0.2593;1.3289;0.4696 ]).The data has been randomly generated with MATLAB R2015b. The eigenvalues of A in increasing order are λ(A) = ( [ 0.0245; 0.1082; 1.0932; 1.4319; 2.8521; 4.3453 ])MATLAB R2015b yields the following solution to Ax = b, x ∈ℝ^m:x = ( [ -33.1807; -56.9574; -42.5687; 2.4589;-3.3323;56.7669 ])Based on the discussion in this section, we may instead solve the complex linear system (M - NM^-1N)z = p - NM^-1p, z ∈ℂ^n, whereM - NM^-1N = ( [0.5756 - 0.0000i0.3906 + 0.2060i -0.1576 - 0.5196i;0.3906 - 0.2060i0.8131 - 0.0000i -0.5220 - 0.6089i; -0.1576 + 0.5196i -0.5220 + 0.6089i0.9137 - 0.0000i ])andp - NM^-1p = ( [ 0.1759 - 1.9830i; 2.1302 + 1.2597i; 1.4658 - 1.7835i ]).This yieldsz = ( [ -33.1807 + 2.4589i; -56.9574 - 3.3323i; -42.5687 +56.7669i ])to be compared with x in (<ref>).The eigenvalues of M - NM^-1N in increasing order areλ(M - NM^-1N) = ( [ 0.0463; 0.2488; 2.0073 ])which can be seen to interlace those of A in (<ref>). As a byproduct, the conditioning number of M - NM^-1N, equal to 43.3843, is less than that of A, equal to 177.3795. We remind the reader that the closer to 1, the better the conditioning of a matrix.§ CONCLUSION We classify linear systems involving complex variables and their conjugates with respect to systems involving only complex variables. Our main tool is the orthogonal projection. As a result, we obtain a new link between linear systems with real variables and linear systems with complex variables. Our findings are illustrated on several numerical examples. A natural question follows: when can algebraic varieties defined by non-holomorphic polynomials be reduced to varieties defined by holomorphic polynomials? This manuscript gives the answer when the polynomials are of degree one. § ACKNOWLEDGEMENTSI wish like to thank the anonymous reviewers for their helpful comments. Many thanks to Ximun Loyatho and Antoine Coutand for the fruitful discussions during my visit to Imswan.siam | http://arxiv.org/abs/1706.00268v1 | {
"authors": [
"Cédric Josz"
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"published": "20170526124951",
"title": "On the Relationship Between Real and Complex Linear Systems"
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firstpage–lastpage 2017Semiclassics in a system without classical limit:The few-body spectrum of two interacting bosons in one dimension Klaus Richter Accepted 2017 June 22. Received 2017 June 21; in original form 2017 April 21 =================================================================================================================== We present an analysis of the height distributions of the different types of supernovae (SNe) from the plane of their host galaxies. We use a well-defined sample of 102 nearby SNe appeared inside high-inclined (i ≥ 85^∘), morphologically non-disturbed S0–Sd host galaxies from the Sloan Digital Sky Survey. For the first time, we show that in all the subsamples of spirals, the vertical distribution of core-collapse (CC) SNe is about twice closer to the plane of host disc than the distribution of SNe Ia. In Sb–Sc hosts, the exponential scale height of CC SNe is consistent with those of the younger stellar population in the Milky Way (MW) thin disc, while the scale height of SNe Ia is consistent with those of the old population in the MW thick disc. We show that the ratio of scale lengths to scale heights of the distribution of CC SNe is consistent with those of the resolved young stars with ages from ∼10 Myr up to ∼100 Myr in nearby edge-on galaxies and the unresolved stellar population of extragalactic thin discs. The corresponding ratio for SNe Ia is consistent with the same ratios of the two populations of resolved stars with ages from a few 100 Myr up to a few Gyr and from a few Gyr up to ∼10 Gyr, as well as with the unresolved population of the thick disc. These results can be explained considering the age-scale height relation of the distribution of stellar population and the mean age difference between Type Ia and CC SNe progenitors.supernovae: general – galaxies: spiral – galaxies: stellar content – galaxies: structure – Galaxy: disc. § INTRODUCTIONThe detailed understanding of the spatial distribution of Supernovae (SNe) in galaxies provides a strong possibility to find the links between the nature of their progenitors and host stellar populations <cit.>. Such studies allow to constrain the important physical parameters of the different SN progenitors like their masses <cit.>, ages <cit.>, and metallicities <cit.>. For a comprehensive review addressing these issues, the reader is referred to <cit.>.According to the properties of SNe progenitors, they are divided into two general categories: core-collapse (CC) and Type Ia (thermonuclear) SNe. CC SNe are the colossal explosions that mark the violent deaths of young massive stars <cit.>,[According to the spectral features in visible light, CC SNe are classified into three basic classes <cit.>: hydrogen lines are visible in the spectra of Type II SNe, but in Types Ib and Ic SNe; helium lines are seen in the spectra of SNe Ib, but in SNe Ic. Subclass IIn SNe are dominated by narrow emission lines, while subclass IIb SNe have transitional spectra closer to SNe II at early times, then evolving to SNe Ib.] while SNe Ia are the explosive end in the evolution of binary stars in which one of the stars is an older white dwarf (WD) and the other star can be anything from a giant star to a WD <cit.>. Type Ia SNe result from stars of different ages <cit.>, with longer progenitors lifetime than the progenitors of CC SNe <cit.>.Usually, the spatial distribution of SNe in S0–Sm galaxies is studied with the reasonable assumption that all CC SNe and the vast majority of SNe Ia belong to the disc, rather than the bulge population <cit.>. Moreover, the distributions of SNe in the disc are studied assuming that the disc is infinitely thin <cit.>. The height distribution of SNe from the disc plane is mostly neglected when studying the host galaxies with low inclinations (close to face-on orientation) assuming that the exponential scale length of the radial distribution is dozens of times larger in comparison with the exponential scale height of SNe <cit.>.Direct measurements of the heights of SNe and estimates of the scales of their vertical distributions in host galaxies with high inclination (close to edge-on orientation) were performed only in a small number of cases <cit.>. Mainly due to the small number statistics of SNe and inhomogeneous data of their host galaxies, the comparisons of vertical distributions of the different types of SNe resulted in statistically insignificant differences. Therefore, while the detailed study of the vertical distributions in edge-on galaxies has allowed to constrain ages, masses and other physical parameters of their components <cit.>, the lack of analogous studies on the distribution of various SN types has prevented the determination of their parent populations via the direct comparison with the nearby extragalacric discs and the thick/thin discs of the Milky Way (MW) galaxy <cit.>.The purpose of this paper is to address these questions properly through an investigation of the vertical distributions of the main classes of SNe in a nearby sample of 102 SNe and their well-defined edge-on S0–Sd host galaxies from the Sloan Digital Sky Survey-III <cit.>.This is the fifth article of the series and the content is as follows. Sample selection and reduction are introduced in Section <ref>. Section <ref> describes the stellar disc model that we use to fit our data. All the results are discussed in Section <ref>. Section <ref> summarizes our conclusions. To conform to values used in our data base <cit.>, a cosmological model with Ω_ m=0.27, Ω_Λ=0.73, and H_0=73 kms^-1 Mpc^-1 Hubble constant <cit.> are adopted in this article. § SAMPLE SELECTION AND REDUCTIONIn this paper, we composed our sample by cross-matching the coordinates of classified Ia, Ibc[`Stripped-envelope' SNe of Types Ib and Ic, including the mixed Ib/c with uncertain subclassification, are denoted as SNe Ibc.], and II SNe from the Asiago Supernova Catalogue[We use the updated version of the catalogue, which includes all classified SNe exploded before 2015 January 1.] <cit.> with the footprint of SDSS Data Release 12 <cit.>. All SNe are required to have equatorial coordinates. We use SDSS DR12 and the approaches presented in Paper to identify the host galaxies and classify their morphological types. It is worth noting that morphological classification of nearly edge-on galaxies is largely based on the visible size of bulge relative to the disc because other morphological properties, such as the shape of spiral arms or presence of the bar, are generally obscured or invisible. The morphologies of galaxies are restricted to S0–Sd types, since we are interested in studying the vertical distribution of SNe in host stellar discs. A small number of Sdm–Sm host galaxies are not selected, because they show no clear discs.From the signs of galaxy–galaxy interactions, we classify the morphological disturbances of the hosts in the SDSS DR12 following the techniques described in detail in <cit.>. We then exclude from this analysis any galaxy disc exhibiting strong disturbances: interacting, merging, and post-merging/remnant.Using the techniques presented in Paper , we measure the apparent magnitudes and the geometry of host galaxies.[Instead using the data from Paper , which is based on the SDSS DR8, for homogeneity we re/measure the magnitudes and the geometry of all host galaxies, with additional new SN hosts included, based only on DR12.] In the SDSS g-band, we first construct isophotes, and then centred at the each galaxy centroid position an elliptical aperture visually fitted to the 25 mag arcsec^-2 isophote. We measure the apparent magnitudes, major axes (D_25), position angles (PA) of the major axes, and elongations (a/b) of galaxies using these apertures. In this analysis, we correct the magnitudes and D_25 for Galactic and host galaxy internal extinction (see Paper ).§.§ InclinationThe main difficulty in measuring the vertical distribution of SNe above the host stellar discs is that we have no way of knowing where along the line of sight the SNe lie. This means that reliable measurements can only be done in discs which are highly inclined, i.e., closer to an edge-on orientation (e.g. 85^∘⩽ i ⩽ 90^∘). In contrast to galaxies with lower inclination, the matter is complicated by the difficulty of making an accurate determination of the inclination angle. For these galaxies, the inclination cannot be measured simply from the major and minor axes because the presence of a central bulge places a limit on the axis-ratio even for a perfectly edge-on galaxy.This problem with the bulge has been solved by using the axial ratio of the exponential disc fits in the g-band provided by the SDSS (from the model with r^1/4 bulge and exponential disc), i.e., . Indeed, real stellar discs are not flat with negligible thicknesses, but have some intrinsic width, and a proper measurement of the inclination depends on this intrinsic ratio of the vertical and horizontal axes of the disc, known as q. Therefore, we calculate the inclinations of SNe host galaxies following the formulacos^2 i= ()^2-q^2/1-q^2,where i is the inclination angle in degrees between the polar axis and the line of sight and q is the intrinsic axis-ratio of galaxies viewed edge-on. According to <cit.>,q= dex[-(0.43+0.053t)]for -1≤ t≤ 7, where t is the morphological type code. Using equations (<ref>) and (<ref>), we restrict the inclinations of host galaxies to 85^∘⩽ i ⩽ 90^∘.All the selected SNe host galaxies are visually inspected because sometimes bright stars projected nearby, strong dust layers, bright nuclear/bulge emission, large angular sizes, etc. do not allow the SDSS automatic algorithm to correctly determine the parameters of galaxies, in particular the axis-ratio . The host discs with a clearly seen dust layer, or without signs of non-edge-on spiral arms, are selected as true edge-on galaxies. In other words, we exclude the discs whose galactic plane is not aligned along the major axis of their fitted elliptical apertures <cit.>. As a result, we select 106 SNe in edge-on host galaxies.In S0–Sd galaxies, all CC SNe and the vast majority of Type Ia SNe belong to the disc, rather than the bulge component <cit.>. Therefore, for the selected 106 SNe in this restricted sample of edge-on galaxies, we perform a visual inspection of the SNe positions on the SDSS images to identify the SNe from the bulge population of host galaxies. The result is that three Type Ia (1990G, 1993aj, and 2003ge) and one Type Ib/c (2005E) SNe may belong to the bulge because of their location. The three SNe Ia are clearly outside the host discs, located far in the bulge population. The Type Ib/c SN is also located far from the host galaxy disc but it is a peculiar, calcium-rich SN whose nature is still under debate and may have a different progenitor from typical CC <cit.>. All these four SNe are excluded from the sample.After these restrictions, we are left with a sample of 102 SNe within 100 host galaxies. The mean distance of this sample is 100±8 Mpc, the median distance and standard deviation are 78 Mpc and 84 Mpc, respectively. The mean D_25 of our host galaxies is 108±10 arcsec with the smallest value of 22 arcsec. Table <ref> displays the distribution of all SNe types among the various considered morphological types of host galaxies. Fig. <ref> shows images of typical examples of edge-on host galaxies with marked positions of SNe. §.§ Measurements of the heights of SNeThe heights of SNe above host galactic plane might be calculated by using the simple formulas presented in <cit.> with available SNe offsets from host galaxy nuclei and PA of the galaxies (see also Paper ). However, as demonstrated in Paper , SN catalogs report different offsets with different levels of accuracy. Individual offsets are based on the determination of the positions of the host galaxy nuclei, which might be uncertain and depend on many factors (e.g. colour of image, plate saturation, galaxy peculiarity, incorrect SDSS fiber targeting of the galaxy nucleus, etc.). For more details, the reader is referred to Paper .For this study, using the SN coordinates and its edge-on host galaxy image in the SDSS g-band, we measure the perpendicular distance, i.e., the height, from the major axis of the fitted elliptical aperture of each galaxy to the position of SN. At the same time, using the coordinates of the host galaxy nucleus, we also measure the projected galactocentric radius of SN along the same major axis.[We remind that in comparison with the measured heights, the measurements of projected galactocentric radii of SNe include some minor inaccuracy because of the mentioned uncertain determination of the exact point like positions of host galaxy nuclei. The projected galactocentric radii are only used in Fig. <ref> of Section <ref> for ancillary purposes.] Fig. <ref> schematically illustrates the geometrical location of an SN within an edge-on disc, where v is the height (in arcsec) and u is the projected galactocentric radius (in arcsec) of the SN. A similar technique was also used in <cit.> on the Digital Sky Survey (DSS) images to determine the v and u coordinates of SNe.It is important to note that as in the case of the radial distribution of SNe in face-on galaxies <cit.>, the distribution of linear distances in the vertical direction is biased by the greatly different intrinsic sizes of host discs. Fig. <ref> illustrates the comparison of the heights v of SNe and R_25 of host galaxies in kpc. Also shown are the best fit lines log(v_ Ia) = (-1.10±0.11) + (0.89±0.08)log(R_25), log(v_ CC) = (-1.68±0.15) + (1.07±0.13)log(R_25) with near unity slopes. To check the significance of the correlations, we use the Spearman's rank correlation test, which indicates strong positive trend between the heights and R_25 for Type Ia SNe (r_ s=0.382, P=0.005), while not significant for CC SNe (r_ s=0.166, P=0.255). Therefore, in the remainder of this study, we use only relative heights and projected galactocentric radii of SNe, i.e., normalized to R_25=D_25/2 of host galaxies in g-band.The full data base of 102 individual SNe (SN designation, type, equatorial coordinates, v and u) and their 100 host galaxies (galaxy SDSS designation, distance, morphological type and corrected g-band D_25) is available in the online version (Supporting Information) of this article.§ THE MODEL OF STELLAR DISCIn our model, the volumetric density ρ^ SN(r̃, z̃) of SNe in the host axisymmetric stellar discs is assumed to vary as follows in the radial r̃ and vertical z̃ directions: ρ^ SN(r̃, z̃)=ρ_0^ SNexp(-r̃/h̃_ SN)f(z̃), where r̃=R_ SN/R_25, z̃=z_ SN/R_25 and (R_ SN, z_ SN≡v) are cylindrical coordinates, ρ_0^ SN is the central volumetric density, h̃_ SN = h_ SN/R_25 is the radial scale length, and f(z̃) is a function describing the vertical distribution of SNe.In equation (<ref>), we adopt a generalized vertical distribution f(z̃)= sech^2/n(nz̃/z̃_0^ SN), where z̃_0^ SN = z_0^ SN/R_25 is the vertical scale height of SNe and n is a parameter controlling the shape of the profile near the plane of host galaxy. Following the vertical surface brightness distribution of edge-on galaxies <cit.>, we also assume that the scale height of SNe is independent of projected galactocentric radius <cit.>, i.e., there is no disc flaring.Recent photometric fits to the surface brightness distribution of a large number of edge-on galaxies in near-infrared <cit.> and SDSS g-, r-, and i-bands (, see alsofor other photometric bands) suggest that a value of n=1 is an appropriate model of stellar discs. When n →∞, equation (<ref>) reduces to f(z̃)∼exp(-|z̃|/H̃_ SN), where H̃_ SN=z̃_0^ SN/2 at large heights, and is widely used to successfully fit the dust distribution in edge-on galaxies <cit.>.In linear units, the exponential (exp) form of f(z̃) is used to model the distribution of Galactic stars <cit.>, novae <cit.>, SNe <cit.>, SN remnants <cit.>, pulsars <cit.>, and extragalactic SNe <cit.>, while the sech^2 form is used to fit the vertical distribution of resolved stars <cit.> and CC SNe <cit.> in highly inclined nearby galaxies.Note that sech^2 profile (n=1) is expected for an isothermal stellar population <cit.>, while exp profile (n →∞) can be obtained by a combination of isothermal stellar populations with different “temperatures” (velocity dispersions). While at large heights, sech^2 (x) → 4exp(-2x), at low heights, the sech^2 profile is uniform, while the exp profile is cuspy. § RESULTS AND DISCUSSION§.§ The vertical distribution and scale height of SNe We fit sech^2 and exp forms of f(z̃) profile to the distribution of normalized absolute heights (|z̃|≡|v|/R_25) of SNe, using maximum likelihood estimation (MLE). Here, because of the small number statistics of Type Ibc SNe (see Table <ref>), we group them with Type II SNe in a larger CC SNe sample. Fig. <ref> shows the histograms of the normalized heights with the fitted sech^2 and exp probability density functions (PDFs) for Type Ia and CC SNe in Sa–Sd galaxies.[For this comparative illustration, we do not include S0–S0/a galaxies because they host almost only Type Ia SNe (see Table <ref>). For the sake of visualization, the distribution of Type Ibc SNe is also presented in the bottom panel of Fig. <ref>.] In columns 4, 7, and 10 of Table <ref>, we list the mean values of |z̃| and the maximum likelihood scale heights for both types of SNe in various subsamples of host galaxies. From column 4 of Table <ref>, it immediately becomes clear that in all the subsamples of host galaxies the vertical distribution of CC SNe is about twice closer to the plane of host disc than the distribution of Type Ia SNe. In fact, the two-sample Kolmogorov–Smirnov (KS) and Anderson–Darling (AD) tests,[The two-sample AD test is more powerful than the KS test <cit.>, being more sensitive to differences in the tails of distributions. Traditionally, we chose the threshold of 5 per cent for significance levels of the different tests.] shown in Table <ref>, indicate that this difference is statistically significant in Sa–Sd galaxies, although not significant if only late-type hosts are considered.Note that four Type IIb SNe are included in Type II SNe sample (see Table <ref>). For Sa–Sd galasies, it might be reasonable also to group Types IIb and Ibc SNe as a wider `stripped-envelop' (SE) SN class (13 objects) and compare them with pure Type II SNe (35 objects). However, we find no difference between the vertical distributions of SE and pure Type II SNe (P_ KS=0.401, P_ AD=0.320), resulting in statistically indistinguishable scale lengths between these SN types. Therefore, in the remainder of this study, we will group all these subtypes as the main CC SN sample and compare that with Type Ia SN sample. It is important to note that dust extinction in edge-on SN host galaxies might have an impact on our estimated scale heights.[Another factor, such as a deviation from perfectly edge-on orientation of the host discs, may also affect our estimation of the scale heights, increasing them. However, we are quite confident that our galaxies can vary by a few degrees only from perfectly edge-on orientation (see Section <ref>). In addition, other authors have demonstrated that slight deviations from i=90^∘ have minimal impact on the derived structural parameters of the vertical distributions of different stellar populations <cit.>.] In Paper , we demonstrated that in general there is a lack of SNe host galaxies with high inclinations, which can be explained by a bias in the discovery of SNe due to strong dust extinction <cit.>, particularly in edge-on hosts <cit.>.The vertical distribution of dust in disc galaxies has an exponential profile with about three times smaller scale height in comparison with distribution of all stars <cit.>.[This value can vary from two to four, depending, respectively, on early- and late-type morphology of edge-on spiral galaxies <cit.>.] Analysing the vertical distribution of the resolved stellar populations in nearby edge-on galaxies, <cit.> found that the dust has negligible impact on the distribution parameters of stars at |z| ≳ H_ dust heights <cit.>. Therefore, in Table <ref>, to check the impact of the dust extinction on the obtained scale heights, we also estimate the distribution parameters considering the SNe in Sa–Sd galaxies only at |z̃|>H̃_ dust heights. For the average dust scale height, we use H̃_ dust=0.02, roughly considering that H_ dust≈ H_ Ia/3 <cit.>. In Fig. <ref>, we show the distribution of coordinates of SNe along the major (u/R_25) and minor axes (z̃≡v/R_25) of their Sa–Sd host galaxies with the |z̃|≤0.02 opaque region, and for the sake of visualization, we scale the distribution to the PGC 037591 galaxy (also shown in Fig. <ref>, better known as NGC 3987), which is one of the representatives of the edge-on galaxies with a prominent dust line along the major axis.From columns 7 and 10 of Table <ref> (the subsamples of Sa–Sd hosts labeled with `†' symbols), despite the small number statistics (column 3), we see that the extinction by dust near to the plane of host galaxies does not strongly bias the estimated scale heights of SNe. The scale height of CC SNe with |z̃|>0.02 is almost equal to that with the |z̃|≥0, while the scale height of Type Ia SNe with |z̃|>0.02 is only ∼15 per cent greater (still statistically insignificant) than that with the |z̃|≥0. In the remainder of this study, we will generally use the scale heights of SNe without height-truncation due to the small number statistics and insignificance of the effect, however, if needed, we will emphasize the impact of the dust extinction on the scale heights.To check whether the distribution of SN heights follows the best-fitting profiles, we perform one-sample KS and AD tests on the cumulative distribution of the normalized absolute heights (|z̃|), where the sech^2 and exp models have E(|z̃|) =tanh(|z̃|/z̃_0^ SN) and E(|z̃|) = 1 - exp(-|z̃|/h̃_ z^ SN) cumulative distribution functions (CDFs), respectively. Columns 5, 6, 8 and 9 of Table <ref> show the KS and AD probabilities that the vertical distributions are drawn from the best fitting profile. Cumulative distributions of the heights and CDFs of the fitted forms for Type Ia and CC SNe in Sa–Sd galaxies are presented in the insets of Fig. <ref>.From columns 5, 6, 8 and 9 of Table <ref>, we see that the vertical distribution is consistent with both profiles in most subsamples of Type Ia SNe and in all subsamples of CC SNe. For Type Ia SNe in Sa–Sd (also in S0–Sd) galaxies, the vertical distribution is consistent with the exp profile, but not with the sech^2 one (as seen in the AD statistic but only very marginally in the KS statistic). When we separate SNe Ia between early- and late-type host galaxies, the inconsistency vanishes with only barely AD test significance in early-type spirals (see the P_ AD value in column 6 of Table <ref> for SNe Ia in Sa–Sbc galaxies). The ⟨|z̃|⟩ value (scale heights too) for SNe Ia is ∼25 per cent greater in Sa–Sbc galaxies than that in Sc–Sd hosts (although the difference is not significant, see Table <ref>), while for CC SNe this parameter has a nearly constant value in the mentioned subsamples. This effect can be attributed to the earlier and wider morphological distribution of SNe Ia host galaxies (from S0/Sa to Sd, see Table <ref> and also Papers and ) in comparison with CC SNe hosts, and the systematically thinner vertical distribution of the host stellar population from early- to late-type discs <cit.>.In the first attempts to estimate the mean value of the vertical coordinates of SNe, <cit.> used the distribution of SN colour excesses without precise information on their spectroscopic types and host galaxy morphology in a sample of non-edge-on spirals. No difference was found in the vertical distributions of Type I and II SNe with indication that both types belong to the young population I. However, the inclinations of host galaxies and the uncertain separation[Type Ibc SNe were labelled as `I pec' types during observations before 1986 and included in the sample of Type I SNe.] of SN types might be the reason for the similarity between the vertical distributions of the mentioned SN types. Using a similar colour excess data of the best photometrically studied Type Ia SNe in late-type galaxies, <cit.> showed that these SNe have a considerably broader vertical distribution than the dust discs of their hosts and concluded that SNe Ia are older than the old disc population.Direct measurements of the heights of SNe and estimation of the scales of their vertical distributions were performed only in a small number of cases <cit.>. <cit.> examined the offsets between the major axes of a sample of highly inclined (i ≥ 60^∘) galaxies and the SNe they hosted in an attempt to measure the scale heights of Type Ia and II SNe. Unfortunately, the sample of such objects was quite small (66 galaxies), especially when restricted to galaxies at i ≥ 75^∘, which resulted in statistically indistinguishable vertical distributions (in kpc) between the mentioned types of SNe. <cit.> used data from the ASC to study the vertical distribution (in kpc) of 64 CC SNe in highly inclined (i ≥ 80^∘) Sa–Sd host galaxies. He showed that the distribution can be well fitted by a sech^2 profile. However, these studies only used linear scales to estimate the vertical distribution of SNe. This is somewhat undesirable because the absolute distribution of SN heights (in kpc) is biased by the greatly different intrinsic sizes of host discs (as already shown in Fig. <ref>).Most recently, <cit.> studied the absolute (in kpc) and relative (normalized to radius of host galaxy) vertical distributions of SNe using a sample of 26 Type Ia, 8 Ibc, and 44 II SNe in spiral host galaxies with i ≥ 85^∘. They found that the distributions can be fitted by exp profiles with scale heights H̃_ Ia=0.030±0.006, H̃_ Ibc=0.024±0.006, and H̃_ II=0.029±0.005. The scale heights for Type Ibc and II SNe are in good agreement with our H̃_ CC=0.028±0.003 in Sa–Sd galaxies, while their scale height for Type Ia SNe is much smaller than our H̃_ Ia=0.055±0.007 in the same morphological bin. However, the direct comparison of the scale heights obtained bywith ours is difficult because they used the DSS images for reduction of SNe host galaxies without mentioning the photometric band (we assume that they used B-band), while we use the SDSS g-band to normalize the heights to the 25^ th magnitude isophotal semimajor axes of host galaxies. On the other hand, we are not able to check the consistency between the morphological distributions of edge-on galaxies hosting Type Ia and CC SNe in their and our samples because morphological types were not provided by .To exclude any dependence of scale height of host stellar population on the morphological type, we analyse the vertical distribution of SNe in the most populated morphological bins, i.e., in the narrower Sb–Sc subsample (see Table <ref>).[On the other hand, by selecting these bins we reduce the possible contribution by SNe Ia from central bulges of host galaxies, although the bulge contribution is only up to 9 per cent of the total SN Ia population in Sa–Sd host galaxies (Barkhudaryan et al. in preparation).] In addition, the Sb–Sc subsample is more suitable for comparison of the estimated vertical scale heights of SNe with those of different stellar populations of thick and thin discs of the MW galaxy (see Section <ref>), and to exclude a small number of very thin discs <cit.>, which usually appear in late-type galaxies.From Table <ref>, we conclude that the vertical distributions of Type Ia and CC SNe in Sb–Sc galaxies can be well fitted by both the sech^2 and exp profiles. The vertical distribution of CC SNe is significantly different from that of Type Ia SNe (Table <ref>), being 2.3±0.5 times more concentrated to the plane of the host disc (Table <ref>). This difference also exists when the above-mentioned effect of the dust extinction is considered for the particular subsample (Sb–Sc hosts labeled with `†' symbols in Tables <ref> and <ref>). In Fig. <ref>, we present the comparison of vertical distributions as well as the fitted sech^2 and exp CDFs between both the types of SNe in Sb–Sc host galaxies.It is important to note that Type Ia SNe, because of their comparatively high luminosity <cit.> and the presence of dedicated surveys, are discovered at much greater distances than CC SNe (see Paper ). To check the possible distance biasing on the vertical distribution of SNe, we truncate the sample of Sb–Sc galaxies to distances ≤ 200 Mpc.[It would be more effective to check this with distance-truncation at 150 (100) Mpc (see Papers and ), however the remaining statistics in this case is very low, which destroys any comparison with significance. With the mentioned distance-truncation, we have only 19 (9) Type Ia SNe with ⟨|z̃|⟩=0.071±0.019 (0.086±0.025) and 30 (24) CC SNe with ⟨|z̃|⟩=0.027±0.005 (0.031±0.006).] In Table <ref>, the comparison of ⟨|z̃|⟩, z̃_0^ SN, and H̃_ SN as well as P_ KS and P_ AD values of distance-truncated sample (labeled with `*' symbols) with those of Sb–Sc host galaxies allows to conclude that possible distance biasing in our sample is negligible. Due to the smaller number statistics, we get larger error bars in Table <ref>, and lose only the KS test significance in Table <ref>. Therefore, in the remainder of this study, we will use SNe in Sb–Sc galaxies without distance-truncation.§.§ The thick and thin discsIt is largely accepted that the disc of the MW, one of the well-studied representatives of Sb–Sc classes, is separated into at least three components/populations: (1) the youngest star-forming disc (H̃≲0.01), including molecular clouds and massive young stars; (2) the younger thin disc (H̃∼0.02), which contains stars with a wide range of ages; and (3) the old thick disc (H̃∼0.06), composed almost exclusively of older stars <cit.>. For extragalactic discs of nearby edge-on spirals, the thick and thin components are also resolved <cit.>. In this sense, we may be able to put constraints on the nature of the progenitors of Type Ia and CC SNe by comparing the parameters of their distributions (H̃_ SN or z̃_0^ SN and h_ SN/z_0^ SN or h_ SN/H_ SN) in edge-on Sb–Sc galaxies with those of different stellar populations of thick and thin discs of MW and other similar galaxies. Note that the mean luminosity of our sample of Sb–Sc host galaxies (⟨ M_ g⟩=-20.5±1.0) is in good agreement with that of the MW <cit.>.In Table <ref>, we list the exp scale heights of SNe estimated in this study and the exp scale heights of the MW thick and thin discs derived from star counts (from hundreds of thousands to millions of individual stars) by other authors. As can be seen, the scale height of the vertical distribution of CC SNe is consistent with those of younger stellar population in the thin disc <cit.>, while the scale height of Type Ia SNe is consistent with those of old population in the thick disc <cit.> of the MW galaxy.Note that, in Table <ref>, the MW H̃ values are calculated using the original values of H (in kpc) from the references and assuming R_25^ MW=15±1 kpc, i.e., H̃=H/R_25^ MW, while the ratio of radial to vertical scales (h/H) would be better for a comparison of SNe distribution with the distribution of stars in the MW, avoiding the use of ambiguous value of R_25^ MW.In Paper , we studied the radial distributions of SNe and estimated the scale lengths of Type Ia and CC SNe using a well-defined sample of 500 nearby SNe and their low-inclined (i ≤ 60^∘) and morphologically non-disturbed S0–Sm host galaxies from the SDSS.[At these inclinations, dust extinction has minimal impact on the efficiency of SNe discovery <cit.>, making the estimation of the scale lengths as the most reliable.] In particular, the radial distributions of Type Ia and CC SNe in spiral galaxies are consistent with one another and with an exponential surface density according to exp(-r̃/h̃_ SN) in equation (<ref>) where r̃=R_ SN/R_25 and h̃_ SN=h_ SN/R_25=0.21±0.02. However, to be consistent with the present study, we use the estimation of the scale lengths of SNe restricted to Sb–Sc host galaxies from that sample. Note that the similar determination of the sample of the present paper is not possible because of its extreme inclination. For both types of SNe, we find h̃_ SN=0.20±0.02 using 79 Type Ia and 198 CC SNe.In Table <ref>, we list the ratios of radial to vertical scales of SNe (h_ SN/H_ SN) estimated in this study and the analogous ratios of MW thick and thin discs derived from star counts by other authors. The ratio of scales of CC SNe appears consistent with those of the younger stellar population in the thin disc, while the corresponding ratio of Type Ia SNe is consistent with the old population in the thick disc of the MW (although on the small side).It should be noted that the parameters of the vertical distributions of different stellar populations in the MW are determined using samples dominated by stars relatively near the Sun, not including the sizable population of the disc <cit.>. Therefore, the structural parameters of the MW may be different from those of other galaxies. In particular, <cit.> analysed the vertical distribution of the resolved stellar populations in nearby six edge-on Sc galaxies observed with the Hubble Space Telescope and found that the ratios of radial to vertical scales of young star-forming discs are much smaller (∼ 3-4 times) than that of the MW. In other words, the young star-forming discs of their sample galaxies are much thicker in comparison with that of the MW. Their results are in agreement with those of <cit.>, who analysed the vertical structure of 34 late-type, edge-on, undisturbed disc galaxies using the two-dimensional fitting to their photometric profiles.Interestingly, <cit.> found that the scale height of a stellar population increases with age, which is also correct for the MW galaxy <cit.>. They used colour-magnitude diagrams (CMDs) to estimate the ages of resolved stellar populations <cit.>. The young population in their main-sequence (MS) box of the CMD is dominated by stars with ages from ∼10 Myr up to ∼100 Myr, the intermediate population in the asymptotic giant branch (AGB) box is dominated by stars with ages from a few 100 Myr up to a few Gyr, while the old population in the red giant branch (RGB) box is dominated by stars with ages from a few Gyr up to ∼10 Gyr. In light of this, we compare in Table <ref> the ratios of radial to vertical scales of SNe with those detected from resolved stars in nearby edge-on late-type galaxies <cit.> and from unresolved populations of extragalactic thick and thin discs estimated using the edge-on surface brightness profiles <cit.>.[Here, to be consistent with the original values from the references, we use the h_ SN/z_0^ SN ratios.]In Table <ref>, we see that the ratio of scales of the distribution of CC SNe is consistent with those of the resolved MS-box stars in <cit.> and unresolved stellar population of the thin disc in <cit.>. On the other hand, the h_ SN/z_0^ SN ratio of Type Ia SNe is consistent and located between the values of the same ratios of resolved RGB- and AGB-box stars, respectively <cit.>. In addition, the h_ SN/z_0^ SN ratio of Type Ia SNe is consistent with those of the unresolved population of the thick disc in <cit.> and with the thick+thin disc population in <cit.>.These results are in good agreement with the age-scale height relation of stars in galaxy discs <cit.>, and that Type Ia SNe result from stars of different ages <cit.>, with even the shortest lifetime progenitors having much longer lifetime than the progenitors of CC SNe <cit.>. § CONCLUSIONSIn this fifth paper of a series, using a well-defined and homogeneous sample of SNe and their edge-on host galaxies from the coverage of SDSS DR12, we analyse the vertical distributions and estimate the sech^2 and exp scale heights of the different types of SNe, associating them to the thick or thin disc populations of galaxies. Our sample consists of 100 nearby (the mean distance is 100±8 Mpc), high-inclination (i ≥ 85^∘), and morphologically non-disturbed S0–Sd galaxies, hosting 102 SNe in total.The extinction by dust near to the plane of edge-on host galaxies has an insignificant impact on our estimated SN scale heights, although as was shown previously (e.g. Paper ), it is significantly decreasing the efficiency of SN discovery in these galaxies. We also check that there is no strong redshift bias within our SNe and host galaxies samples, which could drive the observed behaviours of the vertical distributions of the both SN types in host galaxies with edge-on discs.The results obtained in this article are summarized below, along with their interpretations.* For the first time, we show that in both early- and late-type edge-on spiral galaxies the vertical distribution of CC SNe is about twice more concentrated to the plane of host disc than the distribution of Type Ia SNe (Fig. <ref> and Table <ref>). The difference between the distributions of the SN types is statistically significant with only the exception in late-type hosts (Table <ref>).* When considering early- and late-type spiral galaxies separately, the vertical distributions of Type Ia and CC SNe are consistent with both the sech^2 and exp profiles (Table <ref>). In wider morphological bins (S0–Sd or Sa–Sd), the vertical distribution of Type Ia SNe is not consistent with sech^2 profile, most probably due to the earlier and wider morphological distribution of SNe Ia host galaxies in comparison with CC SNe hosts (Table <ref>), and the systematically thinner vertical distribution of the host stellar population from early- to late-type discs.* By narrowing the host morphologies to the most populated Sb–Sc galaxies (close to the MW morphology) of our sample, we exclude the morphological biasing of host galaxies between the SN types and the dependence of scale height of host stellar population on the morphological type. In these galaxies, we find that the sech^2 scale heights (z̃_0^ SN) of Type Ia and CC SNe are 0.096±0.016 and 0.042±0.007, respectively. The exp scale heights (H̃_ SN) are 0.065±0.012 and 0.028±0.003, respectively. In Sb–Sc galaxies, the vertical distribution of CC SNe is significantly different from that of Type Ia SNe (Table <ref>), being 2.3±0.5 times more concentrated to the plane of the host disc (Table <ref>).* In Sb–Sc hosts, the exp scale height (also the h_ SN/H_ SN ratio) of CC SNe is consistent with that of the younger stellar population in the thin disc of the MW, derived from star counts, while the scale height (also the ratio) of SNe Ia is consistent with that of the old population in the thick disc of the MW (Tables <ref> and <ref>).* For the first time, we show that the ratio of scale lengths to scale heights (h_ SN/z_0^ SN) of the distribution of CC SNe is consistent with those of the resolved young stars with ages from ∼10 Myr up to ∼100 Myr in nearby edge-on galaxies and the unresolved stellar population of extragalactic thin discs (Table <ref>). On the other hand, the corresponding ratio for Type Ia SNe is consistent and located between the values of the same ratios of the two populations of resolved stars with ages from a few 100 Myr up to a few Gyr and from a few Gyr up to ∼10 Gyr, as well as with the unresolved population of the thick disc of nearby edge-on galaxies. All these results can be explained considering the age-scale height relation of the distribution of stellar population and the mean age difference between Type Ia and CC SNe progenitors. § ACKNOWLEDGEMENTS We would like to thank the anonymous referee for his/her commentary, and also Massimo Della Valle for his constructive comments on the earlier drafts of this manuscript. AAH, LVB, and AGK acknowledge the hospitality of the Institut d'Astrophysique de Paris (France) during their stay as visiting scientists supported by the Programme Visiteurs Extérieurs (PVE). This work was supported by the RA MES State Committee of Science, in the frames of the research project number 15T–1C129. AAH is also partially supported by the ICTP. VA acknowledges the support from Fundação para a Ciência e Tecnologia (FCT) through national funds and from FEDER through COMPETE2020 by the following grants UID/FIS/04434/2013 & POCI-01-0145-FEDER-007672, and the support from FCT through Investigador FCT contract IF/00650/2015/CP1273/CT0001. This work was made possible in part by a research grant from the Armenian National Science and Education Fund (ANSEF) based in New York, USA. Funding for SDSS-III has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, and the US Department of Energy Office of Science. The SDSS–III web site is http://www.sdss3.org/http://www.sdss3.org/. SDSS–III is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS–III Collaboration including the University of Arizona, the Brazilian Participation Group, Brookhaven National Laboratory, University of Cambridge, University of Florida, the French Participation Group, the German Participation Group, the Instituto de Astrofisica de Canarias, the Michigan State/Notre Dame/JINA Participation Group, Johns Hopkins University, Lawrence Berkeley National Laboratory, Max Planck Institute for Astrophysics, New Mexico State University, New York University, Ohio State University, Pennsylvania State University, University of Portsmouth, Princeton University, the Spanish Participation Group, University of Tokyo, University of Utah, Vanderbilt University, University of Virginia, University of Washington, and Yale University.§ SUPPORTING INFORMATIONAdditional Supporting Information may be found in the online version of this article:PaperVonlinedata.csv Please note: Oxford University Press is not responsible for the content or functionality of any supporting materials supplied by the authors. Any queries (other than missing material) should be directed to the corresponding author for the article. | http://arxiv.org/abs/1705.09626v2 | {
"authors": [
"A. A. Hakobyan",
"L. V. Barkhudaryan",
"A. G. Karapetyan",
"G. A. Mamon",
"D. Kunth",
"V. Adibekyan",
"L. S. Aramyan",
"A. R. Petrosian",
"M. Turatto"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170526154119",
"title": "Supernovae and their host galaxies - V. The vertical distribution of supernovae in disc galaxies"
} |
Super Vertex Algebras, Meromorphic Jacobi Forms and Umbral Moonshine [==================================================================== We define a second-order neural network stochastic gradient training algorithm whose block-diagonal structure effectively amounts to normalizing the unit activations. Investigating why this algorithm lacks in robustness then reveals two interesting insights. The first insight suggests a new way to scale the stepsizes, clarifying popular algorithms such as RMSProp as well as old neural network tricks such as fanin stepsize scaling. The second insight stresses the practical importance of dealing with fast changes of the curvature of the cost.§ INTRODUCTION Although training deep neural networks is crucial for their performance, essential questions remain unanswered.Almost everyone nowadays trains convolutional neural networks (CNNs) using a canonical bag of tricks such as dropouts, rectified linear units (ReLUs), and batch normalization <cit.>. Accumulated empirical evidence unambiguously shows that removing one of these tricks leads to less effective training.Countless papers propose new additions to the canon. Following the intellectual framework set by more established papers, the proposed algorithmic improvements are supported by intuitive arguments and comparative training experiments on known tasks.This approach is problematic for two reasons.First, the predictive value of intuitive theories is hard to assess when they share so little with each other. Second, the experimental evidence often conflates two important but distinct questions: which learning algorithm works best when optimally tuned, and which one is easier to tune.We initially hoped to help the experimental aspects by offering a solid baseline in the form of an efficient and well understood way to tune a simple stochastic gradient (SG) algorithm, hopefully with a performance that matches the canonical bag of tricks. To that effect, we consider reparametrizations of feedforward neural networks that are closely connected to the normalization of neural network activations <cit.> and are amenable to zero overhead stochastic gradient implementations.Invoking the usual second order optimization arguments <cit.> leads to tuning the reparametrization with a simple diagonal or block-diagonal approximation of the inverse curvature matrix. The resulting algorithm performs well enough to produce appealing training curves and compete favorably with the best known methods. However this algorithm lacks robustness and occasionally diverges with little warning. The only way to achieve robust convergence seems to reduce the global learning rate to a point that negates its speed benefits.Our critical investigation led to the two insights that constitute the main contributions of this paper.The first insight provides an elegant explanation for popular algorithms such as RMSProp <cit.> and also clarifies well-known stepsize adjustments that were popular for the neural networks of the 1990s. The second insight explains some surprising aspects of batch normalization <cit.>. These two insights provide a unified perspective in which we can better understand and compare how popular deep learning optimization techniques achieve efficiency gains. This document is organized as follows.Section <ref> describes our reparametrization scheme for feedforward neural networks and discusses the efficient implementation of a SG algorithm. Section <ref> revisits the notion of stepsizes when one approximates the curvature by a diagonal or block-diagonal matrix. Section <ref> shows how fast curvature changes can derail many second order optimization methods and justify why it is attractive to evaluate curvature on the current minibatch as in batch-normalization.§ ZERO OVERHEAD REPARAMETRIZATIONThis section presents our reparametrization setup for the trainable layers of a multilayer neural network.Consider a linear layer[Appendix <ref> discusses the case of convolutional layers.]with n inputs x_i and m outputs y_j∀ j∈Ι1my_j = w_ + ∑_i=1^n x_i w_ij .Let E represent the value of the loss function for the current example. Using the notation g_j = Ey_j and the convention x_0=1, we can write∀ (i,j)∈Ι0n×Ι1m Ew_ij = x_i g_j . §.§ Reparametrization We consider reparametrizations of (<ref>) of the formy_j = β_j ( v_ + ∑_i=1^n α_i (x_i-μ_i) v_ij) ,where v_ and v_ij are the new parameters and μ_i, α_i>0, and β_j>0 are constants that specify the exact reparametrization. The old parameters can then be derived from the new parameters with the relationsw_ij = α_iβ_j v_ij( for i=1… n. )w_ = β_j v_ - ∑_i=1^nμ_i w_ij = β_j v_ - ∑_i=1^nμ_i α_i β_j v_ij .Using the convention z_0=1 and z_i=α_i(x_i-μ_i), we can compactly writey_j = β_j ∑_i=0^n v_ij z_i .Running the SG algorithm on the new parameters amounts to updating these parameters by adding a quantity proportional toδ v_ij = Ev_ij = β_j g_j z_i ,where the notation … is used to represent an averaging operation over a batch of examples. The corresponding modification of the old parameters is then proportional to δ w_ij = α_i β_j δ v_ij = β_j^2 g_jα_i^2 (x_i - μ_i)δ w_ = β_j δ v_ - ∑_i=0^nμ_i δ w_ij = < β_j^2 g_j > - ∑_i=1^nμ_i δ w_ij . This means that we do not need to store the new parameters.We can perform both the forward and backward computations using the usual w_ij parameters, and use the above equations during the weight update. This approach ensures that we can easily change the constant α_i, μ_i, and β_j at any time without changing the function computed by the network.Updating the weights using (<ref>) is very cheap because we can precomputeandin time proportional to n+m. This overhead is negligible in comparison to the remaining computation which is proportional to nm.§.§ Block-diagonal representation The weight updates δ w_ij described by equation (<ref>) can also beobtained by pre-multiplying the averaged gradient vector ∂ E/∂ w_ij=g_jx_i by a specific block diagonal positive symmetric matrix. Each block of this pre-multiplication reads as[ [ δ w_; δ w_1j;⋮; δ w_nj ]] = β_j^2 [ [ 1+∑α_i^2μ_i [-2ex]0pt1ex-α_1^2μ_1 -α_n^2μ_n;-α_1^2μ_1α_1^2;⋮ ⋱ 2ex0;-α_n^2μ_n0 α_n^2 ] ] × [ [ ∂ E / ∂ w_; ∂ E / ∂ w_1j;⋮; ∂ E / ∂ w_nj ]] .This rewrite makes clear that the reparametrization (<ref>) is an instance of quasi-diagonal rescaling <cit.>, with the additional constraint that, up to a scalar coefficient β_j^2, all the blocks of the rescaling matrix are identical within a same layer. §.§ Choosing and adapting the reparametrization constantsMany authors have proposed second order stochastic gradient algorithms for neural networks <cit.>. Such algorithms rescale the stochastic gradients using a suitably constrained positive symmetric matrix. In all of these works, the key step consists in defining an approximation G of the curvature of the cost function, such as the Hessian matrix or the Fisher Information matrix, using ad-hoc assumptions that ensure that its inverse G^-1 is easy to compute and satisfies the desired constraints on the rescaling matrix.We can use the same strategy to derive sensible values for our reparametrization constants. Appendix <ref> derives a block-diagonal approximation G of the curvature of the cost function with respect to the parameters v_ij. Each diagonal block G_j of this matrix has coefficients[G_j]_i = β_j^2 g_j^2×{[ z_i^2[-1.5ex]0pt0ptif i=,; z_iz_if i≠, ].where the expectation · is meant with respect to the distribution of the training examples. Choosing reparametrization constantsμ_i, α_i, and β_j that make this surrogate matrix equal to the identity amounts to ensuring that a simple gradient step in the new parameters v_ij is equivalent to a second order step in the original parameters w_ij. This is achieved by choosingμ_i = x_iα^2_i=1/x_iβ^2_j=1/g_j^2 . It is not a priori obvious that we can continuously adapt the reparametrization constants on the basis of the observed statistics without creating potentially nefarious feedback loops in the optimization dynamics. On the positive side, it is well-known that pre-multiplying the stochastic gradients by a rescaling matrix provides the usual convergence guarantees if the eigenvalues of the rescaling matrix are upper and lower bounded by positive values <cit.>, something easily achieved by adequately restricting the range of values taken by the reparametrization constants α^2_i, μ_i, and β^2_j. On the negative side, since the purpose of this adaptation is to make sure the rescaling matrix improves the convergence speed, we certainly do not want to see reparametrization constants hit their bounds, or, worse, bounce between their upper and lower bounds.The usual workaround consists in ensuring that the rescaling matrix changes very slowly. In the case of our reparametrization scheme, after processing each batch of examples, we simply update online estimates of the moments[𝚖𝚡[𝚒]←λ 𝚖𝚡[𝚒]+(1-λ) x_i; 𝚖𝚡2[𝚒]← λ 𝚖𝚡2[𝚒]+(1-λ) x^2_i; 𝚖𝚐2[𝚓]← λ 𝚖𝚐2[𝚓]+ (1-λ) g_j^2, ]with λ≈0.95, and we recompute the reparametrization constants (<ref>). We additionally make sure that their values remain in a suitable range. This procedure is justified if we believe that the essential statistics of the x_i and g_j variables change sufficiently slowly during the optimization. §.§ Informal comment about the algorithm performanceThis algorithm performs well enough to produce appealing training curves and compete favorably with the best known methods (at least for the duration of a technical paper). The day-to-day practice suggests a different story which is both more important and difficult to summarize with experimental results. Finding a proper stepsize with plain SG is relatively easy because excessive stepsizes immediately cause a catastophic divergence. This is no longer the case with this proposed algorithm: many stepsizes appear to work efficiently, but occasionally cause divergence with little warning. The only way to achieve robust convergence seems to be to reduce the stepsize to a point that essentially negates the initial speed gain. This observation does not seem to be specific to our particular algorithm. For instance, <cit.> mention that, in practice, their diagonal rescaling method reduces the number of iterations by no more than a factor of three relative to plain SG, barely justifying the overhead.§ STEPSIZES AND DIAGONAL RESCALINGThe difficulty of finding good global stepsizes with second order optimization methods is in fact a well-known issue in optimization, only made worse by the stochastic nature of the algorithms we consider. After presenting a motivating example, we return to the definition of the stepsizes and develop an alternative formulation suitable for diagonal and block-diagonal rescaling approaches.§.§ Motivating exampleFigure <ref> represents the apparently benign convex functionF(w_1,w_2) = 1/2 w_1^2 + log(e^w_2+e^-w_2)whose gradients and Hessian matrix respectively are∇ F(w_1,w_2) = [ [ w_1; tanh(w_2) ]] ∇^2 F(w_1,w_2) = [ [10;0 cosh(w_2)^-2 ]] .Following <cit.>, assume we are optimizing this function with starting point (3,3).The first update moves the current point along direction -∇ F≈[-3,-1] which unfortunately points slightly away from the optimum (0,0). Rescaling with the inverse Hessian yields a substantially worse direction -(∇^2F)^-1∇ F≈[-3,-101]. The large second coefficient calls for a small stepsize. Using stepsize γ≈0.03 moves the current point to (2.9,0). Although the new gradient ∇ F≈[-2.9,0] points directlty towards the optimum, the small stepsize that was necessary for the previous update is now ten times too small to effectively leverage this good situation.We can draw two distinct lessons from this example: a) A global stepsize must remain small enough to accomodate the most ill-conditioned curvature matrix met by the algorithm iterates. This is precisely why most batch second-order optimization techniques rely on line search techniques instead of fixing a single global stepsize <cit.>, something not easily done in the case of a stochastic algorithm. Therefore it is desirable to automatically adjust the stepsize to account for the conditioning of the curvature matrix.b) The objective function (<ref>) is a sum of terms operating on separate subsets of the variables. Absent additional information relating these terms to each other, we can leverage this structural information by optimizing each term separately. Otherwise, as illustrated by our example, the optimization of one term can hamper the optimization of the other terms. Such functions have a block diagonal Hessian.Conversely, all functions whose Hessian is everywhere block diagonal can be written as such separated sums (Appendix <ref>). Therefore, using a block-diagonal approximation of a curvature matrix is very similar to separately optimizing each block of variables. §.§ Stepsizes for natural gradientThe classic derivation of the natural gradient algorithm provides a useful insight on the meaning of the stepsizes in gradient learning techniques <cit.>. Consider the objective function C()=E_ξ(), where the expectation is taken over the distribution of the examples ξ, and assume that the parameter space is equipped with a (Riemannian) metric in which the squared distance between two neighboring pointsand +δ can be written asD(,+δ)^2 = δ^⊤ G() δ + o(δ^2) .We assume that the positive symmetric matrix G() carries useful information about the curvature of our objective function,[This is why this document often refer to the Riemannian metric tensor G() as the curvature matrix. This convenient terminology should not be confused with the notion of curvature of a Riemannian space.]essentially by telling us how far we can trust the gradient of the objective function. This leads to iterations of the formt+1 = t + _δ{δ^⊤< ∇ E(t) > subject to δ^⊤ G(t) δ≤η^2 } ,where the angle brackets denote an average over a batch of examples and where η represents how far we trust the gradient in the Riemannian metric. The classic derivation of the natural gradient reformulates this problem using by introducing a Lagrange coefficient 1/2γ>0,t+1 = t + _δ{δ^⊤<∇ E(t)>+ 1/2γδ^⊤ G(t) δ} .Solving for δ then yields the natural gradient algorithmt+1 = t + γG^-1(t) <∇ E(t)> .It is often argued that choosing a stepsize γ is as good as choosing a trust region size η because every value of η can be recovered using a suitable γ. However the exact relation between γ and η depends on the cost function in nontrivial ways. The exact relation, recovered by solving δ^⊤G(t)^-1δ=η^2, leads to an expression of the natural gradient algorithm that depends on η instead of γ.t+1 = t + η G^-1(t) <∇ E(t)>/√(<∇ E(t)>^⊤G^-1(t) <∇ E(t)>) .Expression (<ref>) updates the weights along the same direction as (<ref>) but introduces an additional scalar coefficient that effectively modulates the stepsize in a manner consistent with Section <ref>.a. A similar approach was in advocated by <cit.> for the TRPO algorithm used in Reinforcement Learning. The next subsection shows how this approach changes when one considers a block-diagonal curvature matrix in a manner consistent with Section <ref>.b.§.§ Stepsizes for block diagonal natural gradientWe now assume that G() is block-diagonal. Let _j represent the subset of weights associated with each diagonal block G_jj(). Following Section <ref>.b, we decouple the optimization of the variables associated with each block by replacing the natural gradient problem (<ref>) by the separate problems∀ j _jt+1 = _jt + _δ_j{δ_j^⊤<∇_j E(_t)> subject to δ_j^⊤G_jj(t) δ_j≤η^2 } ,where ∇_j represents the gradient with respect to w_j. Solving as above leads to∀ j _jt+1 = _jt + η G_jj^-1(t) <∇_j E(t)>/√(<∇_j E(t)>^⊤G_jj^-1(t) <∇_j E(t)>) .This expression is in fact very similar to (<ref>) except that the denominator is now computed separately within each block, changing both the length and the direction of the weight update.It is desirable in practice to ensure that the denominator of expression (<ref>) or (<ref>) remains bounded away from zero. This is particularly a problem when this term is subject to statistical fluctuations induced by the choice of the batch of examples. This can be addressed using the relation<∇_j E(t)>^⊤G_jj^-1(t) <∇_j E(t)>≈ ∇_j E(t)^⊤G_jj^-1(t)∇_j E(t)≤ ∇_j E(t)^⊤G_jj^-1(t) ∇_j E(t) .Further adding a small regularization parameter μ>0 leads to the alternative formulation∀ j _jt+1 = _jt + η G_jj^-1(t) <∇_j E(t)>/√(μ+∇_j E(t)^⊤G_jj^-1(t) ∇_j E(t)) . §.§ Recovering RMSprop Let us first illustrate this idea by considering the Euclidian metric G=I. Evaluating the denominator of (<ref>) separately for each weight and estimating the expectation (∇_jE)^2 with a running averageR_jt = (1-λ)R_jt-1+λ(∂ E/∂ w_j)^2 ,yields the well-loved RMSProp weight update <cit.>:w_jt+1 = w_jt -η/√(μ+R_jt)< ∂ E/∂ w_j> .§.§ Recovering a well-known neural network trickWe now consider a neural network using the hyperbolic tangent activation functions as was fashionable in the 1990s <cit.>. Using the notations of Section <ref>, we consider block-diagonal curvature matrices whose blocks G_jj are associated to the weights _j=(w_0j… w_nj) of each unit j. Because this activation function is centered and bounded, it is almost reasonable to assume that the x_i have zero mean and unit variance. Proceeding with the approximations discussed in Appendix <ref>, and further assuming the x_i are uncorrelated,[G_jj]_i ≈ g_j^2x_i x_0pt2.2ex ≈ {[g_j^2if i=;0 otherwise. ].We can then evaluate the denominator of (<ref>), with μ=0, under the same approximations:√(∑_i=1^n x_i g_j g_j x_i/g_j^2) ≈ √(∑_i=1^n x_i^2g_j^2/g_j^2) = √(n) .Although dividing the learning rate by the inverse square root of the number n of incoming connections (the fanin) is a well known trick for such networks <cit.>, no previous explanation had linked it to curvature issues.Figure <ref> (left) illustrates the effectiveness of this trick when training a typical convolutional network[ ] on the CIFAR10 dataset. Although our network uses ReLU instead of hyperbolic tangent activations, the experiment shows the value of dividing the learning rates by √(n× 𝒮), where n represents the fanin and where the weight sharing count 𝒮 is always 1 for a linear layer and can be larger for a convolutional layer (see Appendix <ref>).In both cases we use mini-batches of 64 examples and select the global constant stepsize that yields the best training loss after 40 epochs. § WHITENING REPARAMETRIZATIONSince the zero-overhead reparametrization of Section <ref> amounts to using a particular block-diagonal curvature matrix, we can apply the insight of the previous section and optimize the natural gradient problem within each block. Proceeding as in Section <ref>, we use the reparametrization constantsμ_i = x_iα^2_i=1/x_iβ^2_j=1/√(n× 𝒮) ,The only change relative to (<ref>) consists in replacing the original β^2_j=1/g_j^2 by an expression that depends only on the geometry of the layer (the fanin n and the sharing count 𝒮). Meanwhile the constants α^2_i and μ_i are recomputed after each minibatch on the basis of online estimates of the input moments as explained in Section <ref>.An attentive reader may note that we should have multiplied (instead of replaced) the original β^2_j by the scaling factor 1/√(n× 𝒮). In practice, removing the g_j^2 term from the denominator makes the algorithm more robust, allowing us to use significantly larger global stepsizes without experiencing the occasionnal divergences that plagued our original algorithm (the cause of this behavior will become clearer in Section <ref>.)§.§ Comparison with batch normalization Batch normalization <cit.> is an obvious point of comparison for our reparametrization approach. Both methods attempt to normalize the distribution of certain intermediate results. However they do it in a substantially different way.The whitening reparametrization normalizes on the basis of statistics accumulated over time, whereas batch normalization uses instantaneous statistics observed on the current mini-batch. The whitening reparametrization does not change the forward computation. Under batch normalization, the output computed for any single example is affected by the other examples of the same mini-batch. Assuming that these examples are picked randomly, this amounts to adding a nontrivial noise to the computation, which can be both viewed as a nuisance and as a useful regularization technique.§.§ Cifar10 experiments Figure <ref> (right plot) compares the evolution of the training loss of our CIFAR10 CNN using the whitening reparametrization or using batch normalization on all layers except the output layer. Whereas batch normalization shows a slight improvement over the unnormalized curves of the left plot, training with the whitening reparametrization quickly drives the training loss to zero.From the optimization point of view, driving the training loss to zero is a success. From the machine learning point of view, this means that we overfit and must compensate by either adding explicit regularization or reducing the size of the network. As a sanity check, we have verified that we can recover the batch normalization testing error by adding L2 regularization to the network trained with the whitening reparametrization. The two algorithms then reduce the test error with similar rates.[Using smaller networks would of course yield better speedups. A better optimization algorithm can conceivably help reduce our reliance on vastly overparametrized neural networks <cit.>.]§.§ ImageNet experiments In order to appreciate how the whitening reparametrization works at scale, we replicate the above comparison using the well known AlexNet convolutional network <cit.> trained on ImageNet (one million 224×224 training images, 1000 classes.)The result is both disappointing and surprising. Training using only 100,000 randomly selected examples in ImageNet reliably yields training curves similar to those reported in Figure <ref> (right). However, when training on the full 1M examples, the whitening reparametrization approach performs very badly, not even reaching the best training loss achieved with plain stochastic gradient descent. The network appears to be stuck in a bad place. §.§ Fast changing curvatureThe ImageNet result reported above is surprising because the theoretical performance of stochastic gradient algorithm does not usually depend on the size of the pool of training examples. Therefore we spend a considerable time manually investigating this phenomenon.The key insight was achieved by systematically comparing the actual statistics x_i and x_i, estimated on a separate batch of examples, with those estimated with the slow running average method described in Section <ref>. Both estimation methods usually give very consistent results. However, in rare instance, they can be completely different. When this happens, the reparametrization constants α^2_i and μ_i are off. This often leads to unreasonably large changes of the affected weights.When the bias of a particular unit becomes too negative, the ReLU activation function remains zero regardless of the input example, and no gradient signal can correct this in the future. In other words, these rare events progressively disable a significant fraction of the neural network units.How can our slow estimation of the curvature be occasionally so wrong? The only possible explanation is that the curvature can occasionally change very quickly. How can the curvature change so quickly? With a homogenous activation function like the ReLU, one does not change the neural network output if we pick one unit, multiply its incoming weights by an arbitrary constant κ and divide its outgoing weights by the same constant. This means that the cost function in weight space is invariant along complex manifolds whose two-dimensional slices look like hyperbolas. Although the gradient of the objective function is theoretically orthogonal to these manifolds, a little bit of numerical noise is sufficient to cause a movement along the manifold when the stepsize is relatively large.[In fact such movements are amplified by second-order algorithms because the cost function has zero curvature in directions tangent to these manifolds.This is why we experienced so many problems with the β^2_j=1/g_j^2 scaling suggested by the naïve second-order viewpoint.]Changing the relative sizes of the incoming and outgoing weights of a particular unit can of course dramatically change the statistics of the unit activation.This observation is important because most second-order optimization algorithms assume that the curvature changes slowly <cit.>. Batch normalization does not suffer from this problem because it relies on fresh mean and variance estimates computed on the current mini-batch. As mentioned in Section <ref> and detailled in Appendix <ref>, computing α_i and μ_i on the current minibatch creates a nefarious feedback loop in the training process. Appendix <ref> describes an inelegant but effective way to mitigate this problem. § CONCLUSION Investigating the robustness issues of a second-order block-diagonal neural network stochastic gradient training algorithm has revealed two interesting insights. The first insight reinterprets what is meant when one makes a block-diagonal approximation of the curvature matrix. This leads to a new way to scale the stepsizes and clarifies popular algorithms such as RMSProp as well as old neural network tricks such as fanin stepsize scaling. The second insight stresses the practical importance of dealing with fast changes of the curvature. This observation challenges the design of most second order optimization algorithms. Since much remains to be achieved to turn these insights into a solid theoretical framework, we believe useful to share both the path and the insights.Acknowledgments Many thanks to Yann Dauphin, Yann Ollivier, Yuandong Tian, and Mark Tygert for their constructive comments.plainnat § DERIVATION OF THE CURVATURE MATRIXFor the sake of simplicity, we only take into account the parameters =̌(… v_ij…) associated with a particular linear layer of the network (hence neglecting all cross-layer interactions).Each example is then represented by the layer inputs x_i and by an additional variable ξ that encode any relevant information not described by the x_i.For instance ξ could represent a class label.We then assume that the cost function associated with a single example has the formE(;̌ξ, x_1… x_n)= -log( φ(ξ, y_1… y_m) )= -log( φ(ξ, …β_j ∑_i=0^n v_ij x_i …) ),where the function φ encapsulates all the layers following the layer of interest as well as the loss function.This kind of cost function is very common when the quantity φ can be interpreted as the probability of some event of interest.The optimization objective C()̌ is then the expectation of E with respect to the variables ξ and x_i,C()̌ = E(ξ,x_1… x_n) .Its derivatives are∂ C/∂ v_ij = - 1/φ∂φ/∂ y_jβ_j z_i = g_j β_j z_i and the coefficients of its Hessian matrix are∂^2 C/∂ v_ij∂ v_ = ( 1/φ^2∂φ/∂ y_j∂φ/∂ y_ - 1/φ∂^2 φ/∂ y_j y_)β_j β_ z_i z_ .Our first approximation consists in neglecting all the terms of the Hessian involving the second derivatives of φ, leading to the so-called Generalized Gauss-Newton matrix G <cit.> whose blocks G_j have coefficients[G_j]_i = 1/φ^2∂φ/∂ y_j∂φ/∂ y_β_j β_ z_i z_ = β_j β_ g_j g_ z_i z_Interestingly, this matrix is exactly equal to a well known approximation of the Fisher information matrix called the Empirical Fisher matrix <cit.>.We then neglect the non-diagonal blocks and assume that the squared gradients g_j^2 are not correlated with either the layer inputs x_i or their cross products x_ix_. See <cit.> for a similar approximation. Recalling that z_0=1 is not correlated with anything by definition, this means that the g_j^2 is not correlated with z_iz_ either.[G_jj]_i = β_j^2 g_j^2z_iz_ .Further assuming that the layer inputs x_i are also decorrelated leads to our final expression[G_jj]_i = β_j^2 g_j^2×{[z_i^2 if i=,;z_iz_ otherwise. ]. The validity of all these approximations is of course questionable. Their true purpose is simply to make sure that our approximate curvature matrix G can be made equal to the identity with a simple choice of the reparametrization constants, namely,μ_i=x_iα^2_i=1/x_iβ^2_j=1/g_j^2 .§ REPARAMETRIZATION OF CONVOLUTIONAL LAYERSConvolutional layers can be reparametrized in the same manner as linear layers (Section <ref>) by introducing additional indices u and v to represent the two dimensions of the image and kernel coordinates. Equations (<ref>) and (<ref>) then becomey_ju_1u_2 =w_0j+∑_i=1^n ∑_v_1v_2 x_i(u_1+v_1)(u_2+v_2) w_ijv_1v_2= β_j( v_0j +∑_i=1^n ∑_v_1v_2α_i(x_i(u_1+v_1)(u_2+v_2)-μ_i) v_ijv_1v_2 ) ,and the derivative of the loss E with respect to a particular weightinvolves a summation over all the terms involving that weight:∂ E/∂ v_i j v_1 v_2=β_j∑_u_1u_2g_ju_1u_2 z_i(u_1+v_1)(u_2+v_2) .Following Appendix <ref>, we write the blocks G_jj' of the generalized Gauss Newton matrix G,[G_jj']_iv_1v_2,i'v_1'v_2' = ∂ E/∂ v_i j v_1 v_2∂ E/∂ v_i'j'v_1'v_2' .Obtaining a convenient approximation of G demands questionable assumptions such as neglectingnearly all off-diagonal terms, and nearly all possible correlations involving the z and g variables. This leads to the following choices for the reparametrization constants, where the expectations andvariances are also taken across the image dimension subscripts (“∙”) and where theconstant 𝒮 counts the number of times each weight is shared, that is, the number of applications of the convolution kernel in the convolutional layer.μ_i=x_i∙∙α^2_i=1/x_i∙∙β^2_j=1/ 𝒮 g_j∙∙^2 . § COORDINATE SEPARATIONIt is obvious that a twice differentiable functionf : (x_1… x_k)∈^n_1×…×^n_k ⟼ f(x_1,…,x_k)∈that can be written as a sumf(x_1… x_k) = f_1(x_1)+…+f_k(x_k)has a block diagonal Hessian everywhere, that is,∀ (x_1… x_k)∈^n_1×…×^n_k∀ i ≠ j ∂^2 f/∂ x_i ∂ x_j = 0 .Conversely, assume the twice differentiable function f satisfies (<ref>), and writef(x_1… x_k) - f(0… 0)= ∑_i=1^k f(x_1… x_i,0… 0) - f(x_1… x_i-1,0… 0) = ∑_i=1^k∫_0^1 x_i^⊤ ∂ f/∂ x_i(x_1… x_i-1, t x_i, 0… 0) dt .Then observe∂ f/∂ x_i(x_1… x_i-1, r, 0, … 0)- ∂ f/∂ x_i(0… 0,r,0… 0) = ∫_0^1 ∑_j=1^i-1 x_j^⊤ ∂^2 f/∂ x_j∂ x_i(tx_1… tx_i-1,r,0… 0)dt = 0 .Therefore property (<ref>) is true becausef(x_1… x_k) = f(0… 0) + ∑_i=1^k∫_0^1 x_i^⊤ ∂ f/∂ x_i(0… 0, t x_i, 0, … 0) dt . § COUPLING EFFECTS WHEN ADAPTING REPARAMETRIZATION CONSTANTSThe reparametrization constants suggested by (<ref>) and (<ref>) are simple statistical measurements on the network variables. It is tempting use to directly compute estimates α̂_i, μ̂_i, and β̂_j on the current mini-batch in a manner similar to batch renormalization. Unfortunately these estimates often combine in ways that create unwanted biases. Consider for instance the apparently benign case where we only need to compute an estimate μ̂_i because an oracle reveals the exact values of α_i and β_j. Replacing μ_i by its estimate μ̂_i in the update equations (<ref>) gives the actual weight updates δ w_ij performed by the algorithm. Recalling that μ̂_i is now a random variable whose expectation is μ_i, we can compare the expectation of the actual weight update δ w_ with the ideal value δ w_.δ w_ = β_j^2 g_j (1 - ∑_i α_i^2 μ̂_i (x_i-μ̂_i))= β_j^2 ( 1 - ∑_i α_i^2 ( μ̂_i x_i g_j - μ̂_i^2 g_j) ) = δ w_ + ∑_i β_j^2 α_i^2 (μ̂_ig_j + cov[μ̂_i^2, g_j] - cov[μ̂_i, x_i g_j] ) .This derivation reveals a systematic bias that results from the nonzero variance of μ̂_̂î and its potential correlation with other variables.In practice, this bias is more than sufficient to severely disrupt the convergence of the stochastic gradient algorithm.§ MITIGATING FAST CURVATURE CHANGE EVENTSFast curvature changes mostly happens during the first phase of the training process and disappears when the training loss stabilizes. For ImageNet, we were able to mitigate the phenomenon by using batch-normalization during the first epoch then switching to the whitening reparametrization approach for the remaining epochs (Figure <ref>.) | http://arxiv.org/abs/1705.09319v1 | {
"authors": [
"Jean Lafond",
"Nicolas Vasilache",
"Léon Bottou"
],
"categories": [
"cs.LG",
"stat.ML"
],
"primary_category": "cs.LG",
"published": "20170525183324",
"title": "Diagonal Rescaling For Neural Networks"
} |
[2010]Primary 11A41; Secondary 11A51 Wright proved that there exists a number c such that if g_0 = c and g_n+1 = 2^g_n, then ⌊ g_n ⌋ is prime for all n > 0.Wright gave c = 1.9287800 as an example. This value of c produces three primes, ⌊ g_1 ⌋ = 3, ⌊ g_2 ⌋ = 13, and ⌊ g_3 ⌋ = 16381. But with this c, ⌊ g_4 ⌋ is a 4932-digit composite number. However, this slightly larger value of c,c = 1.9287800 + 8.2843 · 10^-4933,reproduces Wright's first three primes and generates a fourth: ⌊ g_4 ⌋ = 191396642046311049840383730258 … 303277517800273822015417418499is a 4932-digit prime.Moreover, the sum of the reciprocals of the primes in Wright's sequence is transcendental.Wright's Fourth Prime Robert Baillie December 30, 2023 =====================§ INTRODUCTION In 1947, Mills <cit.> proved that there exists a number A such that⌊ A^3^n⌋is prime for all n > 0. (Here, ⌊ x ⌋ is the largest integer ≤ x.) Mills did not give an example of such an A.Caldwell and Cheng <cit.> calculate such an A ≈ 1.30637788386308069046 which generates a sequence of primes that begins 2, 11, 1361, 2521008887, and 16022236204009818131831320183. The next prime has 85 digits. Their digits of A are in <cit.>. Their sequence of primes is in <cit.>.In 1951, Wright <cit.>, <cit.> proved that there exists a number c such that, if g_0 = c and, for n ≥ 0, we define the sequenceg_n+1 = 2^g_n ,then⌊ g_n ⌋is prime for all n > 0.This sequence grows much more rapidly than Mills' sequence.The key ingredient in Wright's proof is the relatively elementary fact (Bertrand's postulate) that, for every N > 1, there is a prime between N and 2N <cit.>.Wright gave an example of such a constant: c = 1.9287800 .This value of c produces three primes, ⌊ g_1 ⌋ = ⌊ 3.8073 …⌋ = 3,⌊ g_2 ⌋ = ⌊ 13.9997 …⌋ = 13, and ⌊ g_3 ⌋ = ⌊ 16381.3640 …⌋ = 16381.But with this value of c, ⌊ g_4 ⌋ is a 4932-digit composite number,⌊ g_4 ⌋ = ⌊ 2^2^2^2^c⌋ = 19139664204631104 … 822015417386540 . In 1954, Wright <cit.> proved that the sets of values of such A and c have the cardinality of the continuum, are nowhere dense, and have measure 0.Neither Mills' nor Wright's formula is useful in computing primes that are not already known.In the next section, we show how to compute a value of c slightly larger than Wright's constant, which causes ⌊ g_4 ⌋ to be a prime. Our modified value preserves all of Wright's original seven decimal places.In Section <ref>, we discuss a different modification of c that gives a different fourth prime.What's new in this revision? A little bit of PARI/GP code was added in Section <ref>. The 4932-digit probable prime discussed in Section <ref> was proved prime; Section <ref> has the details. Section <ref>, proving that the sum of the reciprocals of Wright's primes is transcendental,and Section <ref>, which illustrates a sequence of smallest possible primes, are new. A few minor wording changes were made. The links in the references were updated, and are valid as of March, 2019.§ EXTENDING WRIGHT'S CONSTANT TO PRODUCE FOUR PRIMESHere is all the Mathematica code for this section:c = 1 + 92878/10^5Floor[2^c] , Floor[2^(2^c)] , Floor[2^(2^(2^c))] N[2^c] , N[2^(2^c)] , N[2^(2^(2^c))] (* gives 3.80733100076, 13.9997678682, 16381.3640014 *) g4 = 2^(2^(2^(2^c))) ;(* exact expression for g4 *) Block[MaxExtraPrecision = 6000, N[g4 - Floor[g4], 20] ](* gives 0.95480385178280261500 *) Block[MaxExtraPrecision = 6000, N[g4 - Floor[g4], 20] ] n4 = Floor[2^(2^(2^(2^c)))] ; PrimeQ[n4]N[n4] , N[n4 , 25] , Mod[n4, 10^25](* this gives the first 25 digits rounded, and about the last 25 digits: 1.913966420463110498403837*10^4931, 7517800273822015417386540 *)(* prp4 = NextPrime[n4]; *) (* warning: this takes a while *) prp4 = n4 + 31959; (* to save time, use this previously-computed value *)PrimeQ[prp4]N[prp4, 25] , Mod[prp4, 10^25](* this gives the first few and last few digits of prp4:1.913966420463110498403837*10^4931, 7517800273822015417418499 *) wMin = Log[2, Log[2, Log[2, Log[2, prp4 + 0]]]] ; wMax = Log[2, Log[2, Log[2, Log[2, prp4 + 1]]]] ; Block[MaxExtraPrecision = 6000, N[wMin - c, 20] ] Block[MaxExtraPrecision = 6000, N[wMax - c, 20] ] w =(1 + 92878/10^5) + (8 + 2843/10^4) * 10^-4933 ; prp4a = Floor[2^(2^(2^(2^w)))] ; prp4a - prp4Floor[2^w] , Floor[2^(2^w)] , Floor[2^(2^(2^w))] The calculations here were done in Mathematica. Snippets of Mathematica code are scattered throughout this paper, in a font that looks like this: In Mathematica, if we specify a floating-point value such as 1.9287800, then Mathematica assumes this number has only machine precision. Therefore, it is better for calculations like ours to use the exact value of Wright's constant:With thisc, we can verify that the integer parts ofg_1,g_2, andg_3,are Wright's three primes 3, 13, and 16381.However, the integer part ofg_4is composite:(The trailing semicolon prevents display of lengthy output that we don't need.)returns , son4is not prime.is a strong probable prime test; see <cit.> and <cit.>.The Mathematica commandshows thatn4is about1.913966420463110 · 10^4931. The last few digits ofn4can be found with . These digits are 5417386540, so we can see thatn4is not prime.We can use Mathematica'sfunction to locate the first (probable) prime larger thann4:prp4is displayed in its entirety in Appendix <ref>. The first and last 35 digits ofprp4are19139664204631104984038373025808682 … 26398303277517800273822015417418499 . In Section <ref>, we discuss a proof thatprp4is prime.This differenceis 31959, which is small compared ton4. This suggests that we can compute a new starting value for the sequence, say,g_0 = w, wherewis only slightly larger thanc, which makes⌊ g_4 ⌋ = prp4, and which reproduces Wright's first three primes.Becauseprp4is computed with thefunction, this value ofwmust satisfy the inequalitiesprp4 ≤ 2^2^2^2^w < prp4 + 1. Solving the inequalities for the minimum and maximum possible values ofw, If we attempt to see how much largerandare compared toc, we find thatgives0. · 10^-70and the warning, “ ...” .To remedy this, we use thestructure to do the calculation with plenty of added precision inside the : These differences arewMin - c ≈ 8.2842370595324508541 · 10^-4933wMax - c ≈ 8.2844962818036719650 · 10^-4933 . Any number between these two values, for example,8.2843 · 10^-4933, when added toc, should produce the valueprp4. Note that8.2842 · 10^-4933is too small and that8.2845 · 10^-4933is too large. Also, values with four or fewer significant digits, like8.284 · 10^-4933,8.28 · 10^-4933, or8.29 · 10^-4933, are either too small or too large.As above, we should use the exact value of8.2843 · 10^-4933in our calculations. The value ofwthat should produceprp4, isw = c +(8 + 2843/10^4) · 10^-4933 . A quick check in Mathematica verifies this:This difference is 0, as expected. We can also check that thiswgives Wright's first three primes:These three values are 3, 13, and 16381. Note:prp4is not the closest probable prime ton4. Let's search for the largest probable prime just less thann4:returns 129, son4 - 129is closer ton4than wasprp4. (The negative second argument tocauses Mathematica to search for the largest probable prime less thann4.) z = (1 + 92878/10^5) - (3 + 35/100)*10^-4935 ; q4 = Floor[2^(2^(2^(2^z)))] ; n4 - q4 PrimeQ[q4]Floor[2^z] , Floor[2^(2^z)] , Floor[2^(2^(2^z))]give 3, 13, 16381. Therefore, this z works, but it alters some of Wright's digits Like we did above, we can compute the value ofg_0which starts a sequence such thatg_4 = n4 - 129. The value we get isc - 3.35 · 10^-4935 = c -(3 + 35/10^2 ) · 10^-4935≈ 1.9287799999 ⋯ .Unfortunately, this changes the last two of Wright's original decimal places.If you don't have Mathematica.The Wolfram Alpha website <http://www.wolframalpha.com> can do some of the calculations shown here. First, put the entire calculation into one expression, such asand paste it into that webpage. On the main Wolfram Alpha page, this returns no (that is, the number is not prime). This works because we are computingof an even number, so this takes very little time to evaluate.However,prp4is probably prime, sotakes a while to compute. So, this expression forprp4remains unevaluated. But if you select “Open code”, then press the little button with the arrow to evaluate it, Wolfram Alpha will return .PARI/GP code for these calculations. The following PARI/GP<cit.> code is similar to the above Mathematica code. But note that we must increase the precision and define our ownlogfunction to the base 2.§ THE FIFTH PRIME IN WRIGHT'S SEQUENCE What can we say about the fifth prime in Wright's sequence?The fourth termin Wright's sequence isg_4 = 2^2^2^2^w≈ 1.913966420463110 · 10^4931 .The fifth term,g_5 =2^g_4is too large for Mathematica to compute directly. It is also impossible to adjustg_5, like we did above withg_4, to produce a 5th prime.However, we can use base 10 logarithms to calculate how many digitsg_5has, and even to calculate the first few of those digits.SupposeLis the base 10 logarithm ofg_5, that is,L = log_10g_5 = log_102^g_4 = g_4 ·log_102 .Theng_5 = 10^L. Letkbe the integer part ofL, that is,k = ⌊ L ⌋, and letfbe the fractional part ofL, that is,f = L - k. Thenfis between 0 and 1, andg_5 = 10^L = 10^f + k = 10^f · 10^k.The factor10^kdetermines how many digits are ing_5.10^fdetermines what the digits ofg_5are.Here's some Mathematica code. We'll definecandwagain here to make this code be self-contained.The result isL ≈ 5.761613032530158 · 10^4930. Next, extract the integer and fractional parts ofLand display rough approximations to the much more accurate values that are stored internally. The results arek ≈ 5.761613032530158 · 10^4930, andf ≈ 0.77698 85779 22041 44281.kis a very large integer, having 4931 digits. The first few digits ofkare 5761613032. The last few digits ofkcan be obtained from ; they are 8933273637. (Or, we could just displaykitself to see all of its 4931 digits.)So,k = 5761613032 … 8933273637. The number of digits ing_5is the 4931-digit numberk + 1 = 5761613032 … 8933273638. We can also obtain the first few digits ofg_5itself.g_5 = 10^f · 10^k=10^ftimes (a large power of 10) .The digits ofg_5come from10^f, which is about5.9839 58568 58953 9736. So, the first ten digits ofg_5are 5983958568. The “large power of 10” in Equation (<ref>) just moves the decimal point over. Therefore,g_5(rounded to ten digits) is aboutg_5 ≈ 5.983958569 · 10^5761613032 … 8933273637 ,where the exponent has 4931 digits.The first few digits of the fifth prime. So, 5983958568 are the first few digits ofg_5, the fifth term in Wright's sequence. What can we say about the first few digits ofp_5, the fifth prime in Wright's sequence?The reader may wonder if the leading digits we just computed would have to be changed if⌊ g_5 ⌋were not prime, and likeg_4,g_5has to be adjusted to obtain a prime.SupposePis the smallest prime greater thang_5. We will now show that the first few digits ofg_5andPare the same. Dusart <cit.> proved that, for anyx ≥ 396738, there is a primepin the intervalx < p ≤ x (1 + 1/25 (lnx)^2).the 1/25 (lnx)^2 estimate is from 2010. in 2016, dusart published two better estimates: (1) for x ≥ 89693, there is a prime in the interval (Proposition 5.4)x < p ≤ x ( 1 + 1/lnx^3) (2) for x ≥ 468991632, there is a prime in the interval (Corollary 5.5)x < p ≤ x ( 1 + 1/5000 (lnx)^2) “Explicit estimates of some functions over primes”. The Ramanujan Journal. 45: 227251. doi:10.1007/s11139-016-9839-4 in two places in this paper, we use a dusart bound with the following values of logx: logx = 1.32666 * 10^4931 : 1/(25*logx^2) = 2.27269*10^-9864, 1/(logx^3) = 4.28273*10^-14794, 1/(5000*logx^2) = 1.13635*10^-9866 logx = 9.52654 * 10^10 : 1/(25*logx^2) = 4.40747*10^-24, 1/(logx^3) = 1.15663*10^-33,1/(5000*logx^2) = 2.20374*10^-26 in both cases, Proposition 5.4 gives the best estimate (i.e., the largest denominator)in 2016, christian axler published the estimate (theorem 1.5) for x ≥ 58837, there is a primex < p ≤ x*( 1 + 1.1817/(logx^3) ) New Bounds for the Prime Counting Function, Integers, vol. 16, #A22, (2016)in 2017, axler published two further estimates (theorem 1.4) in a preprint New Estimates for Some Functions Defined Over Primes, https://arxiv.org/abs/1703.08032 (May 17, 2017): (3) for x ≥ 6034256, there is a primex < p ≤ x*( 1 + .087/(logx^3) ) (4) for x > 1, there is a primex < p ≤ x*( 1 + 198.2/(logx^4) ) logx = 1.32666 * 10^4931 : .087/(logx^3) = 3.72598*10^-14795, 198.2/(logx^4) = 6.39831*10^-19723 logx = 9.52654 * 10^10 : .087/(logx^3) = 1.00627*10^-34,198.2/(logx^4) = 2.40637*10^-42in 2018, axler published estimates (3) and (4) (theorem 4) in New Estimates for Some Functions Defined Over Primes, Integers, vol. 18, #A52, (2018)for our values of logx, (4) is the strongest, but the original estimate 1/(25*(logx)^2) is good enough. also, the wikipedia article on bertrand's postulate has better estimates, if the reader needs them.Forx = g_5, we havelnx = lng_5 = log_10g_5·ln10. Using the Mathematica variables above, this is , which is about1.32666 · 10^4931. The fraction1/25 (lnx)^2≈ 2.3 · 10^-9864tells us that there is a prime betweenxandx · ( 1 + 2.3 · 10^-9864 ). So, ifxis nearg_5, we need to increasexby only a tiny fraction ofxto reach the next prime larger thanx. A similar argument holds for the largest prime less thanx. We therefore conclude thatp_5 = ⌊ g_5 ⌋also begins with the digits 5983958568. (In fact,g_5andp_5will agree to over 9800 digits, unless one of them has a string of 0's where the other has a string of 9's). § ANOTHER VERSION OF THE FOURTH TERM IN WRIGHT'S SEQUENCEHere is all the Mathematica code for this section:q = 2^16382 - 35411 ;N[q, 25] , Mod[q, 10^25](* this gives the first few and last few digits: 2.974328738393079412714398*10^4931, 8115111756822667490981293 *) zMin = Log[2, Log[2, Log[2, Log[2, q + 0]]]] ; zMax = Log[2, Log[2, Log[2, Log[2, q + 1]]]] ;N[zMin, 20] , N[zMax, 20]Block[MaxExtraPrecision = 6000, N[zMax - zMin, 20] ] Block[MaxExtraPrecision = 6000, N[zMax - zMin, 20] ] zMiddle = (zMin + zMax)/2; N[zMiddle] qTest = Floor[ 2^(2^(2^(2^zMiddle))) ] ; qTest - q (* = 0 *)Floor[2^zMiddle] , Floor[2^(2^zMiddle)]Block[MaxExtraPrecision = 6000, Floor[2^(2^(2^zMiddle))] ] (* restore text coloring in the TeX editor *)As mentioned above, Wright later proved that his original value, 1.9287800, is not the only one that works.In OEIS <cit.>, Charles Greathouse defines the sequence:a_0 = 1, a_n = greatest prime<2^a_n-1+1 . Wright does not say anything about a ”greatest prime ...”, so Greathouse's formulation is slightly different from Wright's.The first three terms in Greathouse's sequence match Wright's three primes a_1 = 3, a_2 = 13, and a_3 = 16381. In Greathouse's sequence, a_4 is the 4932-digit probable prime,q = 2^16382 - 35411 = 29743287383930794127 … 11756822667490981293. q ≈ prp4 · 1.554 > prp4.Samuel S. Wagstaff, Jr. used PARI/GP to prove that q is prime. See Section <ref> for details.We can transform Greathouse's a_1, a_2, a_3 and a_4 into a sequence of the form proposed by Wright. That is, we can find a z such thatq = ⌊ 2^2^2^2^z⌋ .We work backwards from q to estimate z, just as we did above:andare both about 1.928782187150216, which is about c + 2.187150216 …· 10^-6.The differenceis about 1.6680090447391719120 · 10^-4937. The fact thatandare so close together means that, in order to get q as the fourth term in the sequence, we must specify z to at least 4937 decimal places.So, a value of z that produces q isz = 1.9287800 + 2.187150216 …· 10^-6≈ 1.928782187150216 … .We can verify that this z also reproduces Wright's first three primes.q has a form that is easy to write down, which is a very nice feature. However, z is not easy to write. In addition, this q leads to a z whose 6th and 7th decimal places are different from Wright's.§ THE SUM OF THE RECIPROCALS OF WRIGHT'S PRIMES IS TRANSCENDENTAL Is the constantc = 1.9287800 + 8.2843 · 10^-4933 + …that gives rise to an infinite sequence conisting entirely of primes, a transcendental number, or at least irrational? If we knew that an infinite number of the g_n each required a monotonically (much) smaller increment to the value of c, then perhaps we could prove that c is transcendental because these increments might decrease toward 0 sufficiently fast to make c a Liouville number. But for all we know, the value c = 1.9287800 + 8.2843 · 10^-4933 might itself produce only primes. In this case, this c would be rational.However, we can prove that the sum of the reciprocals of Wright's primes,1/3 + 1/13 + 1/16381 + 1/ 1913966420 … 5417418499+ 1/ 5.98 …· 10^57616 … 73637+ …(where the fourth denominator has 4932 digits and the exponent in the last term has 4931 digits), is transcendental because it is a Liouville number.Here is the definition of a Liouville number that we will use <cit.>. A real number x is a Liouville number if for every positive integer m, there is a rational number h_m/k_m such that| x - h_m/k_m| < 1/ (k_m)^m. All Liouville numbers are transcendental <cit.>.Remark. To prove x is a Liouville number, it is sufficient to prove (<ref>) for sufficiently large m. This is because, if (<ref>) holds for some number m, then (assuming k_m > 1) it also holds for smaller m:| x - h_m/k_m| < 1/ (k_m)^m< 1/ (k_m)^m-1< 1/ (k_m)^m-2… < 1/ k_m. Given the initial constant c that generates an infinite number of primes, let p_j be the jth prime in the sequence.We will show that the following sum is a Liouville number:s = ∑_j = 1^∞1/p_j = 0.41031745659057788964 … .Let s_n = h_n/k_n be the sum of the first n terms. The common denominator of the first n terms will bek_n = p_1 · p_2 … p_n.All terms in the series are positive, so x - h_n / k_n is greater than 0, and| s - h_n/k_n| =s - h_n/k_n =1/p_n+1 + 1/p_n+2 + … < 2/p_n+1 .This last inequality is true because the sum of the tail would be exactly2/p_n+1 if the series was a geometric series with p_n+k+1 = 2 · p_n+k. However, the denominators increase much faster than that, so the sum of terms in the tail of the series is less than 2/p_n+1.To see how p_n compares to p_n+1, consider a typical term in Wright's sequence: 13 < g_2 < 14, so p_2 = 13. Because g_3 = 2^g_2, we have2^13 < g_3 < 2^14 .Therefore, p_3 = ⌊ g_3 ⌋ must be somewhere in the range 2^13 < p_3 < 2^14. (In this sequence, g_n+1 = 2^g_n can never be an integer, because otherwise, p_n+1 = g_n+1 would be a power of 2, and p_n+1 would not be prime). In general,2^p_n < p_n+1 < 2 · 2^p_n . In order to use the above definition of Liouville number, we want to prove that2/ p_n+1 < 1/ ( k_n)^nso that (<ref>) would becomes - h_n/k_n < 2/p_n+1 < 1/ ( k_n)^n. Inequality (<ref>) is equivalent to2( k_n)^n < p_n+1 .We will establish (<ref>) by proving the following chain of inequalities for n > 2:2( k_n)^n <( p_n)^2n < 2^p_n < p_n+1 . Table <ref> shows that (<ref>) is true for n ≤ 2:2 · k_1 = 2 · p_1 < p_2and2 · ( k_2)^2 = 2( p_1 · p_2)^2 < p_3. For n = 3,2 · ( k_3)^3 = 2( p_1 · p_2 · p_3)^3= 2( p_1 · p_2)^2 · ( p_1 · p_2 · p_3^3)< p_3 · ( p_1 · p_2 · p_3^3)< p_3 · ( p_3^2 · p_3^3) = p_3^6.The first inequality is just the previous (induction) step for n = 2. The second inequality is true because p_1 and p_2 are both less than p_3.Nowp_3^6 < 2^p_3will be true if p_3 is sufficiently large. How large, exactly? We will discuss the details below, but this inequality will be true as long as p_3 is greater than the largest real root of the equation x^6 = 2^x.That root is x = 29.210 …. But p_3 = 16381, so the inequality (<ref>) is certainly true. Therefore, we have2( k_3)^3 = 2( p_1 · p_2 · p_3)^3 < p_3^6 < 2^p_3 < p_4. We'll do one more step to make the pattern clear:2 · ( k_4)^4= 2( p_1 · p_2 · p_3 · p_4)^4= 2( p_1 · p_2 · p_3)^3 · ( p_1 · p_2 · p_3 · p_4^4)< p_4 · ( p_1 · p_2 · p_3 · p_4^4)< p_4 · ( p_4^3 · p_4^4)= p_4^8because p_1, p_2, and p_3 are all less than p_4. We will havep_4^8 < 2^p_4provided p_4 exceeds the largest real root of the equation x^8 = 2^x. But this root is 43.559 …, and p_4 is much larger. Therefore, we have proved that2( k_4)^4 = 2( p_1 · p_2 · p_3 · p_4)^4 < p_4^8 < 2^p_4 < p_5. To be complete, we formally write the general induction step:2( k_n)^n = 2( p_1 … p_n)^n = 2( p_1 … p_n-1)^n-1· ( p_1 …p_n-1· p_n^n)< p_n ·( p_1 … p_n-1· p_n^n) < p_n ·( p_n^n-1· p_n^n) = p_n^2n < 2^p_n < p_n+1 ,where the first inequality follows from the previous induction step, and p_n^2n < 2^p_n is true as long as p_n > x_2n, where x_2n is the largest real root of the equation x^2n = 2^x.We will now show that, if n > 2, then p_n > x_2n, which will complete the proof of inequality (<ref>). (The very rough estimates in our inequalities are so sloppy that the inequality p_n > x_2n is violated for n ≤ 2: p_1 = 3 < x_2 = 4, andp_2 = 13 < x_4 = 16. But this doesn't matter, because already know from the data that (<ref>)does hold for n ≤ 2.)Already for n = 3, we have p_n = 16381 > x_6 ≈ 29.21. We will now see that the sequence of roots x_2n increases much more slowly than p_n.The equation x^k = 2^x.Suppose x > 0, and let k be a fixed positive integer. It is well known that, as x increases, 2^x will eventually be greater than x^k. If k > 1, this equation has two positive real roots, say r and R, where r < R. The smaller root, r, is slightly larger than 1. Furthermore,for x < r, we have x^k < 2^x;for r < x < R, x^k > 2^x;for all x > R, x^k < 2^x.For selected values of k, Table <ref> shows the value of x_k, the largest real root of x^k = 2^x. (We care only about even k.) The roots x_k increase as k increases. If x > x_k, then we will have x^k < 2^x. Our goal is to show that, for n > 2, p_n is greater than x_2n, so that p_n^2n < 2^p_n, from which it will follow that p_n^2n < p_n+1.Fortunately, there is an infinite family of exact roots that makes the analysis easier: For any value of k of the form k = 2^2^n/2^n, the root x_k is equal to 2^2^n. These x_k clearly increase much more slowly than an exponential tower of n 2's. Finally, putting everything together, we have 2( k_n)^n <( p_n)^2n < 2^p_n < p_n+1, sos - h_n/k_n < 2/ p_n+1 < 1/ (k_n)^n,so s is a Liouville number, and is, therefore, transcendental.Another transcendental number. Consider the sum of reciprocals of the squares of the primes:t = ∑_j = 1^∞1/p_j^2 = 0.11702827460107963588 … .The proof that t is also a Liouville number uses results from the proof above. Let t_n = H_n/K_n be the sum of the first n terms. If we add the first n terms by obtaining a common denominator, that denominator will beK_n = p_1^2 · p_2^2 … p_n^2 =( k_n)^2. t - H_n / H_n > 0, and| t - H_n/H_n|=t - H_n/K_n= 1/p_n+1^2 + 1/p_n+2^2 + … < 2/p_n+1^2 < 4/p_n+1^2 . The reason for including 4/p_n+1^2 will become clear shortly. Similarly to what we proved above in (<ref>), we want to prove that4/ p_n+1^2< 1/ ( K_n)^nso that we would havet - H_n/K_n < 4/p_n+1^2 < 1/ ( K_n)^n,which would prove that t is a Liouville number.Inequality (<ref>) is equivalent to4( K_n)^n < p_n+1^2, But already know from inequality (<ref>) that2( k_n)^n < p_n+1 ,Square both sides of this and use Equation (<ref>):( 2( k_n)^n)^2= 4(( k_n)^2)^n= 4( K_n)^n < p_n+1^2,and we are done!It is easy to see that this proof carries over for any exponent k ≥ 1, so that∑_j = 1^∞1/p_j^kis a Liouville number, and is transcendental.§ SMALLEST PRIMES IN A WRIGHT SEQUENCE Wright's original sequence began with the primes 3, 13, and 16381. Can we find another sequence based on the same recursion, (<ref>), but which has smaller primes?Given the initial constant g_0 and base B, let's considerg_n+1 = B^g_n .If both B and g_0 are greater than 1, the values in the sequence will become arbitrarily large.Wright's original proof <cit.> that his sequence with B = 2 is prime for every n > 0 is based on the fact that, for N > 1, there is always a prime between N and 2N<cit.>. In general, if K < g_n < K + 1, then writing N = B^K, we have N <g_n+1 = B^g_n < B^K+1 = B · N.Because there is a prime between N and 2N, for any B > 2, there would also be a prime in the larger interval between N and B · N. So, Equation (<ref>) could be made to work for any B > 2. However, in this case, the primes would increase much faster than those in Wright's original sequence.What if 1 < B < 2? In this case, if N is small, there might not be a prime between N and B · N. For example, Schoenfeld <cit.> showed that for B = 1 + 1/16597, there is a prime between N and B · N, providedN ≥ 2010760. It follows from the Prime Number Theorem that, if B > 1, then the interval [N, B · N] contains a prime, but only if N is sufficiently large: that is, N > N_B where N_B is a constant that depends on B.So, let's stick with B = 2, which can be made to yield a prime for every term in the sequence. We will first pick a value of g_0 = c so that the first prime is 2, not 3, which was Wright's first prime.Since p_n = ⌊ g_n ⌋, in order for the p_1, first prime in the sequence, to be 2, we must have2 < g_1 = 2^c < 3. Therefore, by taking natural logs,1 < c < ln3/ln2 = 1.58496 … . To get p_2, we have2^2 < g_2 = 2^g_1 < 2^3.We'll choose the smallest prime greater than 2^2: p_2 = ⌊ g_2 ⌋ = 5, which implies that5 < g_2 = 2^2^c < 6,solnln 5 - lnln 2 /ln2< c < lnln 6 - lnln 2 /ln2,so that 1.25132 … < c < 1.37014 ….For the next step: to get p_3, we have2^5 < g_3 = 2^g_2 < 2^6.We'll choose p_3 = 37, the smallest prime greater than 2^5. Then37 < g_3 = 2^2^2^c < 38,soln ( lnln 37 - lnln 2) - lnln 2 /ln2< c <ln ( lnln 38 - lnln 2) - lnln 2 /ln2,so that 1.25164 … < c < 1.25806 ….Continuing with this procedure, we'll choose p_4 to be the smallest prime greater than 2^37, which is 2^37 + 9 = 137438953481. Then2^37 + 9 < 2^2^2^2^c < 2^37 + 10,so 1.25164759779046301759 … < c < 1.25164759779053169453 ….We see that, with c = 1.251647597790463 …, the value of p_4 = 2^37 + 9 is much smaller than the fourth prime in the sequence when Wright's value of 1.9287800 is used.p_5, the next prime in this sequence, is in the interval 2^p_4 < p_5 < 2 · 2^p_4.Knowing that p_4 = 137438953481, a lower bound for p_5 would be 2^p_4. If we want p_5 to be as small as possible, then we would take p_5 to be the smallest prime greater than 2^p_4, which, as we will see, is proportionately only a little larger than 2^p_4. We can estimate 2^p_4 using logarithms:L = log_102^p_4 = 137438953481 ·log_102≈ 41373247570.4475431962398398. 2^p_4 = 10^L, so2^p_4≈ 10^0.4475431962398398· 10^41373247570 = 2.80248435135 …· 10^41373247570 .Therefore, 2^p_4 has 41373247571 digits, the first ten of which are 2802484351.These are also the first ten digits of the very large prime p_5. Here's why: Let x = 2^p_4, so lnx = 137438953481 ·ln2≈ 9.52654 · 10^10. Then, to apply Dusart's estimate (<ref>), we compute1/25( lnx )^2 ≈ 4.4 · 10^-24 ,and (<ref>) says that there is a prime between 2^p_4 and 2^p_4· (1 + 4.4 · 10^-24). So, p_5 is in this interval, p_5 has 41373247571 digits, and all digits shown for 2^p_4 in estimate (<ref>) are also the initial digits of p_5.We can also prove that the sum of the reciprocals of these primess_2 = 1/2 + 1/5 + 1/37 + 1/137438953481 +1/p_5+ … = 0.72702702703430298464 …is a Liouville number.Recall the Remark just after Definition <ref>. For the sequence of primes here, Table <ref> shows that inequality (<ref>) does not hold for n = 2, but that it does appear to hold for n > 2.Aside from that detail, we can pretty much copy the proof in Section <ref> to show that2( k_n)^n < p_n+1 is true for n > 2, so that sum s_2 is a Liouville number.§ PROOF THAT THE 4932-DIGIT PRPS ARE PRIMES The primality proof for prp4 in Section <ref> was kindly carried out by Marcel Martin, the author of Primo, a program that uses ECPP (elliptic curve primality proving) <cit.> to establish primality of large numbers.Mr. Martin has supplied Primo's primality certificate as a 1.5 megabyte text file.It has been uploaded as an ancillary file to Math arXiv along with the for this pdf, and so is publicly available. That file (converted to use PC-style end of line characters) is . The link to the file may be found at <https://arxiv.org/abs/1705.09741>.The Primo website <cit.> allows one to download a free Linux version of Primo. Inside this compressed file is the file , which explains the format of the certificate file. Excerpts of this primality certificate are in Appendix <ref>.Charles Greathouse's 4932-digit probable prime. The software package PARI/GP<cit.> also uses ECPP to produce primality certificates for large numbers. This free software runs on Android, Linux, MacOS, and Windows. In Section <ref>, we discussed Greathouse's probable prime q = 2^16382 - 35411. Samuel S. Wagstaff, Jr. used PARI/GP to prove that q is prime. A script to produce this certificate is in Appendix <ref>. This job required 16 gigabytes of memory and took about 26 hours running on one node of the Rice cluster at Purdue University. PARI/GP produced a file containing a primality certificate for q. The same machine took about 15 minutes for PARI/GP to validate the certificate. The certificate (one long line of text) was also validated on a laptop running Windows 10. The 3.8-megabyte certificate file, , has been uploaded as an ancillary file and can be found at the above link.99CaldwellAndCheng Chris K. Caldwell and Yuanyou Furui Cheng, Determining Mills' Constant and a Note on Honaker's Problem, Journal of Integer Sequences, vol. 8 (2005), Article 05.4.1<https://arxiv.org/abs/1010.4883v2>Dusart Pierre Dusart (2010), Estimates of Some Functions Over Primes Without R.H. <https://arxiv.org/abs/1002.0442>Mills W. H. Mills, A prime-representing function, Bulletin of the American Mathematical Society, vol. 53, no. 6 (1947) p. 604. Available at <https://dx.doi.org/10.1090/S0002-9904-1947-08849-2>NivenBook Ivan Niven, Irrational Numbers, Carus Mathematical Monograph Number 11, Mathematical Association of America, 1967.OEIS_Caldwell_A OEIS Foundation, The On-Line Encyclopedia of Integer Sequences, Sequence A051021,<https://oeis.org/A051021>OEIS_WrightPrimes Charles Greathouse's version of Wright's Constant, OEIS Foundation, The On-Line Encyclopedia of Integer Sequences, Sequence A016104, <https://oeis.org/A016104>OEIS_MillsPrimes OEIS Foundation, The On-Line Encyclopedia of Integer Sequences, Sequence A051254,<https://oeis.org/A051254>PARI/GP PARI/GP, <https://pari.math.u-bordeaux.fr/>primo Primo, <http://www.ellipsa.eu/public/primo/primo.html>Schoenfeld Lowell Schoenfeld, Sharper bounds for the Chebychev Functions θ(x) and ψ(x). II, Mathematics of Computation, vol. 30, no. 134 (April, 1976) pp. 337–360, DOI: 10.2307/2005976.Available through JSTOR at <http://www.jstor.org/stable/2005976> or from sci-hub at <http://www.jstor.org.sci-hub.tw/stable/2005976> or at <https://www.ams.org/journals/mcom/1976-30-134/S0025-5718-1976-0457374-X/>Wiki-BertrandBertrand's Postulate, Wikipedia,<https://en.wikipedia.org/wiki/Bertrand's_postulate>Wiki-ECPPElliptic curve primality, Wikipedia,<https://en.wikipedia.org/wiki/Elliptic_curve_primality>Wiki-WrightFormula for primes - Wright's formula, Wikipedia,<https://en.wikipedia.org/wiki/Formula_for_primes#Wright.27s_formula>Wiki-PRPProbable prime, Wikipedia,<https://en.wikipedia.org/wiki/Probable_prime>Wright E. M. Wright, A prime-representing function, American Mathematical Monthly, vol. 58, no. 9 (November, 1951) pp. 616–618.Available through JSTOR at <http://www.jstor.org/stable/2306356> or from sci-hub at <http://www.jstor.org.sci-hub.tw/stable/2306356>Wright2 E. M. Wright,A class of representing functions, J. London Math. Soc., vol. 29 (1954), pp. 63–71,DOI 10.1112/jlms/s1-29.1.63 .Available from sci-hub at <http://sci-hub.tw/10.1112/jlms/s1-29.1.63>WolframDoc-PrimeQ Wolfram Research, Some Notes on Internal Implementation,<http://reference.wolfram.com/language/tutorial/SomeNotesOnInternalImplementation.html#6849>rjbaillie 'a' with a circle around it, frii dotcom; State College, Pennsylvania§ WRIGHT'S 4932-DIGIT PRIME Here is prp4 from Section <ref>. The first 49 lines have 100 digits each; the 50th line has 32. 1913966420463110498403837302580868256924068401302629071247047560451589953807435264854392127830031342 3720949605721845025408541416289929256457498154990879565082392381927933483828466923960616991247583802 9883619110692151423464455379009608955465329715762183525752181460161562758974828177320048995147265873 2612842019664152281348948186398732693179381636809020721953435180587581344583081883196010622609758369 7767679075913848908389442791706899766927582774282426822260152187401770387233733171089048849946028924 5157524523119653473649546027804630790081490847531422093148584816528706829028167565311356355769106236 3436320845234403098854218594788259973815323356158099262195691733448838182334266034192071091676439686 2265510565742437869510668830397269397683367735888705803587084949626067429633365796074180406455970912 6668629783145246115021331298258625391024527249386102804251972165845252223261489931721363858670767105 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File=/home/primo64/work/Baillie4932.in N=292F...0303 HexadecimalSize=4096 DecimalSize=4932 BinarySize=16382[1] S=12 W=B979...FD56 J=4FE3...51F2 T=2[2] S=39330122B W=12FA...F0F5 J=A137...006F T=1...[537] S=5CB304 W=-AD87C590F88A80304CA A=2 B=0 T=3[538] S=5A Q=3[539] S=379 W=-19D0C7AD9EA43AB5 A=0 B=3 T=1[540] S=5E0A0257FA10 B=2[Signature] 1=06DE9AC57B1F53C2FD64648659604AEF1531E97C9871A932 2=C5858EFD6BB8AADF1C00EA4A566005740310857178FBCE25§ SCRIPT TO CREATE THE PARI/GP PRIMALITY CERTIFICATE This script creates a primality certificate, then reads it back in to validate it. Note that the paths to the files are not specified here.The following script converts the single line, PARI/GP-readable certificate into a multi-line human-readable text file. Warning: some lines in the human-readable file are still very long (9871 characters). | http://arxiv.org/abs/1705.09741v4 | {
"authors": [
"Robert Baillie"
],
"categories": [
"math.NT",
"11A41 (Primary), 11A51 (Secondary)"
],
"primary_category": "math.NT",
"published": "20170526233600",
"title": "Wright's Fourth Prime"
} |
Department of Physics, Yale University, New Haven, CT 06520, USADepartment of Physics, Yale University, New Haven, CT 06520, USADepartment of Physics, Yale University, New Haven, CT 06520, USA We study the energy spectrum and the electromagnetic response of Andreev bound states in short Josephson junctions made of semiconducting nanowires. We focus on the joint effect of Zeeman and spin-orbit coupling on the Andreev level spectra. Our model incorporates the penetration of the magnetic field in the proximitized wires, which substantially modifies the spectra. We pay special attention to the occurrence of fermion parity switches at increasing values of the field and to the magnetic field dependence of the absorption strength of microwave-induced transitions. Our calculations can be used to extract quantitative information from microwave and tunneling spectroscopy experiments, such as the recently reported measurements in Van Woerkom et al. <cit.>. Zeeman and spin-orbit effects in the Andreev spectra of nanowire junctions L. I. Glazman December 30, 2023 ==========================================================================§ INTRODUCTION The Josephson current flowing across a weak link between two superconductors is mediated by Andreev bound states <cit.>, sub-gap states localized at the position of the weak link. Recent years have witnessed the direct observation of Andreev bound states in different types of weak links <cit.>, via either tunneling or microwave spectroscopy, as well as their coherent manipulation <cit.>. Aside from increasing our understanding of mesoscopic superconductivity, these results pave the way to the realization of novel types of qubits <cit.> and superconducting circuits. Among these results, of particular interest is the very recent microwave detection of Andreev bound states in InAs/Al nanowires <cit.>. Such hybrid semiconducting/superconducting systems are under intense investigation <cit.> as platforms for Majorana zero modes <cit.> and, eventually, topological quantum computation <cit.>. In these devices, the study of Andreev bound states may be a prelu,de for the study of Majoranas in microwave circuits and the realization of topological qubits.In view of these exciting applications, the measurement of the magnetic field dependence of the Andreev spectra was among the most interesting aspects of the experiment of Ref. <cit.>. InAs (or InSb) nanowires are characterized by strong spin-orbit coupling and large g-factors: both are necessary ingredients to reach the topological phase with Majorana bound states which is predicted to occur at high magnetic field and low electron density <cit.>. Spectroscopic studies of nanowire Josephson junctions, even if performed in the topologically trivial phase, can bring quantitative understanding about the interplay of Zeeman and spin-orbit couplings needed for the topological applications. For these purposes, an important merit of such experiments is their high degree of tunability. For instance, in the experiment of Ref. <cit.> three separate knobs could be tuned to study the behavior of Andreev bound states: the phase difference ϕ across the Josephson junction, the transparency of the junction (controlled by a local gate underneath the weak link), and the magnetic field B (which was applied parallel to the wire). Thanks to this high tunability, the measurement of the Andreev spectra can allow one to obtain a great wealth of information about the properties of the device.In order to understand existing experiments and design future ones, it would be beneficial to have a detailed theory of the Andreev bound states, describing their behavior as the magnetic field and other system parameters are continuously varied. This work aims at providing such theory, focusing on the simple yet experimentally relevant situation of a short, single-channel nanowire junction placed in a magnetic field parallel to the wire (see Fig. <ref>). Our theory covers all the important regions of the phase diagram, as depicted in Fig. <ref>. We pay particular attention to the behavior of the Andreev bound states in the topologically trivial phase, since such knowledge may be important to assign experimental data to the correct place of the phase diagram. Aside from the Andreev energy spectrum, we also study in detail the magnetic field dependence of the matrix elements which determine the absorption strength of microwave-induced Andreev transitions.The study of Andreev bound states in Josephson junctions with broken time-reversal and/or spin-rotation symmetries is a very rich topic of research, covered by a large and diverse body of previous works <cit.>. In many of the existing studies it is assumed that the time-reversal symmetry breaking is only operative in the weak link, while the effect of the magnetic field in the superconducting parts of the device is disregarded. At the technical level, this means that the effect of time-reversal symmetry breaking is incorporated in the scattering matrix of the junction but not in the description of the superconducting electrodes. In the present context, however, it is crucial to include the effect of the magnetic field on the proximity-induced pairing occurring in the nanowire segments which are in direct contact with the superconductors. Indeed, in experiments aimed at reaching the topological phase, the purpose of the magnetic field is not to influence the local properties of the weak link, but to change the nature of the superconducting pairing induced in the nanowire (whether or not the topological phase is actually reached). The theory of Andreev bound states developed here, therefore, removes the aforementioned assumption and incorporates the effect of the magnetic field in the entire semiconducting nanowire.Let us summarize the main results presented in this work, and at the same time outline the layout of the paper. In Sec. <ref>, we discuss the nanowire model and the different approximations used in this work. We then derive a determinant equation, Eq. (<ref>), which allows us to solve for the discrete part of the spectrum, i.e. to determine the Andreev bound state energies and their wave functions. The bound state equation (<ref>) makes use of the transfer matrix of the junction, unlike the commonly adopted approach based on the scattering matrix <cit.>, but akin to previous examples appearing in the literature <cit.>.In a short single-channel junction, the sub-gap spectrum consists of a doublet of Andreev bound states. In Sec. <ref>, we study the energies of this Andreev doublet by solving the bound state equation both analytically and numerically [The numerical code accompanying this work is available online at <https://github.com/bernardvanheck/andreev_spectra> ]. The Section begins with a review of basic concepts regarding the excitation spectrum of Josephson junctions (Sec. <ref>) and of known results about the Andreev bound states at zero magnetic field (Sec. <ref>). We then discuss the important features of magnetic field dependence of the Andreev bound state energies at both low and high electron density, and in both the topological and trivial phases (Sec. <ref> and Fig. <ref>). In particular, we present analytical results for the effective g-factor which determines the linear energy splitting of the Andreev doublet in a small magnetic field, see Sec. <ref> and specifically Eqs. (<ref>)-(<ref>). We show that the g-factor of the Andreev bound states can be strongly suppressed by spin-orbit coupling and/or high electron density. The resulting g-factor can be much smaller than the g-factor of the conduction electrons in the normal state. Equations (<ref>)-(<ref>) also indicate that a measurement of the Andreev bound state g-factor, for instance by means of tunneling spectroscopy, can provide information about the other relevant system parameters. In Sec. <ref> we discuss the appearance of Fermi level crossings in the Andreev spectrum. The presence of Fermi level crossings is significant because it signals a change of the ground state fermion parity of the junction. These “fermion parity switches” can be used as a signature of the topological phase. Namely, if the nanowire is in the topological (trivial) phase, the number of fermion parity switches occurring as the phase difference ϕ is advanced by 2π must be odd (even). In the topological phase, this leads to the well-known 4π periodicity of the phase dependence of the Andreev bound state energies. The occurrence of fermion parity switches in the trivial phase has so far attracted less attention: here we show that they can appear once the magnetic field crosses a threshold value B_sw, which depends sensitively on the transparency of the junction and on the strength of the spin-orbit coupling (see Fig. <ref>).In Sec. <ref> we turn our attention to the Josephson current carried by the Andreev bound states, introducing the current operator and briefly discussing the magnetic field dependence of the current-phase relation (Fig. <ref>). The matrix elements of the current operator between the Andreev bound states determine not only the equilibrium properties of the junction, but also its response to a microwave field. The microwave irradiation of the junction can induce two types of transitions within the Andreev bound state doublet: both are discussed in Sec. <ref> within the linear response regime, appropriate if the applied microwave field is weak. In the first and most notable type of transition, microwaves resonantly excite a Cooper pair from the superconducting condensate to the Andreev doublet. In the second type of transition, instead, microwaves excite one quasiparticle from the first to the second Andreev bound state. The two transitions are distinguished by the parity of the number of quasiparticles involved and so, for brevity, we will refer to them as the “even” and “odd” transition respectively. The even transition is present already at zero magnetic field and, being very bright, is the most easily observed in experiment. The odd transition requires a quasiparticle to be present in the initial state of the junction, either due to a non-equilibrium population or as a consequence of a fermion parity switch. At zero magnetic field, the odd transition is not observable, since in this case the microwave field cannot induce a transition within the degenerate doublet of Andreev levels. However, it may become visible in the presence of both Zeeman and spin-orbit couplings. The magnetic field dependence of the current matrix elements for the even and odd transitions is studied in Secs. <ref> and <ref> respectively. The study reveals that the odd transition, while characterized by a non-zero current matrix element, remains much weaker than the even transition over a wide range of system parameters, including for magnetic fields B>B_sw. An important consequence of this fact is that, at low temperatures, the absorption spectrum of the junction should exhibit a sudden drop in visibility if the junction is driven through a fermion parity switch by varying the magnetic field or the phase difference (see Fig. <ref>).§ MODEL AND ANDREEV BOUND STATE EQUATIONOur investigation is based on the well-studied model of a one-dimensional (1D) quantum wire with Rashba spin-orbit coupling, a Zeeman field applied parallel to the wire, and a proximity-induced s-wave pairing <cit.>. We consider the Josephson junction geometry shown in Fig. <ref>. In the limit L/ξ→ 0, we can treat the junction as a point-like defect situated at a position x=0. Furthermore, provided that the length of the entire nanowire is much larger than ξ, we can ignore complications arising from its finite size and treat it as an infinite system in the x direction. The effective BCS Hamiltonian of this system is (we set ħ=1)H= 1/2∫ xψ^†(x) [(-_x^2/2m-iα_x σ_z - μ)τ_z . .- 1/2 g μ_B B σ_x - Δ_0 τ_x ^-iϕ (x) τ_z/2 + V δ(x) τ_z] ψ(x) .In this Hamiltonian, the field operator ψ(x) is the usual four-component Nambu spinor, ψ=(ψ_ , ψ_ , ψ^†_ , -ψ^†_)^T. The two sets of Pauli matrices σ_x,y,z and τ_x,y,z act in spin and Nambu space, respectively. Furthermore, m is the effective mass, α is the strength of the Rashba spin-orbit coupling, B is the applied magnetic field, g is the effective g-factor, μ_B is the Bohr magneton, Δ_0 is the proximity-induced pairing gap, and ϕ is the gauge-invariant phase difference across the Josephson junction, and μ is the chemical potential measured from the middle of the Zeeman gap at k=0 (see Fig. <ref>). Finally, the phenomenological parameter V is the strength of a point-like scatterer which models a potential barrier; later on V will be related to measurable properties of the junction. In practice, the scattering term Vδ(x) enters purely in the boundary condition for ψ(x) at the position of the junction, ψ(0^+) =ψ(0^-) , _xψ(0^+)-_xψ(0^-)= 2 m V ψ(0^-) .Being a purely one-dimensional effective model, the Hamiltonian in Eq. (<ref>) does not incorporate all the complexity of real devices. For instance, the orbital effect of the magnetic field is not included in our analysis: this is well justified if the cross-section A of the nanowire is small, so that at a given field B the flux piercing the cross-section is much smaller than a flux quantum (BA≪ h/e) [Nanowires currently studied in experiments have a nominal area of about (100 nm)^2, but the lowest subbands wave functions are most likely extended over a much smaller area ∼ 1000 nm^2 due to gate confinement and band bending. Thus, they are only sensitive to a portion of the total applied flux. One may estimate that the orbital effect of the magnetic field can not be dominant for B≲ 1 T, while the experiment reported in Ref. <cit.> covered a field range smaller than 500 mT.]. The pairing strength Δ_0, the spin-orbit strength α and the g-factor g appearing in Eq. (<ref>) should be viewed as phenomenological parameters. The value of Δ_0, in particular, strongly depends on the transparencyof the semiconductor-superconductor interface.By virtue of simplicity, the Hamiltonian (<ref>) has become a paradigmatic model in the study of Majorana physics in hybrid semiconducting-superconducting system <cit.>. As is well known, it exhibits two distinct topological phases (see Fig. <ref>). At high chemical potential and/or low magnetic fields, the system is in a trivial superconducting phase with a conventional 2π-periodicity of the ground state energy with respect to the phase difference ϕ. At low chemical potential, and provided that the condition 12gμ_B B >(μ^2+Δ_0^2)^1/2 is satisfied, the system is instead in a topological superconducting phase. In the geometry of Fig. (<ref>), the hallmark of the topological phase is the 4π-periodicity of the ground state energy (for a fixed global fermion parity) with respect to ϕ, which is associated with the presence of two coupled Majorana zero modes at the junction. The two phases are separated by a critical line B_c(μ)=2 (μ^2+Δ_0^2)^1/2/(gμ_B) at which the energy gap in the nanowire vanishes, marking a topological phase transition. Note that this criterion is appropriate if the transparency of the semiconductor-superconductor interface is low, so that the coupling between the two materials is weak: in the opposite limit of strong coupling, the critical field B_c may depend only weakly on μ <cit.>. A more complicated phase diagram in the (μ, B) plane emerges in nanowires with multiple transport channels <cit.>, but we do not consider this situation in this paper.For both regions of the phase diagram, we are interested in the Andreev level spectrum. That is, we want to find the discrete spectrum of sub-gap states with energy E<Δ(B), which are localized at the junction via the mechanism of Andreev reflection at the two superconducting interfaces. The energy Δ(B) is the spectral gap of the continuous part of the spectrum [at zero field, Δ(0)≡Δ_0]. In what follows, we will often omit the field argument, Δ(B)≡Δ. In the short junction limit, one expects the number of sub-gap states to be less than or equal to the number of pairs of left/right propagating modes at the Fermi level in the normal state of the nanowire. Thus, the discrete spectrum of the Hamiltonian in Eq. (<ref>) should consist of at most either one or two Andreev levels, depending on whether the system is in the topological or in the trivial phase.An established way to compute the Andreev level spectrum is the scattering approach <cit.>. In this approach, the usual wave function matching problem for bound states is cast in terms of a scattering matrix S_N(E) which characterizes the junction, and a second scattering matrix S_A(E) which describes the Andreev reflection from the superconducting leads. The two matrices can be combined into a determinant equation for the bound state energies, [1-S_A(E)S_N(E)]=0. This approach is particularly advantageous if the following conditions are satisfied. First, the effect of a magnetic field can be neglected in the superconducting leads, so that the gap and the matrix structure of S_A are independent of magnetic field. Second, the normal reflection at the superconducting interface can be neglected – this is the so-called Andreev approximation <cit.>: it requires Δ_0≪ E_F (where E_F is the Fermi energy measured from the bottom of the conduction band) and amounts to linearizing the electron dispersion in the normal state. Third, the junction is short, so that the energy dependence of S_N(E) can be neglected as long as E<Δ_0. When combined, the first and second conditions guarantee that S_A(E) is a simple sparse matrix whose energy dependence enters only as a prefactor, S_A(E)∝exp[i arccos(E/Δ_0)]. This circumstance greatly simplifies the solution of the problem, as exemplified by the fact that the determinant equation can be transformed into a finite-dimensional linear eigenvalue problem for E <cit.>.However, as mentioned in the introduction, for our purposes it is crucial to include the effect of magnetic field in the entire system, rather than in the junction alone. The motivation for doing so is two-fold. To begin with, many recent experiments focused on InAs nanowires with epitaxial Al: in this geometry, a parallel field penetrates uniformly the thin aluminum shell. Furthermore, the study of the evolution of the Andreev levels in the different regions of the phase diagram — and in particular across the topological transition — requires that we solve for the Andreev energies taking into account the magnetic field dependence of the spectral gap of the continuous spectrum. Unfortunately, once a magnetic field is included, the Andreev reflection amplitude is not unique anymore but may depend on the initial and final spin and/or orbital states. As a consequence, the calculation of S_A(E) becomes non-trivial and strongly dependent on the particular Hamiltonian describing the leads. To overcome this complication, we take a slightly different route and derive an equivalent bound state equation for the Andreev spectrum which generalizes to the more complicated cases in a transparent fashion.The first step in the derivation is the linearization of the model in Eq. (<ref>), which we perform in two different limits allowing us to cover all relevant regimes of the phase diagram (see Fig. <ref>). The first limit is that of the high density, μ≫ mα^2, Δ_0, gμ_B B. For such high values of the chemical potential the nanowire will not enter the topological phase in a realistic range of magnetic fields, thus we will use this limit to model a topologically trivial nanowire. The second limit is that of low density, when the Fermi level is inside the Zeeman gap . This is the “helical” regime of the Rashba nanowire: in the normal state, the low-energy theory contains only a pair of counter-propagating modes at finite wave vectors, as well as a gapped pair of modes close to k=0. The line μ=0 in the phase diagram coincides with the optimal point at which the critical field is minimal, B_c=Δ_0 (see Fig. <ref>), so this limit will allow us to study the Andreev spectrum in the topological phase and around the phase transition. In both limits we will require the Andreev approximation to hold. The Andreev approximation is automatically satisfied in the high density regime, when Δ_0≪μ. In the low density regime, the chemical potential is low and the Fermi energy is set by the spin-orbit energy, E_F∼ mα^2. Thus, in this limit we must assume the spin-orbit energy to be the dominating energy scale: mα^2≫Δ_0, gμ_B B, μ.In the two following subsections, we carry out the linearization procedure in these two limits, which will then allow us to derive the bound state equation that we seek. §.§ Linearization for μ≫Δ_0, gμ_B, mα^2In the limit of a high chemical potential, we may linearize the normal state dispersion around the Fermi wave vectors ± k_F=±(2m μ)^1/2. That is, we write the field ψ(x) as a linear superposition of left- and right-moving components,ψ(x) = ^-ik_F x ψ_L(x) + ^ik_Fx ψ_R(x) .Since we are interested in the energy spectrum in a range of energies of order Δ around the Fermi level, we can assume that ψ_L(x) and ψ_R(x) vary over length scales much larger than k_F^-1. We may therefore use Eq. (<ref>) in the Hamiltonian (<ref>), organize the resulting expression as an expansion in powers of k_F^-1, and only keep the largest terms. The last step also includes neglecting quickly oscillating terms ∝^± ik_F x. The result of this procedure can be concisely presented by introducing an eight-component field vector Ψ = (ψ_R , ψ_L)^T. In terms of the slowly-varying field Ψ(x), the low-energy Hamiltonian of the nanowire isH ≈1/2 ∫ xΨ^†(x) [-i v_F τ_z s_z _x + α k_F τ_z s_z σ_z ..- 12 g μ_B B σ_x - Δ_0 τ_x ^-iϕ (x) τ_z/2] Ψ(x) .with v_F=k_F/m. Here, we have introduced a new set of Pauli matrices s_x,y,z which act in the space of left- and right-movers. Let us now describe the low-energy modes described in this linearized Hamiltonian.As illustrated in Fig. <ref>a, around each Fermi point there are two branches in the spectrum of the normal state. The two branches are separated in energy by an amount 2 [α^2k_F^2 + (12gμ_B B)^2]^1/2 due to the combined effect of spin-orbit and Zeeman coupling. At finite B, the spin of each propagating mode is rotated with respect to its orientation at B=0 (see arrows in Fig. <ref>a). The rotation angle isθ =arccos α k_F/[α^2k_F^2 + (12gμ_B B)^2]^1/2 .and the rotation plane is defined by the Rashba and Zeeman fields. The spin rotation is clockwise (counter-clockwise) for left (right) movers and it can be incorporated in the definition of the field Ψ via a unitary transformation S (see for instance Ref. <cit.>),Ψ_S(x) = S Ψ(x) ,S=exp [-i (θ/2) τ_zs_zσ_y]This rotated basis diagonalizes the homogeneous Hamiltonian of the wire in the normal state. When we express the Hamiltonian in terms of the rotated field Ψ_S, we findH≈1/2∫ xΨ_S^†(x) [-i v_Fτ_zs_z _x + α k_Fθ τ_z s_z σ_z ..- Δ_0 (cosθ τ_x + sinθ τ_ys_zσ_y) ^-iϕ (x) τ_z/2] Ψ_S(x) .This form of the Hamiltonian reveals how the tilting of the modes' spin affects the pairing. At B=0, the s-wave pairing does not mix the inner and outer branches of the spectrum since it requires the spins of the two paired electrons to be anti-parallel. At finite field, however, a pairing coupling with strength Δ_0sinθ≈Δ_0(gμ_B B)/(α k_F) emerges between the inner and outer branches, due to the fact that the spin tilts in opposite directions for left and right movers.To complete the linearization procedure, we must provide the boundary conditions for the field Ψ which are due to the scattering term V δ(x) τ_z in the original model of Eq. (<ref>). The boundary conditions for Ψ can be derived by using Eq. (<ref>) in Eqs. (<ref>) and neglecting terms ∝_xψ_L,R with respect to terms ∝ k_F. The resulting boundary conditions can be arranged in the following form,Ψ(0^+) = T Ψ(0^-) ,withT = 1 - i (V/ v_F) s_z + (V/v_F) s_y .The matrix T is, in fact, the transfer matrix associated with the point-like scatterer Vδ(x) in the original model, computed at the Fermi level. The term ∝ s_y is a backscattering term, while the term ∝ s_z corresponds to forward scattering. The transmission probability τ through the junction in the normal state is related to the dimensionless parameter V/v_F,τ = 1/1+(V/v_F)^2 .The transfer matrix obeys a “pseudo-unitarity” property,T^† = s_z T^-1 s_z ,which is the equivalent of the most universally known unitarity of the scattering matrix.For the rotated field Ψ_S, we must use a rotated transfer matrix T_S=STS^†,T_S= 1 - i (V/ v_F) s_z+ (V/v_F) cosθ s_y-(V/v_F) sinθ τ_zs_zσ_y .We see that, similar to the pairing, the backscattering terms are changed when projected to the basis of momentum eigenstates of the nanowire. At zero field, only a single backscattering channel is open for each mode, because scattering preserves spin. At a finite field B, two backscattering terms appear, due to the fact that the spin of each left-moving mode has non-zero projection on the spin of both right-moving modes. §.§ Linearization for mα^2 ≫Δ_0, gμ_B B, μWhen the chemical potential is low, the linearization procedure must take into account that the position of the Fermi points strongly depends on the spin orientation, since the Fermi points are shifted by the Rashba spin-orbit coupling. Specifically, the Fermi points for modes with spin down (up) are situated at k=2 mα (k=0) for right-movers and at k=0 (k=-2mα) for left-movers; see Fig. <ref>b. The linearization of the model therefore begins by writing the field in the following form <cit.>,ψ(x) = ^-imα x (1+σ_z) ψ_L(x) + ^imα x(1-σ_z) ψ_R(x) .Note the presence of the spin-dependent factors in the exponentials, which take into account the dependence of the Fermi points on spin. From here, we proceed as in the previous subsection: assuming that the left- and right- moving fields ψ_L(x) and ψ_R(x) vary over length scales much larger than (mα)^-1, we replace Eq. (<ref>) in Eq. (<ref>) with μ=0, and neglect all quickly oscillating terms ∝^± 2imα x. Note that, in doing this, it is essential to assume that the spin-orbit energy dominates over the other energy scales. In other words, the spin-orbit length (mα)^-1 takes the role of the Fermi wavelength as the microscopic length scale of the model.As a result we obtain the following linearized Hamiltonian of freely propagating modes,H ≈1/2∫ x Ψ^†(x) [-i ατ_zs_z _x - 14gμ_B B(s_x σ_x - s_y σ_y) ..- μτ_z - Δ_0 τ_x^-iϕ (x) τ_z/2] Ψ(x) .Here, as in Eq. (<ref>), Ψ=(ψ_R, ψ_L)^T and the set of Pauli matrices s_x,y,z acts in the grading of left- and right-movers.When written in terms of the components of the vector Ψ, the Zeeman term in Eq. (<ref>) is proportional to ψ^†_Rψ_L. It is a mass term which gaps out the two counter-propagating modes with opposite spin crossing at k=0 (see Fig. <ref>). Note that, once this Zeeman gap is formed at the Fermi level, the presence of a scattering impurity may lead to Fano resonances <cit.>. The Fano resonances are due to the formation — in the normal state — of quasi-bound states with a characteristic decay length α/gμ_B B. The quasi-bound states originate from the inverted part of the parabolic spectrum close to k=0, and in principle they can lead to a strong dependence of the transmission of the junction in the normal state on energy <cit.>. We may neglect complications associated with their presence by assuming that the junction is short enough so that L≪α/gμ_B B. With this assumption, boundary conditions for Ψ can also be derived as in the previous subsection. We obtain the same transfer matrix T of Eq. (<ref>), except with the velocity v_F replaced by α; the transmission probability is now τ=1/(1+V^2/α^2). §.§ Bogoliubov-de Gennes equations and bound state determinant condition At this point, in either of the two limits μ≫Δ_0 and μ=0, our task is reduced to the solution of a system of Bogoliubov-de Gennes (BdG) equations[-ivτ_zs_z_x+O_N-Δ_0τ_x^-iϕ(x)τ_z/2]Φ = EΦ ,for an eight-component Nambu wave function Φ(x) [Note that we use the letter Φ for the BdG wave functions, and Ψ for the corresponding second-quantized fields], to be solved with the boundary condition Φ(0^+)=TΦ(0^-). The pseudo-unitarity of the transfer matrix T, Eq. (<ref>), guarantees that the kinetic energy in the BdG equations remains a Hermitian operator when removing the point x=0 from its domain. In Eq. (<ref>), O_N=α k_F τ_z s_z σ_z - 12 gμ_B σ_x for μ≫Δ_0, gμ_B B, mα^2 while O_N =- 12gμ_B B12(s_x σ_x - s_y σ_y)-μτ_z for mα^2≫Δ_0, gμ_B B, μ. The velocity v is a placeholder for v_F in the former case, and for α in the latter.As is well known <cit.>, the BdG equations are inherently equipped with a particle-hole symmetry represented by an anti-unitary operator 𝒫. The particle-hole symmetry dictates that for each solution Φ of Eq. (<ref>) at energy E there must be an orthogonal solution 𝒫Φ at energy -E. In our case, 𝒫=τ_ys_xσ_y 𝒦, with 𝒦 the complex conjugation operator. The presence of particle-hole symmetry, and the corresponding doubling of the spectrum, is a consequence of the unphysical doubling of the Hilbert space coming from the introduction of Nambu indices; more fundamentally, it is a consequence of the mean-field approximation which allowed us to express the Hamiltonian (<ref>) as a quadratic form of ψ and ψ^† <cit.>. Once the complete spectrum {±E_n} of the BdG equations is known, the field operator Ψ(x) can be written in the eigenmode expansionΨ(x) = ∑_n Γ_n Φ_n(x) + Γ^†_n [𝒫Φ_n(x)] .Here, Γ_n and Γ_n^† are Bogoliubov annihilation and creation operators, obeying fermionic anticommutation relations. They diagonalize the Hamiltonian,H = ∑_n E_n(Γ^†_nΓ_n - 12) .Our goal is to find the bound state solutions of Eq. (<ref>), which have E<Δ. In order to do so, we first bring the BdG equations to a more convenient form by a change of variable, Φ(x) =^i ϕ(x)τ_z/4 Φ̃(x) .The role of this transformation is to make the spatial dependence of the superconducting phase more easily tractable. The wave function Φ̃(x) satisfies a modified boundary condition at the origin,Φ̃(0^+) = ^-iϕτ_z/2 T Φ̃(0^-) .Next, we define the Green's function G(x, E) by[E-H_BdG(_x)] G(x, E) = i v τ_z s_z δ(x) ,where the operator H_BdG(_x) is the linearized BdG Hamiltonian of the translationally invariant superconducting wire,H_BdG(_x) = -ivτ_zs_z_x +O_N-Δ_0 τ_x .Note that by definition G(0^+, E)-G(0^-, E)=1. Now, using the boundary condition (<ref>), we may writeΦ̃(x) = G(x, E) M Φ̃(0^-) ,withM = (^-iϕτ_z/2 T-1) .Equation (<ref>) holds for any x≠ 0; the wave function is discontinuous at x=0. The Green's function can be computed asG(x, E) = -∫_-∞^∞ v q/2π i^iq x/E-H_BdG(q) τ_z s_z,with H_BdG(q) the Fourier transform of Eq. (<ref>). When E<Δ(B), the poles in the integrand of G(x, E) lie away from the real axis and Eq. (<ref>) can be computed via a contour integral closing on the upper (lower) half of the complex plane for x>0 (x<0). Requiring that a non-trivial solution Φ̃(0^-) exists, we obtain from Eq. (<ref>) the following determinant equation for the bound state spectrum:[1 - G(0^-, E) M ]=0 .This bound state equation for the Andreev levels is cast in terms of a transfer matrix T and a Green's function G(0^-, E) for the superconducting leads, rather than in terms of scattering matrices. As a consequence of the short junction limit considered in this work, the energy dependence of Eq. (<ref>) is entirely contained in G(0^-, E), while T is independent of energy. Furthermore, as mentioned at the end of Section <ref>, the matrix T contained in Eq. (<ref>) is the same in both linearization limits when expressed in terms of the transmission probability τ:T = 1 - i √(1-τ/τ) s_z +√(1-τ/τ) s_yThus, the differences in the subgap spectrum between the two regimes all arise from G(x, E). In the following, we compute G(x, E) for the two regimes of interest. In doing so, we also derive the magnetic field dependence Δ(B) of the continuum gap. §.§ Green's functions and magnetic field dependence of the induced gap §.§.§ Green's function forμ≫ mα^2, gμ_B B, Δ_0 In order to obtain G(x, E), we must first invert the 8× 8 matrixE-H_BdG(q)= E - v_Fq τ_zs_z - α k_F τ_z s_z σ_z +12 gμ_B B σ_x + Δ_0 τ_x.This task is simplified by the fact that E-H_BdG(q) is a real matrix and thus its inverse must also be real. The result is1/E-H_BdG(q) = A_0 + A_1v_Fq + A_2 (v_Fq)^2 -τ_z s_z (v_F q)^3/v_F^4 (q^2-q_0^2)(q^2-q_1^2) .Here, A_0, A_1, A_2 are 8× 8 matrices which do not depend on q. Their detailed expressions are:A_0= -(E + α k_F τ_z s_zσ_z)[Δ_0^2-E^2+(α k_F)^2 + (12gμ_BB)^2] + 12gμ_BB [(α k_F)^2 + (12gμ_BB)^2-Δ_0^2 - E^2] σ_x+Δ_0 [Δ_0^2 - E^2 + (α k_F)^2 - (12gμ_BB)^2]τ_x + gμ_B B Δ_0 (E τ_xσ_x-α k_F τ_y s_z σ_y) ,A_1= [(α k_F)^2 + (12gμ_BB)^2 + E^2 -Δ_0^2] τ_zs_z + 2E (α k_F σ_z - 12gμ_BB τ_z s_z σ_x) - 2 Δ_0 α k_F τ_x σ_z ,A_2= α k_F τ_z s_z σ_z - 12gμ_BB σ_x + Δ_0 τ_x - E . There are four simple poles ± q_0, ± q_1 appearing on the right side of Eq. (<ref>), given byv_F^2 q_0,1^2= E^2-Δ_0^2 + (α k_F)^2 + (12gμ_BB)^2 ± 2i √((α k_F)^2 (Δ_0^2 - E^2)-(12gμ_BB)^2 E^2) .In order to complete the calculation of the Green's function, we must insert Eq. (<ref>) in Eq. (<ref>) and perform the integral over q. Let us choose q_0 and q_1 to be the two poles with negative imaginary part. Then, using the residue theorem and some simple algebra, we obtain the following expression for the Green's function:G(x,E) = 1/21/v_F^2(q_0^2-q_1^2) ∑_n=0,1(-1)^n ^-iq_n x/v_F q_n [A_0 - (x) A_1 v_F q_n + A_2 (v_Fq_n)^2 + (x) τ_z s_z (v_F q_n)^3] τ_z s_z.From Eq. (<ref>) we can easily extract the magnetic field dependence of the continuum gap. The gap Δ(B) is determined by the smallest value of E such that the poles q_0,1 have zero imaginary part. A few lines of algebra give the following answer:Δ(B) =Δ_0 α k_F[(α k_F)^2 + (12gμ_BB)^2]^1/2 if√(12gμ_BB(Δ_0-12gμ_BB))<α k_F or12gμ_BB>Δ_0 ,[(α k_F)^2 + (Δ_0-12gμ_BB)^2]^1/2 if√(12gμ_BB(Δ_0-12gμ_BB))>α k_F . The behavior of Δ(B) is discussed in detail in Fig. <ref>. Here we only note that Δ(B) is a smooth function of B, and never reaches zero provided that spin-orbit is present (so that α≠ 0). These results are true if we assume no suppression of the gap in the parent superconductor which induces the proximity effect in the nanowire. In the case of InAs nanowires with epitaxial Al, this is justified by the smallness of Al g-factor and shell thickness.§.§.§ Green's function for mα^2≫ g μ_B, Δ_0, μ At low chemical potentials, we must repeat the same calculation but starting from the BdG Hamiltonian contained in Eq. (<ref>). We must first invert the matrixE- H_BdG(q)=E - α q τ_zs_z + 14gμ_B B (s_x σ_x - s_y σ_y) + μτ_z + Δ_0 τ_x .In this case, we may simplify the calculation by noting the presence of the unitary symmetry [E-H_BdG(q), s_zσ_z]=0. This symmetry is a consequence of the fact that the inner (k≈ 0, s_zσ_z=1) and outer (k≈± 2 mα, s_zσ_z=-1) branches of the linearized spectrum are decoupled in the homogeneous wire (although they are coupled by scattering at the junction). Furthermore, the outer branches are not coupled to the magnetic field in the linearized Hamiltonian of Eq. (<ref>), and so for these modes s_z and σ_z are also separately conserved operators. These facts allow use to separate the inverse of Eq. (<ref>) as a sum of two parts,1/E- H_BdG(q) =1-s_zσ_z/2 [1+s_z/2 (μ-α q) τ_z s_z + Δ_0τ_x-E/Δ_0^2-E^2+(μ-α q)^2+ 1-s_z/2 (μ+α q) τ_z s_z + Δ_0τ_x-E/Δ_0^2-E^2+(μ+α q)^2]+1+s_zσ_z/2 B_0 + B_1 α q + B_2 (α q)^2 - τ_z s_z (α q)^3/α^4 (q^2-q_0^2)(q^2-q_1^2)This time, the poles q_0, q_1 appearing in Eq. (<ref>) are given byα^2 q_0,1= E^2+μ^2-(12gμ_B B)^2-Δ_0^2± 2 i √(μ^2(Δ_0^2-E^2)-Δ_0^2 (12gμ_B B)^2) ,while the matrices B_0, B_1, B_2 are B_0 = E [E^2-Δ_0^2-(12gμ_B B)^2] - 12gμ_B B[Δ_0^2 + E^2 -(12gμ_B B)^2] s_xσ_x + Δ_0[Δ_0^2 - E^2 -(12gμ_B B)^2] τ_x + gμ_B B Δ_0 E τ_x s_x σ_xB_1 = [E^2 -Δ_0^2-(12gμ_B B)^2]τ_zs_z - gμ_B B Δ_0 τ_y s_x σ_y ,B_2 = - E + 12gμ_B B s_x σ_x + Δ_0 τ_x . From these expressions, we may compute the Green's function in this regime:G(x,E) = 1-s_zσ_z/2 i ^-iμ x s_z/α^-√(Δ_0^2-E^2)x/α/2√(Δ_0^2-E^2) [Δ_0τ_x - E -i (x) τ_z s_z √(Δ_0^2-E^2)] τ_z s_z+1+s_zσ_z/21/2α^2(q_0^2-q_1^2)∑_n=0,1(-1)^n ^-iq_n x/α q_n[B_0 - (x)B_1α q_n + B_2 (α q_n)^2 + (x)τ_z s_z(α q_n)^3] τ_z s_z .We can again extract the magnetic field dependence of the proximity-induced gap looking at the energy dependence of the poles in Eq. (<ref>). In general, the minimal gap is dictated by the competition between that of the inner and outer modes. The spectral gap for the inner modes, which we denote Δ^(k=0)(B), is given byΔ^(k=0)(B)Δ_0^2 √(1-(12gμ_B B)^2/μ^2) if12gμ_B B<μ^2/√(μ^2+Δ_0^2) ,√(Δ_0^2+μ^2)-12gμ_B B if12gμ_B B>μ^2/√(μ^2+Δ_0^2) .The gap of the outer modes is not influenced by the magnetic field to the leading order in the ratio B^2/mα^2, thus in our effective model it is equal to Δ_0 at all fields. The spectral gap is thus given byΔ(B) = min{Δ^(k=0)(B) , Δ_0}At a fixed value of μ, after a slow initial decrease the proximity-induced gap decreases linearly with field and, as already mentioned, closes at B=B_c(μ), at which point the topological transition takes place (see Fig. <ref>). Increasing B further, the gap Δ(B) reopens, growing linearly in field until the gap at k=0 becomes larger than that at finite momentum. The gap at finite momentum is equal to Δ_0, while it is well known that this gap has in fact a weak field dependence: it is quadratically suppressed with increasing B if corrections of the order (gμ_B B/mα^2)^2, not included in our approximation, are taken into account. This limitation is inconsequential for our purposes, since we are mainly interested in the Andreev spectrum in the range of magnetic fields for which the relevant gap is the one at k=0. § PROPERTIES OF THE ANDREEV SPECTRUMIn this Section we discuss in detail the magnetic field and phase dependence of the Andreev bound state energies. We begin with a review of the basic notions underpinning the understanding of the excitation spectrum of a Josephson junction. §.§ Andreev levels, excitation spectrum, and fermion parity switchesSolving the determinant equation derived in the previous Section, Eq. (<ref>), allows us to determine the subgap spectrum of the BdG equations (<ref>). Since we are dealing with a purely 1D model in the short junction limit, we expect that the subgap spectrum consists of (at most) two distinct Andreev levels. That is, taking into account the doubling of the spectrum enforced by the particle-hole symmetry, the subgap spectrum of the BdG equations consists of (at most) four solutions {± E_1, ± E_2}. Without loss of generality, we fix a hierarchy 0≤E_1≤E_2≤Δ(B).Once the Andreev levels are determined, the many-body Hamiltonian can be expanded asH = E_1 (Γ^†_1 Γ_1 - 12) + E_2 (Γ^†_2Γ_2 - 12)+ …where the dots represent the omission of states coming from the continuous part of the spectrum, with energies higher than Δ(B). Neglecting the presence of these states, we can limit ourselves to considering just four many-body eigenstates: the vacuum state |V⟩, which is annihilated by both Γ_1 and Γ_2; two single-particle states |1⟩ = Γ^†_1|V⟩ and |2⟩ = Γ^†_2|V⟩; and finally the state with a pair of quasiparticles, |P⟩=Γ^†_1Γ^†_2|V⟩. The fermion parity of the junction, which is a global symmetry of the Hamiltonian, is even in the states |V⟩ and |P⟩, and odd in the states |1⟩ and |2⟩. Up to a common constant, the energies of these four many-body eigenstates are simply related to the Andreev levels E_1 and E_2 via Eq. (<ref>), see the table in Fig. <ref>.Note that, so far, we have not specified the sign of the energies E_1 and E_2 appearing in Eq. (<ref>). In fact, this choice is arbitrary: as can be seen in Fig. <ref> the many-body spectrum is invariant under a change of sign of E_1 and E_2. This is, again, a consequence of the particle-hole symmetry of the model. Conventionally, one chooses E_1 and E_2 to be positive in Eq. (<ref>). In this case, the ground state of the system is identified with the even parity state |V⟩. The states |1⟩, |2⟩ and |P⟩ are excited states with excitation energies E_1, E_2 and E_1 + E_2 respectively.Although the initial choice of the sign of E_1 and E_2 in Eq. (<ref>) is conventional and does not have measurable consequences, a change in the sign of E_1 is physical, and it has measurable and important consequences. Such a change in sign can occur as some of the parameters of the system are varied, typically the magnetic field B or the phase ϕ. To fix the ideas, let us assume that E_1 is initially positive and that it can be tuned through the point E_1=0 by changing a parameter — a so-called Fermi level crossing (see green arrow on the left panel of Fig. <ref>). When E_1=0, the states |V⟩ and |1⟩ are degenerate in energy: the energy cost to add a quasiparticle to the junction vanishes (see green arrow on the right panel of Fig. <ref>). Furthermore, when E_1 becomes negative, the odd-parity state |1⟩ becomes the ground state of the junction. This ground state transition driven by a Fermi level crossing is commonly referred to as a fermion parity switch.Fermion parity switches can be generically expected in Josephson junctions with broken time-reversal symmetry <cit.>, and can drastically affect the thermodynamic and transport properties of the junction. The Yu-Shiba-Rusinov states associated with magnetic impurities in s-wave superconductors <cit.> provide an early example of this type of phenomenon. A fermion parity switch is also at the basis of the 4π-periodic Josephson effect associated with Majoranas <cit.>. In this case, the peculiarity is that there is an odd number of fermion-parity switches in a 2π phase interval, a signature of the presence of a fermion-parity anomaly in the low-energy theory of the junction (only an even number of fermion parity switches in a 2π phase interval is allowed in a topologically trivial phase). Later in this Section, we will investigate the occurrence of fermion parity switches in the model under study, both in the trivial and topological phases. Before doing so, we provide an overview of the features of the Andreev spectrum of the model, starting from the well-known case in which B=0. §.§ Solution at zero magnetic fieldAt zero magnetic field, an analytic solution leads to a well-known universal result for the Andreev levels <cit.>. The Andreev levels form a degenerate doublet, E_1=E_2≡ E_A withE_A = Δ_0 [1-τsin^2(ϕ/2)]^1/2 .This result is valid independently on the values of chemical potential μ and spin-orbit coupling α, provided that the Andreev approximation is applicable.While the solution (<ref>) is already well-known, it is instructive to reproduce this result from Eq. (<ref>). At B=0, the Green's function G(0^-,E), which can be deduced from Eqs. (<ref>) or (<ref>), takes a particularly simple form:G(0^-, E)=i/2Δ_0/√(Δ_0^2-E^2) [τ_x - ^i β(E) τ_z s_z] τ_z s_z ,with β(E)=arccos(E/Δ). There are three meaningful facts about the above expression. First, it is valid for both limits μ≫ mα^2, Δ_0 and mα^2≫μ, Δ_0, so we already see that the solutions of the determinant equation (<ref>) will be common to the two cases. Second, in both limits the right hand side of Eq. (<ref>) is independent of the spin-orbit coupling strength α. This is a consequence of the fact that spin-orbit coupling can be removed from the Hamiltonian via a local gauge transformation, and so the Green's function evaluated at a single point can be made independent of α. Thus, the independence of E_A on α can also be explained as a consequence of the short junction limit. Third, the right hand side of Eq. (<ref>) is proportional to the unit matrix in the spin grading, which leads to the anticipated double degeneracy of the solutions. Plugging the Green's function from Eq. (<ref>) and the transfer matrix from Eq. (<ref>) into the determinant equation (<ref>), we obtain the solution (<ref>).It is also possible to write down the bound state wave functions explicitly. In order to do this, the first step is to solve the system of linear equations G(0^-, E_A) M Φ̃(0^-)=Φ̃(0^-) to find the wave functions at the position x=0^-. Then, using the knowledge of G(x, E_A) at arbitrary x, one can reconstruct the entire wave function using Eq. (<ref>). Carrying out this procedure, one finds two solutions Φ_1(x) and Φ_2(x), which are written out in detail in Appendix <ref>. In our model, spin along the z direction is a good quantum number at B=0, and Φ_1(x) and Φ_2(x) are identical except for the fact that they carry opposite spin. As anticipated in the previous paragraph, the spin-orbit interaction is not effective in separating the two Andreev levels with opposite spins in energy. This would be true even in a model where the spin-orbit interaction takes a more general form and breaks the spin rotation symmetry completely. The degeneracy can not be explained by invoking Kramers' theorem either: the Kramers partner of Φ_1(x,ϕ) is Φ_2(x,-ϕ), so that the two wave functions form a true Kramers doublet only at the time-reversal invariant points ϕ=0,π. Rather, the degeneracy of the Andreev levels is a consequence of the short-junction limit. It is removed by spin-orbit coupling if corrections of order (L/ξ) are taken into account <cit.>, or even in the short junction limit in the case of a multi-terminal junction <cit.>. §.§ Magnetic field dependence of the spectrum: qualitative featuresWhen a finite magnetic field is present, in general we find that the Andreev level spectrum cannot be found analytically. Thus, away from simple limits, we resort to a numerical search of the roots of the determinant Eq. (<ref>). In total, once one of the two linearization limits is taken, there are four parameters which determine the spectrum: the magnetic field B, the phase ϕ, the transparency of the junction τ, and either the spin-orbit coupling α (when μ≫Δ_0, mα^2, 12gμ_B B) or the chemical potential μ (when mα^2≫Δ_0, μ, 12gμ_B B). We focus in particular on the field and phase dependence of E_1 and E_2, since these are the two parameters which are varied systematically in experiment.Let us first discuss the simple situation in which spin-orbit coupling is absent, α=0. In this case, spin is a good quantum number and the Zeeman interaction is separable from the rest of the Hamiltonian. One simply obtains a linear Zeeman splitting, E_1 = E_A - 12gμ_B B and E_2 = E_A + 12gμ_B B, with the same g-factor as that of the continuum states (see inset in the right panel of Fig. <ref>). Note that by increasing the magnetic field one reaches a field value B_sw(ϕ)= 2 E_A(ϕ)/gμ_B at which E_1 changes sign: a Fermi level crossing occurs. Because E_A(ϕ) has a minimum at ϕ=π, this is the value of the phase at which the Fermi level crossing occurs first upon increasing the magnetic field. After this point, i.e. for B>B_sw(π), a pair of fermion parity switches is nucleated symmetrically around ϕ=π (see for instance the top right panel of Fig. <ref>). This behavior is consistent with the fact that, in a topologically trivial phase, the number of fermion parity switches in a 2π phase interval must be even. While E_1 decreases with field, the other Andreev level E_2 increases and merges with the continuum of states with opposite spins at a field value B_cross=2(Δ_0-E_A)/gμ_B. This crossing of the Andreev level with the continuum is protected by spin conservation. The magnetic field dependence of Andreev level spectrum is qualitatively different in the presence of spin-orbit coupling. The typical behavior of the Andreev spectra at fixed τ, ϕ and α is shown in Fig. <ref> in both linearization limits. Before entering into the quantitative details of the features of the Andreev levels, let us discuss the important qualitative features.We begin by discussing the case μ≫ mα^2, Δ_0, gμ_BB, illustrated in the left panel of Fig. <ref>. For small magnetic fields, the two Andreev levels E_1 and E_2 split linearly. The lowest-lying level E_1 maintains its approximately linear behavior in B up to the occurrence of a Fermi level crossing. Similarly to the zero spin-orbit coupling case discussed earlier, Fermi level crossings first appear at ϕ=π upon increasing the magnetic field and are then nucleated in pairs around this point. The field B_sw(π) at which the Fermi level crossing first occurs depends on α and τ: this dependence is investigated in detail later. The energy E_2 of the second Andreev level increases with B, but bends down at B≳ B_cross, when E_2 becomes close in energy to the continuum gap Δ(B), which is decreasing in field. This is due to the fact that, in the presence of both Zeeman and spin-orbit couplings, there are no symmetries in the model which protect the crossing of the Andreev level with the continuum. This avoided crossing between the Andreev level and the continuum leads to a non-monotonic dependence of E_2 on B. Such a non-monotonic dependence is the cause of the suppression in B of the transition frequency ω_even=E_1+E_2 between the two junction states with even parity, a fact which we used to explain the observed absorption spectra of an InAs/Al Josephson junction in Ref. <cit.> (see also Sec. <ref>).In the low chemical potential regime, shown in the middle panel of Fig. <ref>, the two Andreev levels also split linearly for small magnetic fields. However, their behavior at large fields is drastically different from that at high chemical potential, due to the different behavior of the gap Δ(B). The two Andreev levels merge in rapid sequence with the continuum of states – whose gap is linearly decreasing – right before the topological transition at B=B_c. In the topological phase at B>B_c, we find that the subgap spectrum consists of a single pair of Andreev levels ± E_1. We may see the energy E_1 as the result of the coupling between two Majorana zero modes located at the two interfaces of the junction. This notion is accurate in particular for τ≪ 1, when the two interfaces are weakly coupled.The phase dependence of E_1 in the topological phase is shown in the bottom right panel of Fig. <ref> for two different values of the junction transparency τ. In both cases, and for any B>B_c, the energy spectrum displays a single Fermi level-crossing at ϕ=π. (This behavior should be contrasted with that of the topologically trivial phase, where, as discussed earlier, Fermi level crossingsappear in pairs, see top right panel of Fig. <ref>). The pinning of the position of the Fermi level crossing at ϕ=π for B>B_c is due to a symmetry of our particular model. Under the combined operation 𝒮=σ_x ℛ, where ℛ is the operator of spatial inversion x↦ -x, the Hamiltonian in Eq. (<ref>) is mapped to itself up to the change ϕ↦ -ϕ <cit.>. This dictates that the Andreev spectrum must be symmetric around ϕ=0, i.e. that E_1(ϕ)=E_1(-ϕ). If, additionally, we recall that the entire spectrum must be 2π periodic in ϕ, the only allowed point where E_1 can vanish is indeed ϕ=π (this consideration holds in the case that only a single Fermi level crossing is present in a 2π period.) A Josephson junction with more transport channels or a denser Andreev spectrum may exhibit a higher number of Fermi level crossings <cit.>, and in a model where there are no constraints coming from spatial inversion the position of the Fermi level crossing may be in general different from π.In the rest of this Section, we investigate in more detail the different qualitative features of the Andreev level spectrum described so far. §.§ Behavior at small field: Zeeman splitting of the Andreev levelsWe have seen that in both linearization limits the Andreev levels split starting from infinitesimally small magnetic fields. The linear-in-B splitting can be captured by standard degenerate perturbation theory applied to the zero-field wave functions presented in Appendix <ref>. This procedure is valid as long as |12gμ_B B|≪Δ_0 - E_A, so that the discrete Andreev levels are distant from the continuum part of the spectrum.Thus, the results presented in this section are most relevant for 1-τ≪1 and ϕ-π≪π, i.e. when the energy E_A is much lower than the gap Δ_0.It is useful to cast the result of the perturbation calculation in terms of an effective g-factor which is the linear coefficient of the expansion of E_1 and E_2 around B=0,E_1= E_A - 12 g_A μ_B B + …E_2= E_A + 12 g_A μ_B B + …We find that the Andreev level g-factor g_A is different from the “bare” value, g, which determines the size of the Zeeman gap at k=0 in the homogeneous wire, and that g_A can depend strongly on the system parameters.At perfect transmission, τ=1, the zero-field Andreev bound state wave functions are eigenstates of the velocity operator s_z [see Eq. (<ref>)]. Therefore, only the part of the Zeeman coupling which mixes co-propagating modes is effective in splitting the Andreev levels (see Appendix <ref> for a discussion). In this case, it is possible to derive an expression for g_A which is valid at any value of the ratio μ/m α^2, provided that max(μ, mα^2)≫Δ_0:g_A/g = Δ_0^2 sin^2(ϕ/2)/Δ_0^2 sin^2(ϕ/2) + mα^2 (2μ+mα^2) ,(τ=1).The equation above can be derived by using a linearization procedure which interpolates between the twolimits μ≫ mα^2 and mα^2≫μ used in Sec. <ref> and <ref> respectively; the derivation is contained in Appendix <ref>. When mα^2=0, Eq. (<ref>) yields g_A=g independently on the value of all other parameters. At any finite value of mα^2 the Andreev bound state g-factor g_A is reduced with respect to the bare value, g. The suppression is the strongest when mα^2≫μ, Δ_0, in which case Eq. (<ref>) yields g_A≪ g. A finite spin-orbit coupling also makes g_A dependent on the phase difference ϕ, with a maximum at ϕ=π.The considerations in the previous paragraph, based on Eq. (<ref>), remain qualitatively valid also for τ<1. In the presence of scattering, the Andreev bound state wave functions are superpositions of states with opposite velocity. In this case, the magnetic field mixes the counter-propagating components (originating from modes close to k=0) as well as the co-propagating ones (originating from modes at finite k) - see the discussion in Appendix <ref>. We may write the Andreev level g-factor as a sum of two terms, g_A=g_⇉+g_⇄, corresponding to these two different contributions. The co-propagating contribution is given byg_⇉/g = Δ_0^2-E_A^2/Δ_0^2-E_A^2 + mα^2 (2μ+mα^2) ,of which Eq. (<ref>) is a special case. Equation (<ref>) is the dominant contribution to the g-factor when μ≫ mα^2, Δ_0, in which case g_⇉≫g_⇄ and g_A≈ g_⇉ for any value of the transmission τ.The counter-propagating contribution g_⇄ becomes relevant in the opposite regime mα^2 ≫μ, Δ_0. Indeed, in this regime the dominant mixing introduced by a small magnetic field is the one between the counter-propagating modes at k=0, which both participate in the formation of the Andreev bound states provided that τ<1. In the limit μ/mα^2→ 0 and Δ_0/mα^2→ 0, the one treated in Sec. <ref>, we findg_⇄/g = τ√(1-τ)/2μΔ_0 sin(ϕ/2)/Δ_0^2-E_A^2+μ^2-1-τ/2Δ_0^2-E_A^2/Δ_0^2-E_A^2+μ^2 ,which is illustrated in Fig. <ref>. Equation (<ref>) is the leading contribution to the total g-factor g_A = g_⇉+g_⇄ at low chemical potential, except for the vicinities of τ= 1 and μ=μ_0, with μ_0=Δ_0√(1-τ)sinϕ/2. In these narrow regions of the parameter space, Eq. (<ref>) is vanishing and thus the g-factor is determined by the co-propagating contribution g_⇉, in spite of its smallness. Furthermore, note that g_⇉ and g_⇄ have competing signs when 0<μ<μ_0, and so in this region higher-order corrections in the parameter μ/mα^2 may be crucial to determine the g-factor (including its overall sign). However, as discussed in Appendix <ref>, the correction to Eq. (<ref>) due to a finite ratio μ/mα^2 cannot be easily computed within a linearized spectrum approximation, since such a calculation necessarily involves the electronic state close to the bottom of the parabolic bands of Fig. <ref>. Finally, we note that Eqs. (<ref>) and (<ref>) agree in predicting a ∼1/μ suppression of g_A when μ≫Δ_0.The value of g_A is not directly accessible in microwave absorption spectroscopy, since the microwave transition frequency ω_even=E_1+E_2 is insensitive to the linear splitting in B. However, it is observable in tunneling spectroscopy, which can access E_1 and E_2 individually. The analysis contained in the above paragraphs suggests that a systematic investigation of g_A may be valuable to obtain information about the electron density and the strength of the spin-orbit coupling in the nanowire. This investigation can be carried out at very small values of the field and may be helpful in predicting or understanding the high-field behavior of the system. §.§ Occurrence and position of Fermi level crossingsEarlier in the text, we have seen that, in the high chemical potential regime μ≫ mα^2, gμ_B B, Δ_0, Fermi level crossings may occur at a field B=B_sw (see the left panel of Fig. <ref>). In Fig. <ref> we study in more detail the dependence of B_sw, computed at ϕ=π, on spin-orbit coupling strength and transmission.There are two notable trends. First, the switching field B_sw decreases upon increasing the transmission τ at fixed spin-orbit strength. This is due to the fact that the larger τ is, the closer to zero is E_A, and thus a smaller field is required to induce the Fermi level crossing. Second, when increasing the spin-orbit strength at fixed transmission, the field B_sw increases. This is due to the suppression of the Andreev level g-factor g_A with increasing spin-orbit strength or chemical potential, see Eq. (<ref>), which leads to a slower decrease of E_1 with B. Our numerical results suggest that there is a value (α k_F)_max above which the Fermi level crossings are absent: that is, the curves in Fig. <ref> have an asymptote at finite α k_F at which B_sw diverges. Qualitatively, a strong spin-orbit coupling may prevent the Fermi level crossing to occur because of the level repulsion between the Andreev level E_1 and the negative image of the rest of the spectrum. Judging from the numerical data shown in Fig. <ref>, (α k_F)_max depends on the transmission τ, and it grows with increasing τ→ 1. We attribute this behavior to the fact that, in the limit τ→ 1, E_A(π)→0: thus, a Fermi level crossing appears already at an infinitesimally small field, and it becomes prohibitive to remove it. Finally, we notice that numerical calculations do not reveal the presence of Fermi level crossings in the opposite regime mα^2≫μ, gμ_B B, Δ_0. We attribute this behavior to the fact that, in this regime, g_A≪ g. Therefore, the decrease in energy of the Andreev bound states is much slower than that of the continuum states (see for instance the middle panel of Fig. <ref>), preventing the occurrence of a Fermi level crossing at a field B<B_c.In a tunneling spectroscopy experiment, the closing of the excitation gap of the junction at a Fermi level crossing in the regime μ≫Δ_0 may be naively mistaken for a bulk topological transition. Indeed, a typical magnetic field scale for a fermion parity switch is B_sw∼500 mT <cit.>, not dissimilar from that of the critical field B_c <cit.>. The strong dependence of B_sw on τ (as well as ϕ), however, should allow to discriminate easily between the two cases.§ CURRENT OPERATOR AND THE EQUILIBRIUM CURRENTIn this Section, we evaluate the temperature and magnetic field dependence of the equilibrium current. It is known that, in a short Josephson junction not subject to magnetic field, the current is carried almost entirely by the Andreev bound states <cit.>.This conclusion remains true also in the presence of magnetic field (with or without spin-orbit coupling), as we will now argue following the discussion from Ref. <cit.>.On one hand, the energies of the Andreev bound states vary by an amount ∼Δ upon varying the phase ϕ, and thus they provide a finite contribution to the current in the limit L/ξ→ 0. On the other hand, the contribution of the continuous spectrum to the current density comes from states within the energy range Δ < E < E_Th. Here, E_Th is the Thouless energy, i.e. the energy scale associated with the flight time of quasiparticles across the junction; in a short quasi-ballistic junction, the Thouless energy is large, E_Th/Δ∼ξ/L≫ 1. The spectral density of the current delivered by states with energy ∼ E scales as Δ^2/(E_ThE) for energies in the interval E_Th≳ E≫Δ. It yields a total contribution ∝ (Ł/ξ) ln(ξ/L) to the current, which vanishes in the limit L/ξ→ 0 <cit.>.This argument remains valid even in the presence of a magnetic field or spin-orbit coupling. Therefore, in the following we neglect the contribution of the extended states to the current. We start by finding the current operator j(x) for the junction, and then evaluate the contribution of the many-body eigenstates |V ⟩,|1 ⟩,|2 ⟩, and |P ⟩, see Fig. <ref>.The current operator can be derived from a continuity equation for the electric charge density ρ, which for the original model of Eq. (<ref>) is given by the operatorρ(x) =e/2 ψ^†(x) τ_z ψ(x) .The continuity equation for ρ can be computed using the equation of motion of the field ψ(x) under the Hamiltonian of Eq. (<ref>). It can be cast in the form_t ρ(x) + _x j(x) = s(x) ,with j(x) the quasiparticle current operator, which includes a contribution from the spin-orbit coupling,j(x) = e/2miψ^†(x) ∂_x ψ(x)+ e/2 α ψ^†(x) σ_z ψ(x) ,and s(x) a charge source (or drain) term due to the presence of the superconducting condensate <cit.>,s(x) = e Δ_0 ψ^†(x) τ_ye^-iϕ sgn(x)τ_z/2 ψ(x) .At the position of the junction, x=0, the source term vanishes since there is no proximity-induced pairing Δ_0. Thus, at the junction the equilibrium current can be computed by studying only the quasiparticle contribution coming from j(x). In the superconducting leads, the quasiparticle current is converted into current carried by the condensate over a length ∼ξ. Correspondingly, the contribution of the j(x) term to the equilibrium current decays away from x=0. The decay is compensated by the source term <cit.> to ensure the current conservation along the wire.Using Eq. (<ref>) and Eq. (<ref>) we find the current operator projected to low energies [recall that Ψ = (ψ_R , ψ_L)^T encodes the left- and right-moving envelope fields],j(x) = ev/2Ψ^†(x) s_z Ψ(x) ,where v = v_F in the limit of high chemical potential, and v = α for mα^2≫μ, gμ_BB, Δ_0. Using the pseudounitarity of the transfer matrix, Eq. (<ref>), together with the boundary condition for Ψ at the origin, Eq. (<ref>), one can check that the linearized current operator in Eq. (<ref>) is continuous across the junction, i.e. j(0^-) = j(0^+). We will thus evaluate the current at x=0^- from now on and omit the position argument.The current operator can be expanded in the eigenbasis of the linearized Hamiltonian by using Eq. (<ref>): j =∑_n( Γ_n^†Γ_n-12) j_n,n+ 12∑_n ≠ m (Γ_n^†Γ_mj_n,m+ Γ_n^†Γ_m^†j_n,𝒫m + H.c.) .Here we have introduced the matrix elements of the current operator between BdG eigenstates,j_n,m = e v Φ_n^† s_z Φ_m ,j_n,𝒫m = e v Φ_n^† s_z 𝒫Φ_m .The diagonal matrix elements j_n,n in Eq. (<ref>) give the dissipationless supercurrent. Including, as already discussed, only the contribution from Andreev bound states to the sum in Eq. (<ref>), we findj = (n_1 -12) j_1,1+(n_2-12) j_2,2 ,with · being the quantum expectation value and n_n=Γ^†_nΓ_n the occupation factors for the different quasiparticle states. At thermal equilibrium with temperature T, Γ^†_nΓ_n_eq=f(E_n), where f(E)=[1+exp(E/k_BT)]^-1 is the Fermi-Dirac distribution. At B=0, the diagonal matrix elements have a simple analytic expression which can be computed using the wave function in Eq. (<ref>) in Appendix <ref>,j_1,1=j_2,2= e/2Δ_0^2τsinϕ/E_A(ϕ) ,(B=0) . The result is independent of μ and α, as long as the Andreev approximation is valid, see Sec. <ref>. Plugging the expression above into Eq. (<ref>) immediately leads to the known expression for the Josephson current in a single channel weak link j_eq = e/2ħΔ^2_0/E_A(ϕ) τsinϕ tanh[E_A(ϕ)/2k_BT] .In this zero field case, the fact that j_1,1 = j_2,2 has the consequence that the Josephson current vanishes if the state of junction is one of the two odd parity states. Namely, from Eq. (<ref>) we see that j=0 if j_1,1 = j_2,2 and n_1+n_2=1. This is the so-called “poisoned” state of the junction <cit.>, in which one excess quasiparticle can completely block the passage of current. Note that, if the junction has more than one pair of Andreev bound states, a single excess quasiparticle will not block the current completely, as there will be more contributions to the total equilibrium current.The typical behavior of the equilibrium current-phase relation at finite magnetic field is illustrated in Fig. <ref>. At small fields, the behavior is not qualitatively different from that of Eq. (<ref>). At low temperatures the current-phase relation exhibit the skewed-sine shape typical of weak links, with the skewness being suppressed with increasing temperatures (see upper panel in Fig. <ref>). The behavior is more interesting at higher fields, such that fermion parity switches occur as a function of the phase ϕ, as in the Andreev spectrum in the upper right panel of Fig. <ref>. In this case, at T=0 the current exhibits a discontinuity in correspondence with each fermion parity switch (see bottom panel in Fig. <ref>). At finite temperatures there is no discontinuity, but a remnant of the fermion parity switches remains in the behavior of the current phase relation close to ϕ=π. Finally, we mention that, as expected, the current model does not exhibit the anomalous Josephson effect (i.e., a finite supercurrent at ϕ=0). Indeed, for single-channel nanowire Josephson junction, it is known that the latter requires a component of the magnetic field to be aligned with the spin-orbit field <cit.>. § MICROWAVE ABSORPTIONIn this section we study the microwave absorption spectrum of a short Josephson junction <cit.> (for the opposite case of a long junction, see also Refs. <cit.>). The microwave field is modeled as a monochromatic ac voltage drop V(t) = V_0 cos(ω t) across the junction and is minimally coupled to the electronic field ψ. This leads to the addition of the following time-dependent term to the Hamiltonian of Eq. (<ref>):δ H(t) = j (V_0/ω) sin(ω t) .where j is the current operator evaluated at the junction. We assume that the perturbation δ H is small, e V_0/ω≪ 1. The form of the perturbation δ H(t) remains valid also after the spectrum linearization, since as we have discussed in the previous Section the current matrix elements at the position of the junction remain well defined and continuous, j(0^-)=j(0^+)≡ j. Using standard linear response theory, the expectation value of the current at time t is determined by the response function χ(t) = -i θ(t) [j(t), j(0)]_eq,j(t) = j_eq+V_0/ω∫_-∞^∞ χ(t-t') sin(ωt')t' .In the frequency domain, the response function χ(ω) determines the admittance of the junction, Y(ω) = iχ(ω)/ω. In turn, the real part of the admittance gives the absorption power W of the microwave radiation, W = 12 V_0^2Y(ω) with ω>0. Using Eq. (<ref>) to compute the response function, we find ReY(ω) =π/ω∑_E_n ≥ E_m|j_n,𝒫m|^2δ(ω-(E_m+E_n)) (1-f(E_m)-f(E_n))+π/ω∑_E_n ≥ E_m|j_n,m|^2δ(ω-(E_n-E_m)) (f(E_m)-f(E_n)) +… . The first line in Eq. (<ref>) corresponds to transitions where two quasiparticles are created by breaking a Cooper pair and occupy two energy levels with energies E_n and E_m. The second line corresponds to transitions where a single quasiparticle with energy E_n is excited into a higher state with energy E_m. We will refer to these two types of transition respectively as the “even” or “odd” ones, since they are distinguished by the parity of the number of quasiparticles involved. Note that only transitions in which initial and final states have the same fermion parity are allowed. Transitions between the discrete states, which are accounted for in Eq. (<ref>), produce sharp maxima in the frequency dependence of the absorption coefficient.The omitted terms in the admittance, indicated by dots in Eq. (<ref>), involve unbound quasiparticle states and result in an absorption continuum. We shall consider low frequencies ω < 2Δ, focusing on the transitions between the Andreev bound states. Indeed, at these low frequencies the excitation of Andreev states are the only possible resonant processes (unless the system is close to the critical point separating topological and trivial phases, a case treated in Ref. <cit.>). Transitions between possible above-gap non-equilibrium quasiparticles are very weak and do not result in a sharp absorption line, so we will ignore them. In the case under consideration of a single-channel short junction, with only two Andreev states with energies E_1 and E_2, there is only one relevant term in each sum in Eq. (<ref>). These terms correspond to the two allowed transitions depicted in the bottom right panel of Fig. <ref>: the pair excitation |V⟩→|P⟩ with frequency ω_even=E_1 + E_2 and the single particle excitation |1⟩→|2⟩ with frequency ω_odd=E_2-E_1.As mentioned in the introduction, we call these “even” and“odd” transitions, respectively.The matrix elements j_2,𝒫1(0^-) and j_2,1(0^-) determine the strengths of these transitions; their dependence on the system parameters is discussed next in detail, first for the even transition and then for the odd one. §.§ Visibility of the even transition |V⟩→|P⟩We start by discussing the case B=0. Again by using the wave functions in Eq. (<ref>) of Appendix <ref>, we find the analytical expression for the relevant current matrix elementj_2,𝒫1^2= e^2 (1-τ) τ^2 sin^4(ϕ/2) (Δ_0^4/E_A^2)Equation (<ref>) was previously derived in Ref. <cit.>using a tunneling Hamiltonian formalism, which is in agreement with our current method based on the transfer matrix. Just like the B=0 Andreev energy E_A, it is independent on the chemical potential μ and the spin-orbit coupling α and it generalizes to the case of multiple transport channels with different transparencies. Note that j_2,𝒫1^2 vanishes for τ=1: the absence of scattering at the junction prevents the excitation of the Andreev bound states since in this case the current is a diagonal operator in the eigenbasis of Eq. (<ref>). In the presence of scattering, the Andreev bound states are superpositions of different current eigenstates and microwave-induced transitions become possible <cit.>. Equation (<ref>) has a maximum at ϕ=π, corresponding to the point of greater visibility of the absorption spectral line. The visibility vanishes for small phases. This behavior is in agreement with experiment both in case of nanowire Josephson junctions <cit.> as well as other types of weak links <cit.>.The dependence of j_2,𝒫1 on magnetic field can be determined by finding the wave functions of the Andreev bound states numerically via Eq. (<ref>). We find that a finite magnetic field suppresses the magnitude of the current matrix element, while maintaining its phase dependence qualitatively similar to that described by Eq. (<ref>), see upper panel of Fig. (<ref>). The decrease of j_2,𝒫1 with increasing magnetic field is slow in both regimes mα^2 ≫μ_0, Δ_0 (see the inset of the bottom panel in Fig. <ref>) and μ≫Δ_0, mα^2 (see Fig. <ref>). We attribute this decrease to the suppression of the proximity-induced gap Δ(B) with B (see Fig. <ref>), which makes the Andreev bound states less tightly confined to the junction and thereby decreases the effective coupling to microwaves, j_2,𝒫1∼Δ(B).Finally, at finite fields, the current matrix elements also acquires a weak dependence on the chemical potential, as illustrated in the bottom panel of Fig. (<ref>). §.§ Visibility of the odd transition |1⟩→|2⟩Without magnetic field, B=0, the current matrix element associated with the odd transition (which has anyway zero frequency) vanishes: j_2,1=0. This is due to the fact that the zero-field Andreev bound states have opposite spin [see Eqs. (<ref>) and (<ref>)], while the perturbation Hamiltonian (<ref>) preserves spin. As the magnetic field is increased from zero, the odd transition may become visible depending on the spin-orbit coupling strength. If spin-orbit is absent (or negligible), the two Andreev bound states would develop an opposite spin polarization in the presence of a Zeeman field: therefore, again due to the spin selection rule, the odd transition would remain forbidden. In the presence of both spin-orbit coupling and magnetic field, however, this spin selection rule is no longer applicable: the two Andreev bound states would have a non-zero spin overlap and one may in general expect a non-vanishing matrix element. Indeed, we determine numerically that j_2,1≠ 0 at finite B in the presence of spin-orbit coupling. Importantly, even in this case we find that j_2,1/j_2,𝒫1≲ 0.1, see the bottom right panel of Fig. <ref>. Hence, despite not being forbidden, the dim odd transition may be still much more difficult to observe with respect to the bright even transition. We now discuss the dependence of j_2,1 on the system parameters.As in the case of the even transition, the current matrix element j_2,1 has a maximum when ϕ=π and vanishes for small phase differences; in what follows, we focus on the peak value. The dependence of j_2,1 on chemical potential is shown in the top panel of Fig. <ref> for different values of B. The current matrix elements is non-zero at μ=0, it grows slowly and it reaches a maximum at a small value of μ/Δ_0 before decreasing again. After this point, we find that j_2,1∝ (Δ_0/μ)^2 when μ/Δ_0≫ 1, as shown in the bottom left panel of Fig. <ref>. These considerations are valid when mα^2≫Δ_0, μ. The suppression of j_2,1 for μ≫Δ_0 in this regime matches the numerical results that we obtain for μ≫ mα^2, where we find that the matrix element is zero (within numerical precision) at any value of the spin-orbit coupling. Finally, the numerical data indicate that the current matrix elements grows quadratically in B at small fields: j_2,1∝ B^2, see the inset in the top panel in Fig. <ref>.The smallness of the current matrix element j_2,1 has important consequences for the Andreev spectroscopy of the junction at low temperatures, in case the junction undergoes a fermion parity switch. For instance, suppose that the magnetic field is sweeped from a value B<B_sw to a value B>B_sw, as in the left panel of Fig. <ref>. This change of magnetic field will be accompanied by a dramatic decrease in the visibility of the absorption line corresponding to the even transition at frequency ω_even. Indeed, for B>B_sw the ground state of the junction is the odd parity state |1⟩, and at low temperatures k_B T≪ E_1 the occupation probability of the even parity state |V⟩ is negligible. The low occupation probability of the state |V⟩ and the smallness of the matrix element j_2,1 combine to yield a dramatic dimming of the absorption line taking place at B=B_sw (see Fig. <ref>).§ CONCLUSIONS In this work, we have investigated several consequences of the competition between Zeeman and spin-orbit couplings on the Andreev bound states in semiconducting nanowire Josephson junctions. Overall, as one may have expected, spin-orbit coupling tends to reduce the effect of the Zeeman coupling on the Andreev bound states. We have seen several examples of this general trend. First, as discussed in Sec. <ref>, spin-orbit coupling tends to suppress the g-factor g_A of the Andreev bound states, potentially resulting in very small energy splittings of the Andreev doublet for small magnetic fields. The measurement of g_A, in tunneling or supercurrent spectroscopy experiments, may allow one to estimate the strength of the spin-orbit coupling. Second, spin-orbit coupling also suppresses the occurrence of fermion parity switches in the topologically trivial phase of the nanowire (see Sec. <ref> and Fig. <ref>). As discussed at the end of Sec. <ref>, fermion parity switches should be easily detectable since they are accompanied by a drastic dimming of the absorption spectrum. The knowledge of the switching field B_sw at which fermion parity switches take place can also be used to infer the strength of the spin-orbit coupling. Finally, spin-orbit coupling prevents the occurrence of level crossings between the Andreev bound states and the continuum. Combined with the suppression of the proximity-induced energy gap in a magnetic field, this leads to a non-monotonic dependence of the Andreev bound state energies on B, see the left panel of Fig. <ref>. The bending of the Andreev level E_2 due to the repulsion from the continuum causes a slow decrease of the even transition frequency in magnetic field, ω_even(0)-ω_even(B)∝ B^2 for small B.Our theoretical results are in good agreement with several aspects of the existing experimental data which motivated the development of the work presented here <cit.>. In particular, we elucidated that the quadratic suppression of ω_even with increasing magnetic field can be understood in terms of the interplay of Zeeman and spin-orbit coupling. The occurrence of fermion parity switches is also compatible with the observation that the even transition visibility vanished at a field larger than 300 mT. This threshold can be well understood within our theory assuming reasonable values of g and α <cit.>. At the experimental level, it would be very valuable to study directly the single-particle energy spectrum via either tunneling or supercurrent spectroscopy. This would allow a measurement of the Andreev bound state g-factor g_A as well as a precise determination of the switching field B_sw, both of which can be directly compared to our theory.Our results are all based on the one-dimensional nanowire model of Eq. (<ref>) treated within the Andreev approximation, i.e. by linearizing the normal state dispersion. This approximation amounts to neglecting the normal reflection amplitude in favor of the Andreev reflection amplitude when considering the two interfaces of the S-N-S junction. It requires that either the chemical potential μ or the spin-orbit energy mα^2 are much larger than the induced superconducting gap Δ_0. As an extension of this work, it may be valuable to relax the Andreev approximation. In particular, it would be interesting to study the Andreev spectrum of the model in the regime μ≪ mα^2, Δ_0 and mα^2∼Δ_0, which may be relevant for the Majorana applications of the semiconducting nanowires. This may give more accurate predictions for the Andreev g-factor g_A and the current matrix elements in the regime of low chemical potential.It will also be important to extend this work beyond the model of Eq. (<ref>), in order to capture more accurately the complexity of real devices. Nanowire junctions may naturally host more than one transport channel, and physical effects not included in this work, such as the orbital effect of the magnetic field, may have an important influence on the Andreev bound state properties. In particular, the orbital effect of the magnetic field provides an additional contribution to the reduction of ω_even. Although this contribution could be heuristically ruled out to be the dominant one in the current nanowire experiments, it would be important to have quantitative theoretical estimates.Finally, the magnetic field dependence of the absorption spectrum in the presence of multiple transport channels stands out as a particularly interesting avenue for future research, both theoretically and experimentally. In such a situation, a new type of low-frequency transitions may become visible, in which a Cooper pair is excited to a pair of Andreev levels belonging to different transport channels. In the topological phase, we expect that these inter-channel transitions can carry a signature of the Majorana bound states in the form of a kink in the phase dependence of the absorption spectrum, similar to the effect predicated in long Josephson junctions <cit.>. Notably, this type of measurement is not limited by stringent requirements on fermion parity relaxation times, as opposed to other signatures of topological Josephson junctions.We acknowledge stimulating discussions with A. Geresdi, D.J. van Woerkom, H. Pothier, R. Lutchyn, T. Hyart and S. Park. BvH was supported by ONR Grant Q00704, JV and LG acknowledge the support by NSF DMR Grant No. 1603243. § BOUND STATE WAVE FUNCTIONS AT ZERO FIELDFollowing the procedure outlined in the main text, we find the two following bound state wave functions Φ_1(x) and Φ_2(x) at B=0. They are a tensor product of a position-dependent part and a position-independent spinor in spin grading:Φ_1(x)= ^iϕ(x) τ_z/4 Φ_A(x) ⊗ χ_Φ_2(x)= ^iϕ(x) τ_z/4 Φ_A(x) ⊗ χ_ .Here χ_=(1,0)^T and χ_=(0,1)^T are the eigenspinors of σ_z, and Φ_A(x) is a space-dependent vector in Nambu and left/right gradings: Φ_A(x) = 1/2 ξ_A^1/21/[E_A (E_A-Δ_0√(τ)cosϕ/2)]^1/2 ^-iα k_F x/v_F ^-x/ξ_A [ [-(x) i ^iγ θ(-x)^iβ θ(x) (E_A-Δ_0√(τ)cosϕ/2);^iβ θ(-x)^iγ θ(x) Δ_0 √(1-τ); -(x) i ^iγ θ(-x)^iβ θ(-x) (E_A-Δ_0√(τ)cosϕ/2); ^iβ θ(x)^iγ θ(x) Δ_0 √(1-τ) ]]with:E_A =Δ_0 [1-τsin^2(ϕ/2)]^1/2 , β = arccos(E_A/Δ) , γ = arccos(√(τ)) , ξ^-1_A = 1/v_F √(Δ_0^2-E_A^2) .The expression above is valid for μ≫Δ_0, mα^2 and in the phase interval ϕ ∈ [0,2π]. The wave functions for negative phase can be determined by applying the time-reversal symmetry operator is_x σ_y 𝒦. In the opposite regime mα^2≫μ, Δ_0, the wave functions are identical except that the oscillating term ^-iα k_F/v_F is replaced by ^-iμ x/α s_z and that α replaces v_F in the expression for the coherence length ξ_A of the bound state. The wave functions above are properly normalized to unity: to see this, it is convenient to use the relation Δ_0^2 (1-τ)=(E_A-Δ_0√(τ)cosϕ/2) (E_A+Δ_0√(τ)cosϕ/2).§ DERIVATION OF THE DIFFERENT CONTRIBUTIONS TO THE G-FACTORIn the main text, Secs. <ref> and <ref>, we linearized the spectrum in two limits of either large μ or large mα^2. In this Appendix, we show that both limits can be obtained from a single linearization which is valid on a strip of width μ+mα^2 around the Fermi level. This linearization is achieved by a projectionψ(x) =e^-imx (√(α^2+v_F^2)+ασ_z) ψ_L(x)+e^imx (√(α^2+v_F^2)-ασ_z) ψ_R(x)where the fields ψ_L,R are slowly varying. For example, the kinetic term in the Hamiltonian density becomes ψ(x)^†(-∂_x^2/2m-iα∂_x σ_z-μ)τ_zψ(x)=-i√(α^2+v_F^2)Ψ^†(x)s_zτ_z∂_xΨ(x) + oscillating terms .We used here μ=1/2mv_F^2.When we project the Zeeman term -1/2gμ_BBψ^†σ_xψ to low energies using Eq. (<ref>), we obtain two terms, -1/2gμ_BBψ^†σ_xψ=Ψ^†(O_⇉+O_⇄)Ψ ,where O_⇉=-1/2gμ_BBe^2imα xσ_zσ_x , O_⇄=-1/2gμ_BBe^-2imx(s_z√(α^2+v_F^2)-σ_zα)s_xσ_x .The first term couples co-propagating states only and it is important when spin-orbit strength is not too large, mα^2≲Δ_0. It leads to Eq. (<ref>) of the main text, which can be derived by evaluating the matrix elements of O_⇉ by using the wave functions from Eq. (<ref>).The second term, O_⇄, mixes counter-propagating states and therefore it only contributes to the g-factor in the presence of scattering at the junction. Furthermore, it is important only for states near k=0 and when μ≪ mα^2, in which case the oscillating exponent vanishes. States belonging to the outer branches at finite momentum have opposite s_z and σ_z eigenvalues, and in this case the O_⇄ term oscillates fast and is negligible. In the limit μ/mα^2→ 0, we thus obtainO_⇄=-1/4gμ_BB(s_xσ_x-s_yσ_y) .After calculating the matrix elements of O_⇄ with the wave functions from Appendix <ref>, we find Eq. (<ref>) of the main text. Note that correction of order μ/mα^2 to the matrix elements of O_⇄ cannot be reliably computed within the linearized Hamiltonian. | http://arxiv.org/abs/1705.09671v2 | {
"authors": [
"B. van Heck",
"J. I. Väyrynen",
"L. I. Glazman"
],
"categories": [
"cond-mat.mes-hall",
"cond-mat.supr-con"
],
"primary_category": "cond-mat.mes-hall",
"published": "20170526180419",
"title": "Zeeman and spin-orbit effects in the Andreev spectra of nanowire junctions"
} |
Balanced vertices inrootedtrees]Balanced vertices in labeled rooted trees Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118105, Gainesville, FL, 32611-8105 (USA) [email protected] In a rooted tree, we call a vertex balanced if it is at equal distance from all its descendant leaves. We count balanced vertices in three different tree varieties. For decreasing binary trees, we can prove that the probability that a vertex chosen uniformly at random from the set of all trees of a given size is balanced is monotone decreasing.[ Miklós Bóna December 30, 2023 =====================§ INTRODUCTIONVarious parameters of many models of random rooted trees are fairly well understood if they relate to a near-root part of the tree or to global tree structure.The first group includes the numbers of vertices at given distances from the root, the immediate progeny sizes for vertices near the top, and so on. See <cit.> for a comprehensive treatment of these results.Not surprisingly, the technical details of fringe analysis become quite complex as soon as the focus shifts to layers of vertices further away from the leaves. So while there are explicit results on the (limiting) fraction of vertices at a fixed, small distance from the leaves, an asymptotic behavior of this fraction, as a function of the distance, remained an open problem. Recently, Boris Pittel and the present author have studied this family of questions in <cit.>.Most work in fringe analysis focused on decreasing binary trees, which are also called binary search trees. We will explain the reason for that. However, in this paper, we will discuss questions that we can successfully investigate for other tree varieties as well.We call a vertex v of a rooted tree balanced if all descending paths from v to a leaf have the same length. In other words, if ℓ is any leaf that is a descendant of v, then the unique path from v to ℓ consists of k edges,where k does not depend on the choice of ℓ. The number k is called the rank of v.§ DECREASING BINARY TREESA decreasing binary tree on vertex set [n]={1,2,⋯ ,n} is a binary plane tree in which every vertex has a smaller label than its parent. This means that the root must have label n, every vertex has at most twochildren, and that every child v is either an left child or a right child of its parent, even if v is the only child of its parent.Decreasing binary trees on vertex set [n] are in bijection with permutations of [n]. In order to see this,let π=π_1π_2⋯π_n be a permutation. Thedecreasing binary treeof π, which we denote by T(π), is defined as follows. The root of T(π) is a vertexlabeled n, the largest entry of π.If a is the largest entry of π on the left of n, and b is the largest entry of π on the right of n, then the root will have two children, the left one will be labeled a, and the right one willbe labeled b. If n is the first (resp. last) entry of π, then the root will have only one child, and that is a left(resp. right) child, and it will necessarily be labeled n-1 as n-1 must be the largest of all remaining elements. Define the rest of T(π) recursively,by taking T(π') and T(π”), where π' and π” are the substrings of π on the two sides of n, and affixing them to a and b. See Figure <ref> for an illustration. Note that in the tree T(π) shown in Figure <ref>, all vertices are balanced, except 8, 9, and 6. Also note thatin any decreasing binary tree, vertex v can be balanced only if all its descendants are balanced.§.§ Balanced vertices of a fixed rank Recall that a vertex v is balanced if all descending paths from vto a leaf have the same length. That length is the rank of v. Let a_n,k be the total number of all balanced vertices of rank k in all decreasing binary trees of size n.Let A_k(x)=∑_n≥ 0 a_n,kx^n/n!, andlet r_n,k be the total number of such trees on n vertices whose root is balanced of rank k.The differential equationA_k'(x)=2A_k(x)/1-x + R_k'(x) holds, with the initial condition A_k(0)=0.Note that a_n,k is the number of ordered pairs (v,T), where v is a balanced vertex of rank k in a decreasing binary tree T on vertex set [n], in other words, a_n,k is the number of decreasing binary trees on [n] with a balanced vertex of rank k marked. If the marked vertex v is not the root of T, then removing the root of T, we get, on the one hand, a structure on [n-1] that is counted by A_k'(x), and, on the other hand, a decreasing binary tree and a decreasing binary tree with a balanced vertex of rank k marked. By the Product formula of exponential generating functions, these pairs of trees are counted by the generating function 2 ·1/1-x A_k(x).The factor 2 is needed since the order of the obtained two trees matters, and 1/(1-x) is just the generating functionof the sequence of factorials, hence of the sequence enumerating decreasing binary trees. Finally, if the marked vertex v is the root of tree, then removing it we just get a structure enumerated by R_k'(x). The special case of k=0 counts leaves, which are of rank 0, and are trivially balanced. Indeed, we have R_0(x)=x, so R_0'(x)=1. Therefore (<ref>) reduces to A_0'(x)=2A_0(x)/1-x + 1,with A_0(0)=0. The solution of this differential equation isindeedA_0(x)=1/3·1/(1-x)^2 + x/3 -1/3,which is the generating function for the number of all leaves of all decreasing binary trees on n vertices. If k=1, then R_1(x)=x^2+ x^3/3, since both trees on two vertices, and both trees on three vertices in which the root has two children, have a root that is balanced of rank 1. So (<ref>) reduces toA_1'(x)=2A_1(x)/1-x + 2x+ x^2,with A_1(0)=0. The solution of this initial value problem isA_1(x)=1/5x^5 - x^3 +x^2/(1-x)^2.Further generating functions A_k(x) can theoretically be computed, but one runs out of computing power very fast.A crucial difference from earlier work such as <cit.> is that for all k, the generating function R_k is a polynomial function, since if the root of a tree is balanced and is of rank k, then that tree cannot have more than 2^k-1 vertices. We are now going to show that this implies that for all k, the generating function A_k(x) is always a rational function of denominator (1-x)^2.Let k be a fixed nonnegative integer.Let c_n,k=a_n,k/n! · n be the probability that avertex chosen uniformly from the set of all n· n! vertices of all decreasing binary trees on [n] is balanced, and is of rank k. Then for any fixed k, the limit c_k=lim_n→∞ c_n,kexist. Furthermore,Let A_k(x)=P_k(x)/(1-x)^2, that is, let P_k(x) be the numerator of A_k(x). Thenc_k=P_k(1). Note that the fact that the limits c_k exist can also be proved using the techniques of additive functionals as explained in <cit.>. (The number of balanced vertices in a rooted tree is an additive functional.)However, we provide a self-contained proof here. Solving the linear differential equation (<ref>) for A_k(x), we get thatA_k(x)=∫ (1-x)^2B_k'(x)dx /(1-x)^2 ,where the constant of integration is to be chosen so that the initial condition A_k(0)=0 is satisfied.Recall that B_k(x), and therefore, B_k'(x), is a polynomial function. Therefore, the numerator P_k(x) of the right-hand side of (<ref>) is a polynomial function. Now let P_k(x)=P(x)(1-x)^2 + Q(1-x)+R, where P(x) is a polynomial, and Q and R are complex numbers.Note that P_k(1)=R. ThenA_k(x)=P_k(x)/(1-x)^2 = P(x) +Q/1-x +R/(1-x)^2.If n is larger than the degree of the polynomial P, then equating coefficients of x^n on both sides, we get thata_n,k/n!= Q + (n+1)R.Solim_n→∞ c_n,k = lim_n→∞a_n,k/n! · n = lim_n→∞a_n,k/(n+1)!= R=P_k(1).Note that it is easy to see that c_k>0 for all k. Indeed, c_k is certainly at least as large as the probability that arandomly selected vertex is of rank k and is the root of a perfect binary tree (one in which every non-leaf vertex has exactlytwo children), and even this last probability stays above a positive constant as n goes to infinity. See <cit.>for details. The simple form of A_k(x) is the reason that decreasing binary trees are easier to analyze from this aspect than other tree varieties. Using the above corollary, we get that c_0=1/3, c_1=1/5,c_2=52/567 and c_3=7175243/222660900. This shows that for large n,about 65.7 percent of all vertices of decreasing binary trees are balanced and of rank at most three.More computation shows that for n sufficiently large, about66.62 percent of all vertices are balanced and of rank at most four, and about 66.84 percent are balanced and of rank at most five.§.§ A result about monotonicityAs decreasing binary trees are in bijection with permutations, we have additional tools analyzing them. This enablesto prove the following strong result. We could not prove similar results for other labeled rooted trees.Let P_n be the probability that a vertex chosen uniformly at random from the set of all vertices of all decreasing binary trees on [n]is balanced.Then the sequence P_1,P_2,⋯ is weakly decreasing. Let p_n,k bethe probability that the root of a randomly selected tree on n vertices is balanced, and is of rank k. Set p_0,i=1 for all i.We start by an inequality for the numbers p_n,k for fixed k. For all n≥ 1 and all fixed k≤ n, the inequality p_n+1,k≤ p_n,k holds. We prove the statement by induction on n. The statement is true for all k if n≤ 3, since in that case, p_n,k=1 for all n and all k. Now let us assume that the statement is true for n and prove it for n+1. Let π be a permutation of length n+1. The probability that the largest entry of π is in position i+1 forany i∈ [0,n] is 1/(n+1).The root of T(π) is balanced of rank k if and only if all its children are balanced of rank k-1, so p_n+1,k =∑_i=0^n p_i,k-1p_n-i,k-1/n+1 .Replacing n+1 by n, we get the analogous formula p_n,k =∑_i=0^n-1 p_i,k-1p_n-1-i,k-1/n . The initial difficulty is that while the summands in (<ref>) are smaller than their counterparts in(<ref>), there is one more of them.The crucial observation is that if we remove the smallest summand from the numerator of (<ref>), then the remaining n summands of that numerator can be matched with the n summands of the numerator of (<ref>), so that in each pair, the summand coming from(<ref>) is at least as large as the summand coming from(<ref>). That will prove that thenumerator of (<ref>) is at most (n+1)/n times as large as the numerator of (<ref>).Let j be the index for which p_j,k-1p_n-j,k-1 is minimal, so that last product is the minimal summand in the numerator of (<ref>). First we look at indices smaller than j.Note that by the induction hypothesis, p_n-i,k-1≤ p_n-1-i,k-1, so p_i,k-1p_n-i,k-1≤ p_i,k-1p_n-1-i,k-1. Let us sum these inequalities for 0≤ i≤ j-1, to get∑_i=0^j-1 p_i,k-1p_n-i-1,k-1≤∑_i=0^j-1 p_i,k-1p_n-1-i,k-1 . Now we consider indices larger than j. Again, by the induction hypothesis,p_i,k-1≤ p_i-1,k-1, sop_i,k-1p_n-i,k-1≤ p_i-1,k-1p_n-i,k-1. Let us sum these inequalities for i∈ [j+1,n], to get∑_i=j+1^n-1 p_i,k-1p_n-i,k-1≤∑_i=j+1^n-1 p_i-1,k-1p_n-i,k-1 .Adding (<ref>) and (<ref>), we get an inequality whose right-hand side agrees with the sum in (<ref>),and whose left-hand side is the sum in (<ref>), except the summand of the latter indexed by j. However, that summand was the smallest of the (n+1) summands in the sum in (<ref>), which implies thatnp_n+1,k = n/n+1∑_i=0^n p_i,k-1p_n-i,k-1≤∑_i=0^n-1 p_i,k-1p_n-1-i,k-1 =np_n,k.This is equivalent to our claim. Let p_n be the probability that the root ofa decreasing binary tree on [n] is balanced. Then p_n≥ p_n+1.It follows from our definitions that p_n=∑_k=1^n-1 p_n,k and p_n+1=∑_k=1^n p_n+1,k.As Lemma <ref> shows thatp_n+1,k≤ p_n,k for k≤ n, the only issue that we must consider is that the sum that provides p_n+1 has one more summand than the sum that provides p_n. However, this is not aproblem, since for all n≥ 2,we have p_n,n-1 = 2^n-1/n!, while p_n+1,n-1=2^n-1/(n+1)! and p_n+1,n=2^n/(n+1)!, sop_n,n-1 =2^n-1/n!≥3· 2^n-1/n+1= p_n+1,n-1 + p_n+1,n .This inequality, and applying Lemma <ref> for all k≤ n-2, proves our claim.(of Theorem <ref>.) Induction on n, the initial case of n=1 being obvious. In order to prove that P_n≥ P_n+1, note that a random vertex of a tree of size n has 1/n probability to be the root. Furthermore, ifi∈ [n-1], then there is an i/n^2 probability that a random vertex of a random tree is part of the left subtree of that root, and that left subtree has i vertices. Indeed, the left subtree of the root has i vertices if and only if n is in position i+1 of the corresponding permutation, and each of the i vertices of that left subtree are equally likely to be chosen. The same argument applies for right subtrees. This proves thatP_n=p_n/n+2∑_i=1^n-1 iP_i/n^2=p_n+∑_i=1^n-12iP_i/n/n. Therefore, the inequality P_n≥ P_n+1 is equivalent to the inequalityP_n=p_n+∑_i=1^n-12iP_i/n/n≥p_n+1+∑_i=1^n2iP_i/n+1/n+1=P_n+1.In order to prove (<ref>), note that the first equality in (<ref>) shows that P_n is obtained as the average of the n summands in the numerator of the fraction that is equal to P_n. The average value of a set of real numbers does not change if we add the average value of the set to the set as a new element. In this case, that average value isP_n, proving that P_n=p_n+P_n+ ∑_i=1^n-12iP_i/n/n+1. So (<ref>) will be proved if we can show that P_n =p_n+P_n+ ∑_i=1^n-12iP_i/n/n+1≥p_n+1+∑_i=1^n2iP_i/n+1/n+1=P_n+1.Noting that p_n≥ p_n+1 by Corollary <ref>, it suffices to prove thatP_n+ ∑_i=1^n-12iP_i/n≥∑_i=1^n2iP_i/n+1 ,which simplifies to the inequality2/n(n-1)∑_i=1^n-1 iP_i≥ P_n.Finally, (<ref>) holds, becausethe right-hand side of (<ref>) grows if we replace p_n by P_n,(since in any given tree, the root can only be balanced if all vertices are balanced), which leads to the inequality P_n -2/n+1 P_n ≤∑_i=1^n-12iP_i/n/n+1,which is clearly equivalent to (<ref>). As the sequence P_1,P_2,⋯ is weakly decreasing, its limit L exists.Note that L≥∑_k=0^5c_n,k≈ 0.6684 as we mentioned in Section <ref>. On the other hand, the number of balanced vertices of ranklarger than five is certainly at most as large as the number of all vertices of rank larger than five, and it is known <cit.> that the latter is about0.00125 timesthe total number of vertices. This proves that0.6684 ≤ L ≤ 0.66965. § NON-PLANE 1-2 TREES In these trees, the vertices are still bijectively labeled by the elements of [n],each vertex has a smaller label than its parent, and each non-leaf vertex has one or two children, but "left" and "right" do not matter anymore. See Figure <ref> for an illustration.It is well known <cit.> that the number of such trees is the Euler number E_n, and that the exponential generating function of the Euler numbers is y(x)=∑_n≥ 0E_n x^n/n! = tan x +x.See sequence A000111 in theOn-line Encyclopedia of Integer Sequences <cit.> for the many occurrences of these numbers in Combinatorics.Let A_k(x) be the exponential generating functionfor the numberof all balanced vertices of rank kin all non-plane 1-2 trees on [n], and let R_k(x) be the exponential generating function for the number of non-plane 1-2 trees on [n]in which the root is balanced and of rank k. The differential equation A_k'(x)- A_k(x)y(x) =R_k'(x) holds, with the initial condition A_k(0)=0. Let (v,T) be an ordered pair in which T is a non-plane 1-2 tree on vertex set [n] and v is a balanced vertex of rank k of T. Then A_k (x) is the exponentialgenerating function counting such pairs.Let us first assume that v is not the root of T,and let us remove the root of T.On the one hand, this leaves a structure that is counted by A_k'(x).On the other hand, this leaves an ordered pair consisting of a non-plane 1-2 tree with a vertex of order k marked, and a non-plane 1-2 tree. By the Product formula of exponential generating functions, such ordered pairs are counted by the generating function A_k(x) y(x). Finally, v was the root of T, then the root of T was balanced and of rank k. Such trees are counted by R_k(x), or, after the removal of their root, by R_k'(x).Crucially, the generating function R_k(x), and hence, its derivative R_k'(x), are polynomials, which enables us to explicitly solve the linear differential equation (<ref>). Indeed, the solution is A_k(x)=∫ R_k'(x)(1-sin x)dx/1-sin x,where the integral in the numerator is an elementary functionsince the integral of x^nsin x is an elementary function for all positive integers n. (The constant of integration is chosen so that the initial condition A_k(0)=0 is satisfied.) Note that we are not able to count all verties of rank k in a similar fashion, since thegenerating function for the number of non-plane 1-2 trees in which the root is of rank k is not a polynomial, andthe solutions analogous to (<ref>) will not be elementary functions for k≥ 1. Even ∫ x tan xdx and ∫ xxdx are not elementary functions.§.§ The number of all verticesSo that we could compute the probability that a randomly selected vertex of arandomly selected non-plane 1-2 tree on [n] is balanced and of rank k, we need to know the size of thenumber nE_nof all vertices in all such trees. The asymptotics of the Euler numbers are well-known (see <cit.>, for example), but to keep the paper self-contained, we provide an argument here at the level of precision that we will need.See any introductory textbook on Complex analysis, such as <cit.> for the relevant notions. As y(x)=tan x +x, the dominant singularities of y(x) are at x=π /2 and x=-π/2, so the coefficients of y(x) are of exponential order 2/π. (Note that the singularity at x=-π /2 is removable.)However, we need a little more more precision. The following proposition will provide that. Let H(x)=f(x)/g(x) be a function so that f(x) and g(x) are analytic functions, f(x_0)≠ 0, while g(x)=0 and g'(x)≠ 0. Then |_x_0 =f(x_0)/g'(x_0).We can apply Proposition <ref> to y(x) at x_0=π/2 with f(x)=1+sin x and g(x)=cos x if we note that y(x)=1+sin x/cos x. ThenProposition <ref> implies that Resy(x) |_π/2=2/-1=-2. The singularity of y at x=-π/2 is removable,since lim_x→ -π/2 y(x) =0 exists, so Resy(x) |_-π/2=0.Now observe thatR/x-a=R/-a·1/1-x/a =R/-a∑_n≥ 0x^n/a^n.Applying thisto y(x) with a=π/2 and R=-2, we get that the dominant term of y(x) is of the form 4/π∑_n≥ 0 x^n (2/π)^n, so E_n/n!∼4/π·(2/π)^n . So the total number of all vertices in all non-plane 1-2 trees of size n is nE_n ∼n!( n 4/π·(2/π)^n). §.§ LeavesLet A_0(x) denote exponential generating function for the total number of leaves in all non-plane 1-2trees on vertex set [n]. It is then easy to verify that A_0(x)=x+x^2/2+2x^3/6+ 9x^4/24+⋯. The equality A_0(x)=x-1+cos x/1-sin xholds.Let us apply Theorem <ref> with k=0, to get the linear differential equation A_0'(x)- A_0(x)y(x) = 1.Indeed, R_0(x)=x, since the only tree whose root is balanced and of rank 0 is the one-vertex tree. Recalling that y(x)=tan x +x, and the initial condition A_0(0)=0, we can solve the last displayed linear differential equation to get what was to be proved.Note that x=π /2 is the unique nonremovable singularity of smallest modulus of A_0(x), and that at that point, A_0(x) haspole of order two, since (1-sin x)' =-cos x also has a zero at that point. Therefore, we cannot apply Proposition<ref> directly. Instead, we use the following lemma. Let H(x)=f(x)/g(x) be a function so that f and g are analytic functions, f(x_0)≠ 0,while g(x_0)=g'(x_0)=0, and g”(x)≠ 0.Then H(x)=2f(x_0)/g”(x_0)·1/(x-x_0)^2 + h_-1/x-x_0 +h_0+⋯ . The conditions directly imply that g has a double root, and henceH has a pole of order two, at x_0. In order to find the coefficient that belongs to that pole, letg(x)=q(x)(x-x_0)^2. Now differentiate both sides with respect to x, to getg”(x)=q”(x)(x-x_0)^2 + 4q'(x)(x-x_0) + 2q(x).Setting x=x_0, we getg”(x_0)=2q(x_0). By our definitions, in a neighborhood of x_0, the function H(x) behaves likef(x)/q(x)(x-x_0)^2,and our claim follows by (<ref>). Let A_n,0 be the number of all leaves in all non-plane 1-2 trees on vertex set [n]. The equality C_0:=lim_n→∞ A_n,0/nE_n = 1-2/π≈ 0.3633802278holds.In other words, for large n, the probability that a vertex chosen uniformly at random from all vertices of allnon-plane 1-2 trees is a leaf is 1-2/π. Note thatA_0(x) has a unique singularity of smallest modulus,at x=π/2, so the exponential growth rate ofits coefficients is 2/π.Also note that at that point, the denominator of A_0(x) has a double root. Therefore, Lemma <ref> applies, with f(x)=x-1+cos x and g(x)=1-sin x, yielding that the coefficient of the (x-π/2)^-2 term in the Laurent series of A_0(x) about x_0=π/2is 2·(π)/2 -1 +cos (π/2)/sin(π/2)=π-2.Now observe that D/(x-a)^2 = D/a^2·1/(1-x/a)^2=D/a^2·∑_n≥ 0(n+1)x^n/a^n.Applying this to the dominant term of A_0(x) with D=π-2 and a=π/2, we get that A_n,0/n!∼ n (π -2) ·(2/π)^n+2.The proof of our claim is now immediate by comparing formulas(<ref>) and (<ref>).§.§ Balanced vertices of rank 1Let A_1(x) be exponential generating function forthe total number of balanced vertices of rank 1 in all non-plane 1-2 trees on vertex set [n]. Note that such vertices have only leaves as neighbors.The differential equationA_1(x)=1/6·(3x^2+6x-6)cos x -(6x+6)sin x+x^3+3x^2+6/1-sin xholds, with the initial condition A_1(0)=0.Let us apply Theorem <ref> with k=1, to get the linear differential equation A_1'(x)- A_1(x)y(x) = x+x^2/2.Indeed, R_1(x)=x^2/2 + x^3/6, since there are only two trees whose root is balanced and of rank 1; one of them has two vertices and the other one has three vertices. Recalling that y(x)=tan x +x, and the initial condition A_1(0)=0, we can solve the last displayed differential equation to prove our claim. The equality C_1 :=lim_n→∞ A_n,1/nE_n = π^2/24 + π/4-1 ∼0.1966316804holds.In other words, for large n, the probability that a vertex chosen uniformly at random from all vertices of allnon-plane 1-2 trees is balanced and of rank 1 is π^2/24 + π/4-1.Note that A_1(x) has a unique singularity of smallest modulus at x=π/2, and that that singularity is a pole of order two. Therefore, we can apply Lemma <ref> with f(x)=3x^2+6x-6)cos x -(6x+6)sin x+x^3+3x^2+6 and g(x)=6(1-sin x). At x_0=π /2, this yields f(x_0)= π^3/8 + 3π^2/4 -3π. Furthermore,g”(x)=6sin x, so g”(x_0)= 6. Therefore, the coefficient of the 1/(x-π/2)^2 term in the Laurent series ofA_1(x) about x_0=π/2 is 2f(x_0)/g”(x) = 2/6· (π^3/8 + 3π^2/4 -3π)= π^3/24 +π^2/4 -π . Applying (<ref>) with D=π^3/24 +π^2/4 -π and a=π /2, we get that A_n,1/n!∼ n(π^3/24 +π^2/4 -π ) ·(2/π)^n+2. We can now prove our claim by comparing formulae (<ref>) and (<ref>). §.§ Balanced vertices of higher rankFor any fixed k, we can compute the probability that a vertex selected from all vertices of all non-plane 1-2 trees uniformly at random is balanced and of rank k. For instance, for k=2, we get that lim_n→∞ A_n,2/nE_n= π^6/32256 +π^5/2304 -7π^4/1920-3π^3/64 +π^2/3 +9π/4-8 -2/π≈ 0.0759013197 ,where A_n,2 denotes the number of balanced vertices in all non-plane 1-2 trees on [n]. It is a direct consequence of (<ref>), the well-known fact that (easy to prove by induction)that∫ x^n sin xdx = H(x)sin x +I(x) cos x,for some polynomials H(x) and I(x), and Lemma <ref> that for all positive integers n, there exists a polynomial J_n with rational coefficients so that C_k:=lim_n→∞ A_n,k/nE_n= J_n(π)/π . § PLANE 1-2 TREESPlane 1-2 trees are similar to non-plane 1-2 trees, except that the children of each vertex are linearly ordered, left to right. The difference between plane 1-2 trees and decreasing binary trees is that in plane 1-2 trees, if a vertex has only onechild, then that child has no "direction", that is, it is not a "left child" or a "right child".See Figure <ref> for anillustration. So plane 1-2 trees are "in between" decreasing binary trees (where left or right matters for every child) and non-plane 1-2 trees (where left or right does not matter for any vertex). Indeed, in plane 1-2 trees, left or right matters, except for vertices that have no siblings.Let Z(x)=∑_n≥ 0z_nx^n/n! be the exponential generating function for the number of plane 1-2 trees on vertex set [n]. Then Z(x) satisfies the differential equationZ'(x)=Z^2(x) - Z(x)+1, with initial condition Z(0)=1. Indeed, removing the root of a plane 1-2 tree T that has more than one vertex, that treefalls apart to the ordered set of two such trees, except when the root of T has only one child. See sequence A080635 in <cit.> for many other combinatorial problems whose solution involves the power series Z(x). Solving (<ref>), we get the explicit formulaZ(x)=√(3)/2tan ( √(3)/2x+π/6) +1/2 .Noting that tan a = sin a / cos a, and that the summand 1/2 at the end does not influence the growth rate of the coefficients of Z(x), we can proceed as in Section <ref>. That is, we can use Proposition<ref> to compute that the number z_n of plane 1-2 trees on vertex set [n] satisfiesz_n ∼ n!( 3√(3)/2π)^n+1 . Therefore, the total number of all vertices of all such trees satisfies nz_n ∼ n! · n( 3√(3)/2π)^n+1 .Let 𝐀_k(x) be the exponential generating functionfor the numberof all balanced vertices of rank kin all plane 1-2 trees on [n], and let 𝐑_k(x) be the exponential generating function for the number of plane 1-2 trees on [n]in which the root is balanced and of rank k.The differential equation 𝐀_k'(x)=2𝐀_k(x)Z(x) - 𝐀_k(x) + 𝐑_k'(x) holds, with the initial condition 𝐀_k(0)=0. Let (v,T) be an ordered pair in which T is aplane 1-2 tree on vertex set [n] and v is a balanced vertex of rank k of T. Then 𝐀_k (x) is the exponentialgenerating function counting such pairs.Let us first assume that v is not the root of T,and let us remove the root of T.On the one hand, this leaves a structure that is counted by 𝐀_k'(x).On the other hand, this leaves an order pair consisting of a plane 1-2 tree with a a balanced vertex of rank k marked, and a plane 1-2 tree. If the tree without the marked vertex was not empty, then by the Product formula of exponential generating functions, such ordered pairs are counted by the generating function 2𝐀_k(x) Z(x), since the order of the two trees matters. If the tree without the marked vertex was empty, then there is only one way to "order" the 1-element set ofsubtrees of the root, consisting of the subtree with the marked vertex. This results in the correction term-𝐀_k'(x). Finally, v was the root of T, then the root of T was balanced and of rank k. Such trees are counted by 𝐑_k(x), or, after the removal of their root, by 𝐑_k'(x). §.§ LeavesSetting k=0 in (<ref>), noting that 𝐀_0(0)=0, and 𝐑_0'(x)=1, then solving the resulting differentialequation we get the following result for the number of all leaves. The equality𝐀_0(x) =√(3)/6sin(√(3)x+π/3 )+ x/2 -1/4/cos ^2( √(3)x/2+π/6)holds. We can apply Lemma <ref> to the numerator and the denominator of 𝐀_0(x) in (<ref>) tocompute the growth rate of the coefficients of that power series. Indeed, 𝐀_0(x) has a unique singularity of smallest modulus at x_0=2π/(3√(3)), and that point the numerator of 𝐀_0(x) is nonzero,the denominator, and its first derivative are zero, while the second derivative of the denominator is not 0.Then a computation analogous to that immediately following the proof of Theorem <ref> shows thatif 𝐚_n,0 is the number of all leaves in all plane 1-2 trees on vertex set [n], then 𝐚_n,0∼ ( -1/3 + 4/27√(3)π) · n( 3√(3)/2π)^n. Comparing (<ref>) with (<ref>), we obtain the following result.The equality𝐂_0: = lim_n→∞𝐚_n,0/nz_n =2/3 - √(3)/2π≈ 0.391002219holds. In other words, the probability that a vertex selected uniformly at random from all plane 1-2 trees of size n will be a leaf is about 0.391.§.§ Vertices of higher rankNote that remarkably, we can obtain an explicit formula for 𝐀_k(x) for every k. This is in contrast tothe case when we want to compute the generating function of all vertices of a given rank, where we run intonon-elementary functions for k≥ 2. Indeed, the standard form of (<ref>) is 𝐀_k'(x)+(1-2Z(x)) 𝐀_k(x)= 𝐑_k'(x).In order to solve this linear differential equation, first we multiply both sides by the integrating factor μ(x) =exp ( ∫ (1-2Z(x))dx)= exp ( -ln (^2( √(3)x/2+π/6)) )=cos ^2( √(3)x/2+π/6),which is an elementary function. After that multiplication, we need to integrate both sides of the obtained equation to get 𝐀_k(x). However, we are able to do so, since on the right hand side, we have μ(x) 𝐑_k'(x), that is, a cosine function times a polynomial function, and it is well known that such products have elementary integrals.§ FURTHER DIRECTIONSThe method that we used in this paper may well be applicable to count balanced vertices in other tree varieties, as long as the number of children each vertex can have is bounded. It seems intuitively very likely that in any tree variety, as n goes to infinity, the probability that a vertex chosenuniformly at random from all vertices of all trees of size n is balanced will be monotone decreasing. Still in the one case where we could prove this, the case of decreasing binary trees, our proof heavily depended on the simple bijection between these trees and permutations. New ideas are needed for other tree varieties. We mentioned that plane 1-2 trees are "in between" the other two tree varieties studied in this paper. So it is perhaps interesting that their vertices are the most likely to be leafs. Indeed, a random vertex of a decreasing binary tree has a one-third chance to be a leaf, while the same probability is about 36.3 percent for non-plane 1-2 trees, and39.1 percent for plane 1-2 trees. Understanding this phenomenon could lead to new insights.99 protectedM. Bónak-protected vertices in binary search trees. Adv. in Appl. Math. 53 (2014), 1–11. pittel M. Bóna, B. Pittel, On a random search tree: asymptotic enumeration of vertices by distance from leaves. J. Appl. Prob. to appear. Preprint available at atarXiv:1412.2796. anacomb Bóna, M. (2016) Introduction to Enumerative and Analytic Combinatorics, CRC Press, 2016.churchill J. Brown, R. V. Churchill, Complex Variables and Applications, 9th edition. McGraw-Hill, 2013.flajolet P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, Cambridge, UK, 2009. holmgren S. Janson and C. Holmgren, Limit laws for functions of fringe trees for binary search trees and recursive trees. Electronic J. Probability 20 (2015), no. 4, 1-51. oeis Online Encyclopedia of Integer Sequences, online database, <www.oeis.org>. | http://arxiv.org/abs/1705.09688v2 | {
"authors": [
"Miklos Bona"
],
"categories": [
"math.CO",
"05A05, 05A15"
],
"primary_category": "math.CO",
"published": "20170526191955",
"title": "Balanced vertices in labeled rooted trees"
} |
A group law is said to be detectable in power subgroups if, for all coprime m and n, a group G satisfies the law if and only if the power subgroups Gm and Gn both satisfy the law. We prove that for all positive integers c, nilpotency of class at most c is detectable in power subgroups, as is the k-Engel law for k at most 4. In contrast, detectability in power subgroups fails for solvability of given derived length: we construct a finite group W such that W2 and W3 are metabelian but W has derived length 3. We analyse the complexity of the detectability of commutativity in power subgroups, in terms of finite presentations that encode a proof of the result. * N. Tax, N. Sidorova, R. Haakma, W.M.P. van der Aalst December 30, 2023 ======================================================== § INTRODUCTION This article studies the following broad question: what can be deduced about a group G by examining its power subgroups Gn = g^n : g ∈ G? In particular, can one infer which laws G satisfies?Let F_∞ = F(x_1, x_2, …) be the free group on the basis {x_1, x_2, …}. A law (or identity) is a word w ∈ F_∞, and we say a group G satisfies the law w if φ(w) = 1 for all homomorphisms φ F_∞→ G. For notational convenience, when we require only variables x_1 and x_2 we will instead write x and y. We can also think of a law w on k variables x_1, …, x_k as a function w G ×…× G_k times→ G, written w(g_1, …, g_k) := φ(w) for a homomorphism φ F_∞→ G such that φ(x_i) = g_i.Laws give a common framework for defining various group properties; basic examples include commutativity (corresponding to the law [x, y]), having exponent m (the Burnside law x^m), being metabelian (the law [[x_1, x_2], [x_3, x_4]]), and nilpotency of class at most c (the law [[[…[x_1, x_2], x_3], …, x_c], x_c+1]).A group law w is detectable in power subgroups if, for all coprime m and n, a group G satisfies w if and only if the power subgroups Gm and Gn both satisfy w. A subgroup of G will satisfy all the laws of G, but in general it is possible even for coprime m and n that the power subgroups Gm and Gn satisfy a common law that G does not; for example, the holomorph G = 7⋊6 (where 6≅Aut7 acts faithfully) was shown to have this property in <cit.> (and is in fact the smallest such group). A concrete example of a law that holds in G2 and G3 but not G is [[x^2, y^2]^3, y^3]. Another basic example is the holomorph G = 9⋊6, which does not satisfy the law [x^2, x^y] although G2 and G3 do. The law x^r is detectable in power subgroups.This basic example is immediate: for every g ∈ G, if (g^m)^r = 1 and (g^n)^r = 1, then g^r = 1 as m and n are coprime.A classical theme in group theory is the study of conditions that imply that a group is abelian. This was recently revived by Venkataraman in <cit.>, where she proved that commutativity is detectable in power subgroups for finite groups. We can extend this to infinite groups using residual finiteness of metabelian groups (a theorem of P. Hall <cit.>); it appears that this result is folklore.In this article we prove that this result generalizes to the nilpotent case: Let m and n be coprime and let c ≥ 1. Then a group G is nilpotent of class at most c if and only if Gm and Gn are both nilpotent of class at most c.Fitting's Theorem (see <ref> below) readily implies a weak form of the “if” direction, namely that G is nilpotent of class at most 2c, but it is much less obvious that the precise nilpotency class is preserved.Detectability of laws in power subgroups has an elegant formulation in the language of group varieties, which we develop in Section <ref>. The reader unfamiliar with varieties should not be deterred, as our use of this language is simply a means of expressing our reasoning in a natural and general setting. In particular, our treatment of varieties is essentially self-contained, and no deep theorems are called upon.Let _c denote the variety of nilpotent groups of class at most c and let _m denote the `Burnside' variety of groups of exponent m. Employing the notion of product varieties (Definition <ref>), we can restate the conclusion of Corollary <ref> as _c _m ∩_c _n = _c. We prove this as a corollary of a stronger result:Letbe a locally nilpotent variety and let m and n be coprime. Then_m ∩_n = .A variety is locally nilpotent if its finitely generated groups are nilpotent, or, equivalently, if its groups are locally nilpotent. A topic with a rich history, dating to work of Burnside, is that of Engel laws. The k-Engel law is defined recursively by E_0(x, y) = x and E_k+1(x, y) = [E_k(x,y), y]. For example, the 3-Engel law is [[[x, y], y], y]. Havas and Vaughan-Lee <cit.> proved local nilpotency for 4-Engel groups, so we have the following: Let m and n be coprime and let k ≤ 4. A group G is k-Engel if and only if Gm and Gn are both k-Engel.It is an open question whether a k-Engel group must be locally nilpotent for k ≥ 5. Recently A. Juhasz and E. Rips have announced that it does not have to be locally nilpotent for sufficiently large k.The class of virtually nilpotent groups plays an important role in geometric group theory, dating back to Gromov's seminal Polynomial Growth Theorem: a finitely generated group is virtually nilpotent if and only if it has polynomial growth <cit.>. Because of this prominence, we also prove that virtual nilpotency is detectable in power subgroups (Corollary <ref>).In contrast, solvability of a given derived length is not detectable in power subgroups; this fails immediately and in a strong sense as soon as we move beyond derived length one, that is, beyond abelian groups.Letdenote the variety of metabelian groups. Then_2 ∩_3 ≠.Indeed, there exists a finite group W such that W2 and W3 are both metabelian but W is of derived length 3. The construction of W is rather involved and ad hoc, and does not have an obvious generalization. The smallest such W has order 1458.This is yet another example of the chasm between nilpotency and solvability. Other properties that we lose when crossing from finitely generated nilpotent groups to finitely generated solvable groups include the following: residual finiteness, solvability of the word problem, polynomial growth, and finite presentability of the relatively free group.As the free nilpotent group of class c is finitely presented, we know a priori that Corollary <ref> will be true for fixed m and n if and only if it is provable in a very mechanical way, namely via a finite subpresentation of a canonical presentation for the free group of rank c+1 in the variety _c _m ∩_c _n (in a way which we make precise in Section <ref>). Since such a finite presentation `proving' the theorem for those m and n exists, it is natural to ask what such a presentation looks like: what is the minimum number of relators needed, does that number depend on m and n, and how must the specific relators change with m and n?We analyse in detail the abelian case, where the answer to all of these questions is: surprisingly little. Let m and n be coprime. The following is a presentation of ℤ×ℤa, b[a^m, b^m], [a^m, (ab)^m], [b^m, (ab)^m], [a^n, b^n], [a^n, (ab)^n], [b^n, (ab)^n] .The structure of this article is as follows, and reflects the structure of the introduction we have just given. In Section <ref> we set up some basic theory. We prove positive results, including Theorem <ref>, in Section <ref>. We then prove the negative result Theorem <ref> in Section <ref>. The complexity analysis with Theorem <ref> follows in Section <ref>, and finally we record some open problems in Section <ref>.§ BASIC NOTIONSIn this section we develop some basic tools which will be helpful, including aspects of the theory of group varieties. We also probe the definition of detectability in power subgroups: why specifically power subgroups, and what about the non-coprime case?For the first question, there are easy examples showing that we cannot in general determine if a group law is satisfied just by examining two arbitrary subgroups, even if they are assumed to be normal and to generate the whole group: it is essential that we examine the characteristic, “verbally defined” power subgroups. For instance, the integral Heisenberg group x,y,z[x,y]=z, [x,z] = [y, z] = 1 is the product of the two normal subgroups x,z and y,z, which are both isomorphic to ×, however the whole group is not abelian.We now turn to the question of coprimality. For a propertyof groups, we say a group G hascoprime power subgroups if there exist coprime m and n such that Gm and Gn both have the property . For example, using this terminology we can state the theorem of <cit.> as: a finite group with abelian coprime power subgroups is abelian.Notation. We write conjugation g^h = h^-1 g h and commutator [g, h] = g^-1 h^-1 g h.An easily proved property of power subgroups (and verbal subgroups in general, see below for the definition) is the following: Let φ GQ be a surjective group homomorphism and m an integer. Then φ(Gm) = Qm and Gm≤φ^-1(Qm). Power subgroups pick up torsion elements: Suppose that g has finite order r coprime to m. Then g ∈g^m.There exist integers x and y such that xr + ym = 1. Nowg = g^xr + ym = (g^r)^x (g^m)^y = (g^m)^y. §.§ VarietiesWe give a self-contained treatment of some basics from the theory of varieties of groups. For further details, the reader is referred to Hanna Neumann's classic book <cit.>. A variety of groups is the class of all groups satisfying each one of a (possibly infinite) set of laws.Corresponding to the examples of laws given above (immediately before Definition <ref>), we have the following examples of varieties: *– the variety of abelian groups* _m – the `Burnside' variety of groups of exponent m (or exponent dividing m, depending on the definition of exponent used)*– the variety of metabelian groups.* _c – the variety of nilpotent groups of nilpotency class at most c A variety is closed under the operations of taking subgroups, quotients, and arbitrary Cartesian products. In fact, every class of groups which is closed under these operations is a variety (see <cit.>).Letandbe varieties of groups. We define the product varietyto be the class of groups which are an extension of a group fromby a group from . That is, G ∈ if there exists NG such that N ∈ and G / N ∈. We define the product of two classes of groups similarly.Letanddenote the varieties of abelian and metabelian groups, respectively. Then =. We check that the product variety is indeed a variety as follows. Let (G) ≤ G denote the verbal subgroup of G corresponding to , that is, the subgroup generated by the images of the defining laws ofunder all maps F_∞→ G. Thus G ∈ if and only if the verbal subgroup (G) = 1. As defining laws forwe take the images of the defining laws ofunder all maps F_∞→(F_∞). Let G be a group and suppose NG, with qG → G / N the natural homomorphism. Then (G/N) = q((G)) (cf. Lemma <ref>), so the quotient G / N is inif and only if (G) ≤ N. Thus G ∈ if and only if N = (G) is in the variety . Every map F_∞→(G) factors through some map F_∞→(F_∞), so we see that (G) ∈ if and only if it satisfies every law which is the image of a defining law ofin (F_∞). We will explore this further in Section <ref> The product of varieties of groups is associative. Thus the varieties form a monoid under product, and the unit is the variety 1 consisting of only the trivial group. We introduce a more restrictive notion of product for two classes of groups. Letandbe classes of groups. We define the normal product ofand , denoted , to be the class of groups G with normal subgroups C ∈ and D ∈ such that G = CD. In particular, = ⊆∩. This last inclusion can be proper, for example, ⊂_2 (by Theorem <ref>, due to Fitting), whereas =.Let G be a group,a variety of groups, and m an integer. Recall that _m denotes the Burnside variety of exponent m. The power subgroup Gm∈ if and only if G ∈_m.As in the proof that a product variety is a variety, we see that G ∈_m if and only if the verbal subgroup _m(G) ∈, and _m(G) = Gm. With this proposition in hand, we define a varietyto be detectable in power subgroups if, for all coprime m and n, we have _m ∩_n = (the intersection of varieties is simply the intersection as classes of groups). In this article, we mostly encounter varieties that are finitely based, that is, that can be defined by finitely many laws, and thus by a single law (the concatenation of these laws written in distinct variables x_i); in this case, detectability of the variety is simply detectability of such a single defining law.It will be useful for us to understand how taking products of varieties interacts with taking intersection. Although we do not have left-distributivity, we do have some upper and lower bounds, as the next proposition indicates.For all varieties , ,we have(∩) ≤∩⊆ () (∩).(We write ⊆ as the last term is not a variety in general.) The first inclusion is immediate, as ∩≤ implies that (∩) is contained in , and similarly in .Now suppose that G ∈∩. This means G has normal subgroups N_, N_∈ such that G / N_∈ and G / N_∈. Let N = N_ N_ G. The group G / N will be a common quotient of G / N_ and G / N_, and thus in ∩, as varieties are closed under taking quotients. The kernel N_ N_ is then a product of normal subgroups in , so it is in the class . Let m and n be coprime. Then for every variety ,_m ∩_n ≤. Set = _m, = _n in Proposition <ref> and note _m ∩_n = _gcd(m,n) = 1. In contrast, we do have right-distributivity of product of varieties over intersection: For all varieties , , we have(∩)= ∩. The inclusion “≤” is immediate.Suppose G ∈∩, with N_, N_ G such that N_ is in , N_ is in , and both quotients G/N_, G/N_ are in . We have a (generally non-surjective map)G → G/N_× G/N_with kernel N_∩ N_. That is, the kernel is in ∩, and the quotient is in , since a variety is closed under Cartesian product and subgroups. Varieties are determined by their finitely generated groups: Letandbe varieties. Let _f denote the subclass of groups G ∈ such that G is finitely generated and define _f similarly. Then = if and only if _f = _f.Clearly = implies _f = _f. Supposeis not contained in , so there is a law w ∈ F_∞ which is satisfied by all groups in , but there is some G ∈ and φ F_∞→ G with φ(w) ≠ 1. The law w is a word on finitely many letters x_1, …, x_n in the basis for F_∞, and we can assume φ(x_i) = 1 for all i > n. The subgroup G_0 ≤ G generated by φ(x_1), …, φ(x_n) is an element of _f. We can consider φ as a map F_∞→ G_0 and so G_0 does not satisfy the law w. Thus _f is not contained in , so in particular _f is not contained in _f. Recall a well-known fact about torsion groups, which we will apply several times. A finitely generated solvable torsion group is finite.§.§ CoprimalityThe notion of detectability of a law in power subgroups does not make sense in general if one allows m and n for which gcd(m,n) = d > 1; this could only say something about Gd≤Gm∩Gn and not the whole group G. For example, in a group G of exponent d the power subgroups are trivial and satisfy all laws w ∈ F_∞, whereas G does not if it is non-trivial. A more extreme example is provided by the free Burnside groups of exponent d for large odd d, which are infinite by the celebrated work of Novikov and Adian, and thus are not even solvable (by Proposition <ref>).A precise formulation of this idea is the following: For every varietyand for all integers m and n, we have⋆_m ∩_n ≥_gcd(m,n).Suppose further thatis detectable in power subgroups, that is, we have equality in (<ref>) for the case of coprime m and n. Then we have equality in (<ref>) for all m and n.The inclusion (<ref>) is immediate, as both _m and _n contain _gcd(m,n).Suppose now thatis detectable in power subgroups, and let d = gcd(m,n), m' = m/d, n' = n/d, so that m' and n' are coprime. We have _m ≤_m'_d (in general this inclusion may be strict), and similarly for _n, and thus_m ∩_n ≤_m'_d ∩_n'_d = (_m'∩_n') _dvia right-distributivity of the product over intersection (Proposition <ref>), and implicitly using associativity of the variety product. By assumption of detectability, this last term is just _d. The reader is referred to <cit.> for more on the fascinating topic of products of Burnside varieties.[More than two powers] The notion of detectability is unchanged if we replace the two powers m and n with powers m_1, m_2, …, m_k that aremutually (not necessarily pairwise) coprime, that is, gcd(m_1, m_2, …, m_k) = 1. This follows by an easy induction, which can be expressed conveniently using the characterization of Proposition <ref>. § LOCALLY NILPOTENT VARIETIES ARE DETECTABLEThe starting point for this section is a desire to generalize the result that commutativity is detectable in power subgroups to the nilpotent case. For instance, can power subgroups detect whether a group is nilpotent of class at most 2? We are carried quite a way towards our goal by Fitting's Theorem.Let M and N be normal nilpotent subgroups of a group G. If c and d are the nilpotency classes of M and N, then L = MN is nilpotent of class at most c + d. However, this will only tell us, for instance, that if the power subgroups are nilpotent of class at most 2, then our group of interest is nilpotent of class at most 4. We first lay some foundations towards proving the general Theorem <ref>, then see in Theorem <ref> how we can reduce the bound of 2c to c, as in Corollary <ref>. By proving the general theorem, we will also be able to conclude that certain Engel laws are detectable.Let m and n be coprime. Letdenote the class of nilpotent, locally nilpotent, solvable, or locally solvable groups. Then _m ∩_n =.In each of the first three cases, this is an application of Corollary <ref> together with the corresponding standard result that the appropriateis equal to : Fitting's Theorem for the nilpotent case, the Hirsch–Plotkin Theorem <cit.> for the locally nilpotent case, and the solvable case is elementary. (This in fact shows the result still holds after replacing _m and _n with two arbitrary varieties with trivial intersection.) However, a group which is the product of two normal locally solvable subgroups need not be locally solvable, as shown by P. Hall <cit.>, so for the remaining case we exploit the power subgroup structure. This argument also allows us to conclude the locally nilpotent case from Fitting's theorem, without the need to invoke Hirsch–Plotkin.Assume now that G ∈_m ∩_n is finitely generated, so that its quotient G / Gm is finitely generated and of exponent m. By the second isomorphism theorem,G / Gm≅Gn / (Gm∩Gn)and so since Gn is locally solvable, its finitely generated quotient G / Gm is solvable. Now G / Gm is a finitely generated solvable torsion group, and thus finite (Proposition <ref>). Hence the subgroup Gm G is of finite index, so it is finitely generated, and since groups inare locally solvable, Gm is in fact solvable. Similarly, Gn is solvable. Thus G = GmGn is solvable. Let G be a finitely generated nilpotent group and let m and n be coprime. If Gm and Gn both satisfy a law w, then G satisfies w. In other words, the variety generated by G is the intersection of the varieties generated by Gm and Gn. (The variety generated by a group is the intersection all varieties containing it.) Suppose for the sake of contradiction that there is a homomorphism φ F_∞→ G with φ(w) ≠ 1. Since G is finitely generated and nilpotent, it is residually finite <cit.>, so there is a map qGQ for some finite group Q such that q(φ(w)) ≠ 1. As G is nilpotent, so is Q, and thus Q is the direct product of its Sylow subgroups <cit.>. We compose q with a projection onto a Sylow subgroup in which q(φ(w)) has non-trivial image, to get q_pGQ_p. Without loss of generality, p does not divide m so that Q_pm = Q_p (Lemma <ref>). This gives a contradiction, as Q_pm = q_p(Gm) (Lemma <ref>), and Gm satisfies the law w.Letbe a locally nilpotent variety and let m and n be coprime. Then_m ∩_n = . By Proposition <ref>, it suffices to consider finitely generated G ∈_m ∩_n. Sinceis locally nilpotent, Proposition <ref> guarantees that G is locally nilpotent. As G is in fact finitely generated, we can now apply Theorem <ref> to conclude that G satisfies every law which holds in both Gm and Gn. Since Gm and Gn are in the variety , we conclude that G ∈. The nilpotent groups of class at most c form the variety _c, so the following corollary is immediate. Let m and n be coprime and let c ≥ 1. Then a group G is nilpotent of class at most c if and only if Gm and Gn are both nilpotent of class at most c. This means that the precise nilpotency class of G is the maximum of the precise nilpotency classes of Gm and Gn. Let m and n be coprime and let k ≤ 4. A group G is k-Engel if and only if Gm and Gn are both k-Engel.The variety of 4-Engel groups was shown to be locally nilpotent by Havas and Vaughan-Lee <cit.>. This also implies the (previously known) k ≤ 3 cases, as it is clear from the definition E_k+1(x,y) = [E_k(x, y), y] that a k-Engel group is also (k+1)-Engel.Gruenberg proved that a locally solvable k-Engel group is locally nilpotent <cit.>, so the generality achieved in Proposition <ref> would not help to establish detectability of a Engel law beyond the locally nilpotent case. For a survey on Engel groups, the reader is referred to <cit.>. For the sake of completeness, and motivated by its importance in geometric group theory, we show that the class of virtually nilpotent groups (groups with a nilpotent subgroup of finite index) is detectable in power subgroups. We first prove a more general result, and then use the structure of subgroups as specifically power subgroups to argue that the precise `virtual nilpotency class' is preserved.Let G be a group with normal subgroups A and B such that G = AB. Suppose that A and B are virtually nilpotent. Then G is virtually nilpotent.To invoke Fitting's Theorem, we require nilpotent subgroups that are normal in G. Let A_0 be the normal subgroup of A which is nilpotent and of minimal finite index. Such an A_0 exists as A is virtually nilpotent, and it is unique by Fitting's Theorem, as the product of two finite index normal nilpotent subgroups of A is then a normal nilpotent subgroup of smaller index. (A_0 is the `nilpotent radical' or `Hirsch–Plotkin radical' of A.) Now A_0 is characteristic in A, and thus normal in G. We define B_0 similarly.As A_0 and B_0 are both nilpotent and normal in G, their product A_0 B_0 is nilpotent. Since A_0 and B_0 are finite index in A and B respectively, and normal in G, we conclude that A_0 B_0 is finite index in AB = G. That is, G is virtually nilpotent. Let G be a finitely generated group and let m and n be coprime. If Gm and Gn both have finite index subgroups which are nilpotent of class at most c, then so does G.By Proposition <ref>, G is virtually nilpotent. Thus G / Gm has a finite index subgroup which is nilpotent, and moreover finitely generated and of exponent m, and hence finite (by Proposition <ref>). Now Gm is finite index in G, so its finite index subgroups are finite index in G. § DERIVED LENGTH IS NOT DETECTABLEIn this section we show by explicit example that one cannot extend the above results for the nilpotent case to the solvable case. Of course, Proposition <ref> tells us that a group with solvable coprime power subgroups is itself solvable: the class of solvable groups is closed under extensions. The point is that we do not have the precise control over derived length which we did for nilpotency class. Letdenote the variety of metabelian groups. Then_2 ∩_3 ≠.Indeed, there exists a finite group W such that W2 and W3 are both metabelian but W is of derived length 3.Let H_3 denote the mod-3 Heisenberg group, which is the non-abelian group of order 27 and exponent 3. It admits the presentationH_3 =x, y, zx^3 = y^3 = 1, z = [x, y], [x, z] = [y, z] = 1 .Write n for the cyclic group of order n. The group W is constructed asW := (9×9) ⋊_φ (H_3 ×2)where, letting 2 = t, the action is defined by φ H_3 →_2(9) which mapsx ↦1-13-2,y ↦-2004,t ↦-100-1. We check that this is a well-defined group action. (It is in fact faithful, however this is – while easily verified – unnecessary for the proof.) Let X := φ(x), Y := φ(y), T := φ(t), and Z := [X, Y] = φ(z). As T is order 2 and central in _2 (9), we only need to check the map H_3 →_2 (9). We see first thatX^2 = -21-31,Y^2 = 400-2and thus X^3 = Y^3 = 1. AsX Y = -2-4-6-8 = -2-431andX^-1 Y^-1 = -21-31400-2 = -8-2-12-2 = 1-2-3-2we see thatZ = 1-2-3-2-2-431 = -8-6010 = 1301so that Z has order 3, with Z^-1 = 1-301.We now determine the conjugation action of Z:Z^-1abcd Z = a-3cb-3dcd Z = a-3cb + 3(a-d)cd+3cas 9c = 0, so the matrix abcd is in the centralizer of Z precisely when both a-d and c lie in 3 9. This is true for both X and Y, so we have [X,Z] = [Y,Z] = 1 as required, which completes the verification that φ is a well-defined group homomorphism (or in other words, X, Y and T generate a subgroup of _2(9) isomorphic to H_3 ×2, modulo the injectivity of φ which we will not verify here).We claim that W2 = (9×9) ⋊ H_3 and W3 = (9×9) ⋊2. The “≤” inclusion is Lemma <ref>, and the other inclusion is not necessary for the proof and so is left to the curious reader (if it were not the case, it would only make the task at hand easier). The group W3 is obviously metabelian, as it is exhibited as the semidirect product of one abelian group and another abelian group. On the other hand, W2 will require the following basic computations.Recall first the following:Let G = N ⋊ K. Then the derived subgroup G' = (N' [N, K]) ⋊ K'. One can prove the lemma by verifying that every commutator in G lies in the subgroup generated by N', [N, K] and K', and then noting that the action of K on N restricts to an action on N' [N,K].In the present case of W2 = (9×9) ⋊ H_3, since N = 9×9 is abelian we simply have [N, K] ⋊ K'. The subgroup [N, K] is generated by (I-X)n and (I-Y)n for n ∈9×9. AsI - X = 01-33,I - Y = 300-3,we see that [N, K] = 9× 3 9 = [u; 3v ] u,v ∈9.Now H_3' = z≅3, and we see that the set of invariants for Z is auspiciously none other than 9× 3 9. Thus [N, K] ⋊ K' is abelian, so W2 is metabelian.On the other hand, for the negation action of 2 on N = 9×9 we have [N, 2] = 2N = N, so we see that W' = N ⋊ H_3', which has derived length 2, so W has derived length 3 as claimed.The group W constructed above has order 4374 = 2 × 3^7. It is also possible to construct a group W satisfying the requirements of the theorem as (3×9) ⋊ (H_3 ×2), of order 1458 = 2 × 3^6 (the action is not simply a restriction or quotient of φ). An exhaustive search with<cit.> revealed that this is in fact the smallest non-metabelian group with metabelian coprime power subgroups. (There are two such groups of order 1458, and their ID pairs in the Small Groups Library are (1458, 1178) and (1458, 1192).)We cannot extend this construction in an obvious way from the case of p = 3 to other primes. In particular, it appears to depend on the existence of a matrix of order p in _2(), which only has torsion elements of order at most 6.One could ask for a finitely generated infinite group W that shows that being metabelian is not detectable in power subgroups. However, the failure will still be only up to finite index: such a group is solvable, and thus its power subgroups are of finite index (as used in the proof of Proposition <ref>). § COMPLEXITY ANALYSISBy complexity analysis, we are not referring to analysis of algorithms and complexity classes such asand , but the flavour is similar: we wish to quantify the complexity of detectability of given laws (in a sense we shall make precise), and understand the asymptotic behaviour of this complexity when the powers m and n vary.We can formulate detectability of commutativity using an infinitely presented group having the appropriate universal property. The detectability of commutativity in power subgroups is equivalent to the fact that, for all coprime m and n, the group G_m,n defined by the infinite presentationG_m,n := a, b[u^m, v^m], [u^n, v^n] u, v ∈ F(a,b)is isomorphic to ×.If G_m,n is non-abelian then it is a counterexample. Suppose G_m,n is abelian, and let H be a group with Hm, Hn abelian. Then for every pair of elements g,h ∈ H there is a homomorphism from G_m,n to ⟨ g, h ⟩≤ H defined by a ↦ g, b ↦ h, since all the relators have trivial image, and thus g and h commute. Therefore H is abelian. Moreover, as all relators in the presentation of G_m,n are a consequence of the commutativity of the two generators, G_m,n is in fact isomorphic to the free abelian group on 2 generators, ×. Thus for coprime m and n, the word [a,b] is expressible in the free group as a product of conjugates of terms of the form [u^m, v^m] and [u^n, v^n]. Such an expression gives a proof that _m ∩_n =. At this point, it is natural to ask how many different such terms are needed to encode such a proof. Before giving a very succinct choice of such terms, we phrase the set up in greater generality. §.§ General framework Let XR be a group presentation. We call XS a subpresentation of XR if S ⊆ R. If moreover S_F(X) = R_F(X), then we call XS a core of XR.A core not only defines an isomorphic group: there is a natural isomorphism induced by the identity map on the free group F(X). We recall the following standard result.Let XR be a presentation of a group that admits a finite presentation, and assume that X is finite. Then XR admits a finite core.Briefly, the proof is the following (see also <cit.>). Fix an isomorphism to the group defined by a finite presentation YS, with the isomorphism and its inverse induced by ϕ F(X) → F(Y) and ψ F(Y) → F(X) respectively. These data also give an isomorphism for the group G defined by a subpresentation XR' provided that x =_G ψ(ϕ(x)) for x ∈ X and ψ(s) =_G 1 for s ∈ S. These X + S relations will each be a consequence of a finite subset R; the union of these gives us a finite choice for R'.Letbe a finitely based variety, endowed with a chosen finite setof defining laws. We only need finitely many variables for the laws in ; suppose that each law is in F_k ≤ F_∞. Suppose further that the relatively free group F_k / (F_k) is finitely presented (as an abstract group). For example, ifis nilpotent (or locally nilpotent) then this relatively free group is finitely generated and nilpotent, thus finitely presented. Now _m ∩_n is equal toif and only if its relatively free group of rank k is (naturally) isomorphic to F_k / (F_k). That is, if and only if the infinite presentation^_k,m,n =x_1, …, x_kw(u_1, …, u_k), w(v_1, …, v_k)forw ∈, u_i ∈F_km, v_i ∈F_knis a presentation for the finitely presented group F_k / (F_k). If this is the case, then = ^_k,m,n admits a finite core. Thus there is a partial algorithm that will decide ifdoes indeed present F_k / (F_k): enumerate larger and larger finite subpresentations of(that is, a filtration of the set of relators by finite sets) and attempt for longer and longer for each finite subpresentation to find a proof of isomorphism, proceeding in a diagonal fashion (we “diagonalize” the filtration and the isomorphism search). In general, the isomorphism problem is partially decidable (that is, there is an algorithm that will succeed in proving two input groups are isomorphic if they are isomorphic, but may fail otherwise), but our situation is easier, as we can fix the identity map on the generators (assuming our finite presentation for the relatively free group has k generators). So we only require a partial algorithm for the word problem: for instance, at the r-th attempt we can determine all words in F_k which are a product of conjugates of at most r relators or inverses of relators in the finite subpresentation, by words of length at most r, and freely reduce to see whether all the relators of the finite presentation for F_k / (F_k) appear. Thus we have established the following: Letbe a finitely based variety, and suppose that it admits a finite set of defining laws such that each law is on at most k variables. Suppose that the relatively free group F_k / (F_k) is finitely presented as an abstract group. Then the set of coprime integers m and n for which_m ∩_n =is a recursively enumerable set. That is, there is an algorithm which, given as input a pair m, n, will output YES and terminate in finite time if and only if the varieties are equal. However, we have only demonstrated the existence of such an algorithm: to actually implement the algorithm, we require additionally a finite presentation of F_k / (F_k). For example, a presentation of the free 2-generator 4-Engel group was obtained by Nickel <cit.>. To use Nickel's (polycyclic) presentation, where clearly only a_1 and a_2 are needed to generate the group, we use the obvious Tietze moves to remove the other generators; in general, given a finite presentation of F_k / (F_k) on more than k generators, we could either enumerate presentations of the same group (in a blind search, via Tietze moves) until we construct one with k generators, or replace the partial algorithm for the word problem with a partial algorithm for the isomorphism problem (thereby deferring the difficulty).The argument establishing Proposition <ref> works in greater generality. In fact, all that we used about the varieties _m and _n is that they admit a basis which is a recursively enumerable subset of F_∞ (for example, a finite set). This is what implies that F_km = _m(F_k) is recursively enumerable: for a general variety , each element of (F_k) is a finite product of images of the defining laws, and each defining law has a recursively enumerable set of images in F_k. Under these conditions, we have a corresponding recursively enumerable presentationof the k-generated relatively free group in ∩. Thus, foras in Proposition <ref>, there is a partial algorithm that takes as input the description of two recursively enumerable bases, for varietiesand , and will succeed in determining that ∩ = when this is the case. We do not need the assumption that we made in the “Burnside” case that the varietiesandhave trivial intersection, but if this were not the case we actually could have equality only ifwere the variety of all groups <cit.>. §.§ Complexity of the abelian case For the abelian case, where = defined by the law [x, y], k = 2, and F_2 / (F_2) = F_2 / [F_2, F_2] ≅×, there is an extraordinarily short finite presentation of ^{[x,y]}_2,m,n. Let m and n be coprime. The following is a presentation of ℤ×ℤ :a, b[a^m, b^m], [a^m, (ab)^m], [b^m, (ab)^m], [a^n, b^n], [a^n, (ab)^n], [b^n, (ab)^n] .After proving this theorem, we became aware of another proof <cit.> that groups with abelian power subgroups are abelian, from which one can extract a 2-generator 6-relator core ofwhich defines ×, just as in Theorem <ref>. However, our proof has the advantage of uniformity in the words from the verbal subgroup used, whereas in the other proof the length of the words grows quadratically with m and n.The proof proceeds by first showing that the commutator [a,b] is central; once we know this, the proof that it is trivial is very short.Rather than prove that G_m,n is nilpotent of class 2 directly, we instead prove the stronger result that the group Γ (defined below), an extension of G_m,n, is nilpotent of class 2. This group is moreover a common extension of all the G_m,n, so we see our introducing Γ as abstracting away m and n from the proof. (We will of course prove later that each G_m,n≅×, and so technically × itself is also a common extension a posteriori, but we are constructing a group which is a priori a common extension.) Let the group Γ be defined by the presentationa, b, x, y, z[a, x], [b, y], [ab, z], [x, y], [x, z], [y, z], [ax, by], [ax, abz], [by, abz].The group Γ is an extension of G_m, n, with the quotient map sending a ↦ a and b ↦ b.As m and n are coprime, there exist integers p and q such that pm - qn = 1, that is, pm = qn + 1. Define a map Γ→ G_m, n by a ↦ a, b ↦ b, x ↦ a^qn, y ↦ b^qn and z ↦ (ab)^qn. This is easily checked to be well-defined, as every defining relator for Γ is mapped to a relator of the form [u^k, v^l] for some u and v with [u, v] a defining relator of G_m, n and k, l ∈. The subgroup a,b≤Γ is nilpotent of class 2. The group Γ itself is nilpotent of class 2, with [Γ, Γ] ≅. However, we confine ourselves here to proving Proposition <ref>, which is all that is required for Theorem <ref>. We prove Proposition <ref> in a sequence of lemmas. It will be convenient to know that the symmetry in a and b of G_m, n extends to Γ.Let φ : a ↦ b ↦ a, x ↦ y ↦ x, z ↦ z^a. Then φ is an automorphism of Γ.Since the above also defines an automorphism of the free group F(a,b,x,y,z), it suffices to check that φ is a well-defined group homomorphism. To verify this we now show that the images of the relators are trivial, in the cases where this is not immediate. Note that since [ab, z] = 1, we have z^a = z^b^-1.φ([ab, z])= [ba, z^a] = [ab, z]^a = 1φ([x,z])= [y, z^a]= [y, z^b^-1] = [y, z]^b^-1 = 1φ([y, z]) = [x, z^a]= [x, z]^a = 1φ([ax, abz])= [by, baz^a] = [by, bza] = [by, zab]^b^-1 = [by, abz]^b^-1 = 1φ([by, abz])= [ax, baz^a] =[ax, bza] = [ax, abz]^a = 1The commutator [ab, yx] = 1 in the group Γ.In light of Lemma <ref>, we can instead prove [ba, xy]=1 as follows:[ axybabz = abxz(ab)y; = (ax)(by)(abz) = abx(ab)zy; = (abz)(ax)(by) = abaxbyz; = abzxaby= abaxybz. ]After cancelling on the left and right, we have xyba = baxy. The commutator [a, bzy] = 1 in the group Γ.We have[a(bzy) = by(ab)zb^-1; = (abz)(by)b^-1 = byz(ab)b^-1;= (by)(abz) b^-1 = (bzy)a. ]The commutator [b, zax] = 1 in the group Γ.This follows from Lemma <ref> by symmetry, as we have φ([b, zax]) = [a, z^a b y] = [a, z^(ab)^-1 a b y] = [a, z^b^-1 b y] = [a, b z y] = 1. In the following computations, we will frequently use the basic fact that if [g, hk] = 1 then g^h = g^k^-1.The commutator [b, z^-1x] = 1 in the group Γ.Note first that since abz and a both commute with ax, we have [ax, bz] = 1 and thus (ax)^b = (ax)^z^-1. Now[ abxb^-1= (ab)(xy)y^-1b^-1= (ax)^b; = (xy)(ab)y^-1b^-1Lemma <ref> = (ax)^z^-1; = yxay^-1= (ab)^z^-1 (b^-1x)^z^-1; = b^-1 (by)(ax)y^-1= ab z b^-1 x z^-1; = b^-1 (ax)(by)y^-1 = ab z b^-1 z^-1 x. ]Left-multiplying both sides by z^-1 b^-1 a^-1 gives z^-1 x b^-1 = b^-1 z^-1 x. The commutator [b, [a,z]] = 1 in the group Γ.By Lemma <ref> we have zax ∈ C_Γ(b), the centralizer of b, and by Lemma <ref> we have z^-1x ∈ C_Γ(b). The centralizer thus contains z^-1 x z a x = xax, and thus also[xax, x^-1 z] = [a, z]since x is central in a, x, z. We are now equipped to prove Proposition <ref>.Starting by applying Lemma <ref>, we see[ ab= (yx) ab (yx)^-1 = a^bz b^za Lemmas <ref> and <ref>; = y x a y^-1 b x^-1= (a^bz b^za)^z^-1 LHS ab commutes with z; = y a y^-1 x b x^-1= a^b b^zaz^-1; = a^y^-1 b^x^-1 = a^b b^aLemma <ref> ]Thus b^a = (a^b)^-1 ab = b^-1 a^-1 b a b = b^-1 b^a b. Since b commutes with b^a, it commutes with b^-1 b^a = [b, a] = [a, b]^-1. Applying φ, we see that also [a, [a, b]] = 1. Thus the subgroup a,b≤Γ is nilpotent of class 2.As [a,b] is central in a,b≤Γ (Proposition <ref>) and Γ is an extension of G_m,n (Lemma <ref>), it follows that [a,b] is central in G_m,n. Thus [a^m, b^m] = [a, b]^m^2 and [a^n, b^n] = [a, b]^n^2. Since m and n are coprime, so are m^2 and n^2. Now coprime powers of [a,b] are both trivial, so [a,b] = 1. It is not sufficient to take 5 of the 6 relators of G_m, n (for coprime m, n > 1). Suppose that we omitted [b^n, (ab)^n] (the other cases are analogous). A folklore theorem states that for all integers p, q, r > 1 one can find a finite group containing elements a and b such that a, b, ab have orders p, q, r respectively (for a proof, see <cit.>). Such a group for (p,q,r) = (n, m, m) will be a quotient of the group defined by our truncated presentation, as all defining commutativity relators hold trivially by virtue of one term having trivial image. It cannot be abelian, as otherwise the order of ab would be mn. § OPEN PROBLEMSWhile we have a surprisingly and uniformly small core (subpresentation) ofin the abelian case, we still do not know what is the minimum number of relators needed.For which coprime m and n is there a 4-relator presentation for × that encodes a proof that _m ∩_n =, that is, when does the infinite presentation ^{[a,b]}_2,m,n have a 4-relator core? In particular, when isΔ_m,n =a, b[a^m, b^m], [a^n, b^n], [(ab)^m, (ba)^m], [(ab)^n, (ba)^n]isomorphic to × ? The van Kampen diagram in Figure <ref> proves that the question has a positive answer for (m,n) = (2,3). A van Kampen diagram per se is purely topological; the geometry of the drawing has been chosen such that corners generally delimit the 4 subwords u^-1, v^-1, u and v in a commutator [u,v].Computational experiments using<cit.> and<cit.> provided some evidence for Question <ref> having a positive answer. In particular, we have verified this for m = 2 and odd n < 50. However, we could not answer the question either way for (m,n) = (3,4). A necessary condition is for the four relators to generate the full relation module, that is, the abelianization of the kernel F(a,b)' of the full presentation, as a [×]-module. (This is actually a cyclic module.) This condition is met for (m,n) = (3,4); in fact, it is met for all m ≤ 80 and n = m+1.Note that the argument of Remark <ref> cannot be used to show that the group Δ_m,n has a non-abelian finite quotient: if we ask that (ab)^m = 1 and (ba)^n = 1, then since ab and ba are conjugate we in fact have ab = 1. Any possible non-abelian finite quotient is quite constrained; in particular, if the order of ab is coprime to either m or n then we immediately have [ab, ba] = 1, and similarly if a and b both have order coprime to m then [a, b] = 1. Meanwhile, we have determined computationally for the (m,n) = (3,4) case that if a and b have order dividing 24 then such a quotient is abelian. This computation was performed using Holt's package<cit.> to show that the commutator subgroup of Δ_3,4 / a^24, b^24, an a priori finitely presented group, is trivial. Determine the analogous complexity for the nilpotency lawν_c = [[[…[x_1, x_2], x_3], …, x_c], x_c+1].That is, how does the size of the smallest finite core of ^{ν_c}_c+1,m,n vary with c, m and n? It seems that the following classification problem would require substantial progress.Which laws are detectable in power subgroups?The difficulty is exemplified by the fact that the 4-Engel law is detectable, but it has been claimed that not all k-Engel laws imply the essential local nilpotency that we used. We thus ask in particular:For which k is the k-Engel law detectable in power subgroups.To summarize our knowledge at this time, x^m is detectable in power subgroups, as is every locally nilpotent law (for example, the 4-Engel law [x, y, y, y, y] or a nilpotency law such as [[x_1, x_2], x_3]). On the other hand, [[x_1, x_2], [x_3, x_4]] is not detectable, and neither are some assorted laws for which detectability also fails in finite groups, such as [[x^2, y^2]^3, y^3] and [x^2, x^y]. §.§ AcknowledgementsI thank my supervisor Martin Bridson for guidance, encouragement, and helpful suggestions; Nikolay Nikolov for suggesting the problem of detectability of Engel laws; Peter Neumann for helpful conversations; Geetha Venkataraman for helpful correspondence; Robert Kropholler for helpful comments on a draft; and my officemates L. Alexander Betts and Claudio Llosa Isenrich for helpful conversations.../transfer/halpha | http://arxiv.org/abs/1705.09348v2 | {
"authors": [
"Giles Gardam"
],
"categories": [
"math.GR",
"20E10 (Primary), 20F19, 20F45, 20F05 (Secondary)"
],
"primary_category": "math.GR",
"published": "20170525202020",
"title": "Detecting laws in power subgroups"
} |
Low-energy spin dynamics of orthoferrites AFeO_3 (A = Y, La, Bi) Kisoo Park^1,2, Hasung Sim^1,2, Jonathan C. Leiner^1,2, Yoshiyuki Yoshida^3, Jaehong Jeong^1,2,4, Shin-ichiro Yano^5, Jason Gardner^5,6, Philippe Bourges^4, Milan Klicpera^7,8, Vladimír Sechovský^7, Martin Boehm^8 and Je-Geun Park^1,2 December 30, 2023 ============================================================================================================================================================================================================================================== Given a text T and a pattern P over alphabet Σ, the classic exact matching problem searches for all occurrences of pattern P in text T. Unlike exact matching problem, order-preserving pattern matching (OPPM) considers the relative order of elements, rather than their real values. In this paper, we propose an efficient algorithm for OPPM problem using the “duel-and-sweep” paradigm. Our algorithm runs in O(n + mlog m) time in general and O(n + m) time under an assumption that the characters in a string can be sorted in linear time with respect to the string size. We also perform experiments and show that our algorithm is faster that KMP-based algorithm. Last, we introduce the two-dimensional order preserved pattern matching and give a duel and sweep algorithm that runs in O(n^2) time for duel stage and O(n^2 m) time for sweeping time with O(m^3) preprocessing time.§ INTRODUCTION The exact string matching problem is one of the most widely studied problems.Given a text and a pattern, the exact matching problem searches for all occurrences positions of pattern in the text. Motivated by low level image processing, the two-dimensional exact matching problem has been extensively studied in recent decades. Given a text T of size n × n and a pattern P of size m × m over alphabet Σ of size σ = |Σ|, the exact matching problem on two-dimensional strings searches for all occurrence positions of P in T. Bird <cit.> and Baker <cit.> proposed two-dimensional exact matching using dictionary matching algorithm and Amir and Farach <cit.> proposed an algorithm that uses suffix trees. These algorithms require total ordering from the alphabet and run in O (n ^ 2 logσ) time with O (m ^ 2 logσ) preprocessing time. Amir <cit.> also proposed alphabet independent approach to the problem that runs in O(m^2 logσ) preprocessing time and O(n^2) matching time.Unlike the exact matching problem, order-preserving pattern matching (OPPM) considers the relative order of elements, rather than their real values. Order-preserving matching has gained much interest in recent years, due to its applicability in problems where the relative order is compared, rather than the exact value, such as share prices in stock markets, weather data or musical notes.Kubica <cit.> and Kim <cit.> proposed a solution based on KMP algorithm. These algorithms address the one-dimensional OPPM problem and have time complexity of O (n + m log m). Cho <cit.> brought forward another algorithm based on the Horspool's algorithm that uses q-grams, which was proven to be experimentally fast. Crochemore <cit.> proposed data structures for OPPM. On the other hand, Chhabra and Tarhio <cit.>, Faro and Külekci <cit.> proposed filtration methods which practically fast. Moreover, faster filtration algorithms by using SIMD (Single Instruction Multiple Data) instructions were proposed by Cantone <cit.>, Chhabra <cit.> and Ueki <cit.>. They showed that SIMD instructions are efficient in speeding up their algorithms.In this paper, we propose an algorithm that based on dueling technique <cit.> for OPPM. Our algorithm runs in O(n + mlog m) time which is as fast as KMP based algorithm. Moreover, we perform experiments those compare the performance of our algorithm with the KMP-based algorithm. The experiment results show that our algorithm is faster that KMP-based algorithm. Last, we introduce the two-dimensional order preserved pattern matching and give a duel and sweep algorithm that runs in O(n^2) time for duel stage and O(n^2 m) time for sweeping time with O(m^3) preprocessing time. To the best of our knowledge, our solution is the first to address the two-dimensional order preserving patern matching problem.The rest of the paper is organized as follows. In Section <ref>, we give preliminaries on the problem. In Section <ref>, we describe the algorithm for OPPM problem. In Section <ref> we will show some experiment results those compare the performance of our algorithm with the KMP-based algorithm. In Section <ref>, we extend the algorithm and describe the method for the two-dimensional OPPM problem. In Section <ref>, we conclude our work and discuss future work. § PRELIMINARIESWe use Σ to denote an alphabet of integer symbols such that the comparison of any two symbols can be done in constant time. Σ^* denotes the set of strings over the alphabet Σ. For a string S ∈Σ^*, we will denote i-th element of S by S[i] and a substring of S that starts at the location i and ends at the location j as S[i:j].We say that two strings S and T of equal length n are order-isomorphic, written S ≈ T, if S[i] ≤ S[j] ⟺ T[i] ≤ T[j] for all 1 ≤ i, j ≤ n. For instance, (12, 35, 5) ≈ (25, 30, 21) ≉(11, 13, 20).In order to check order-isomorphism of two strings, Kubica <cit.> introduced [Similar arrays Prev_S and Next_S are introduced in <cit.>.] useful arrays S and S defined byS[i]=jS[j]=max_k < i{S[k] |S[k] ≤ S[i] },S[i]=jS[j]=min_k < i{S[k] |S[k] ≥ S[i] }.We use the rightmost (largest) j if there exist more than one such j.If there is no such j then we define S[i] = 0 and S[i] = 0, respectively. From the definition, we can easily observe the following properties.S[S[i]] = S[i]⟺ S[i] = S[s[i]], S[S[i]] < S[i]⟺ S[i] < S[s[i]]. For a string S, let sort(S) be the time required to sort the elements of S. S and S can be computed in O(sort(S) + |S|) time.Thus, S and S can be computed in O(|S|log |S|) time in general. Moreover, the computation can be done in O(|S|) timeunder a natural assumption <cit.> that the characters of S are elements of the set {1,…,|S|^O(1)}.By using S and S, order-isomorphism of two strings can be decided as follow. For two strings S and T of length n, assume that S[1:i] ≈ T[1:i] for some i < n. Let = S[i+1] and = S[i+1]. Then S[1:i+1] ≈ T[1:i+1] if and only if either of the following two conditions holds. S[] = S[i+1] = S[] ∧ T[] = T[i+1] = T[], S[] < S[i+1] < S[] ∧ T[] < T[i+1] < T[]. We omit the corresponding equalities/inequalities if =0 or =0. Hasan <cit.> proposed a modification to Z-function, which Gusfield <cit.> defined for ordinal pattern matching, to make it useful from the order-preserving point of view.For a string S, the (modified) Z-array of S is defined byZ_S[i] = max_1 ≤ j ≤ |S| - i + 1{j | S[1:j]≈ S[i:i+j-1] } 1 ≤ i ≤ |S|.In other words, Z_S[i] is the length of the longest substring of S that starts at position i and is order-isomorphic with some prefix of S. An example of Z-array is illustrated in Table <ref>. (<cit.>) For a string S, Z-array Z_S can be computed in O(|S|) time, assuming that S and S are already computed. Note that in their original work, Hasan <cit.> assumed that each character in S is distinct. However, we can extend their algorithm by using Lemma <ref> to verify order-isomorphism even when S contains duplicate characters.§ ONE-DIMENSIONAL ORDER-PRESERVING MATCHINGIn this section, we will propose an algorithm for one-dimensional OPPM using the “duel-and-sweep" paradigm <cit.>.In the dueling stage, all possible pairs of candidates “duel” with each other. The surviving candidates are further pruned during the sweeping stage, leaving the candidates that are order-isomorphic with the pattern.Prior to the dueling stage, the pattern is preprocessed to construct a witness table that contains witness pairs for all possible offsets. The one-dimensional order-preserving matching problem is defined as follows, Input: A text T ∈Σ^* of length n and a pattern P ∈Σ^* of length m, Output: All occurrences of substrings of T that are order-isomorphic with P. §.§ Pattern preprocessing Let a > 0 be an integer such that when P is superimposed on itself with the offset a, the overlap regions are not order-isomorphic. We say that a pair ij of locations isa witness pair for the offset a if either of the following holds: * P[i] = P[j]andP[i+a]P[j+a], * P[i] > P[j]andP[i+a] ≤ P[j+a], * P[i] < P[j]andP[i+a] ≥ P[j+a].Next, we describe how to construct a witness table for P, that stores witness pairs for all possible offsets a (0 < a < m). For the one-dimensional problem, the witness table P is an array of length m-1, such that P[a] is a witness pair for offset a. In the case when there are multiple witness pairs for offset a, we take the pair ij with the smallest value of j and i < j. When the overlap regions are order-isomorphic for offset a, which implies that no witness pair exists for a, we express it as P[a] = m+1m+1. For a pattern P of length m, we can construct P in O(m) time assuming that Z_P is already computed.Remind that Z_P[k] is the length of the longest prefix of P[k:m] that is order-isomorphic with a prefix of P.For each 1 < k < m, we have two cases. Case 1 Z_P[k] = m - k + 1 : Since P[1:m - k + 1] ≈ P[k:m], there is no witness pair for offset k - 1. Case 2 Z_P[k] < m - k + 1 : Let j_k = Z_P[k] + 1, = P[j_k], and = P[j_k]. Then P[1:j_k-1]≈ P[k:k+j_k-2] and P[1:j_k]≉P[k:k+j_k-1], by the definition of Z_P[k]. By Lemma <ref>, neither condition (<ref>) nor (<ref>) holds. If P[] = P[j_k] then P[j_k] = P[] by property (<ref>), so that P[k +- 1]P[k+j_k-1] ∨ P[k+j_k-1]P[k +- 1] holds by condition (<ref>). Otherwise, i.e. P[] < P[j_k], we have P[j_k] < P[] by property (<ref>), so that P[k +- 1] ≥ P[k+j_k-1] ∨ P[k+j_k-1] ≥ P[k +- 1] holds by condition (<ref>). Therefore, j_k is a witness pair if the leftside of condition (<ref>) or (<ref>) holds, and j_k is a witness pair if rightside of condition (<ref>) or (<ref>) holds. Algorithm <ref> describes the procedure. Clearly it runs in O(m) time.§.§ Dueling stage A substring of T of length m will be referred to as a candidate. A candidate that starts at the location x will be denoted by T_x. Witness pairs are useful in the following situation.Let T_x and T_x+a be two overlapping candidates and ij be the witness pair for offset a. Without loss of generality, we assume that P[i] < P[j] and P[i+a] > P[j + a]. * If T[x + a + i -1] > T[x + a + j -1], then T_x ≉P. * If T[x + a + i -1] < T[x + a + j -1], then T_x+a≉P.Based on this information, we can safely eliminate either candidate T_x or T_x+a without looking into other locations. This process is called dueling. The procedure for the dueling is described in the Algorithm <ref>.[tbp] (*[h]Construct the witness table P) compute the Z-array Z_P for the pattern P k=2 m-1 j = Z_P[k] + 1 j = m - k + 1 P[k - 1] = m + 1m + 1 P[P[j]] = P[j] = P[P[j]] P[k+j-1]P[k + P[j] - 1] P[k-1] = P[j]j P[k-1] = P[j]j P[k+j-1] ≤ P[k + P[j] - 1] P[k-1] = P[j]j P[k-1] = P[j]j Algorithm for constructing the witness table P Next, we prove that the consistency property is transitive. Suppose T_x and T_x+a are two overlapping candidates. We say that T_x and T_x+a are consistent with respect to P if P[1:m-a] ≈ P[a+1:m]. Candidates that do not overlap are trivially consistent. For any a and a' such that 0 < a < a + a' < m, let us consider three candidates T_x, T_x + a, and T_x + a + a'. If T_x is consistent with T_x+a and T_x+a is consistent with T_x + a + a', then T_x is consistent with T_x + a+ a'. Since T_x is consistent with T_x+a, it follows that P[1:m - a] ≈ P[a+1:m], so that P[a' + 1:m - a] ≈ P[(a + a')+1:m]. Moreover, since T_x+a is consistent with T_x+a +a', it follows that P[1:m - a'] ≈ P[a'+1:m], so that P[1:m - a' - a] ≈ P[a'+1:m - a]. Thus, P[1:m - (a + a')] ≈ P[(a + a') + 1:m], which implies that T_x is consistent with T_x+a+a'. During the dueling stage, the candidates are eliminated until all remaining candidates are pairwise consistent. For that purpose, we can apply the dueling algorithm due to Amir <cit.> developed for ordinal pattern matching. The dueling stage can be done in O(n^2) time by using . [t] (*[h]Duel between candidates T_x and T_x+a)T_x, T_x + a ij = P[a] P[i] = P[j] T[x + a + i -1]T[x + a + j -1] return T_x+a return T_x P[i] < P[j] T[x + a + i -1] > T[x + a + j -1] return T_x+a return T_x P[i] > P[j] T[x + a + i -1] < T[x + a + j -1] return T_x+a return T_x Dueling §.§ Sweeping stageThe goal of the sweeping stage is to prune candidates until all remaining candidates are order-isomorphic with the pattern. Suppose that we need to check whether some surviving candidate T_x is order-isomorphic with the pattern P. It suffices to successively check the conditions (<ref>) and (<ref>) in Lemma <ref>, starting from the leftmost location in T_x. If the conditions are satisfied for all locations in T_x,then T_x ≈ P. Otherwise, T_x ≉P, and obtain a mismatch position j.A naive implementation of the sweeping will result in O(n^2) time. However, if we take advantage of the fact that all the remaining candidates are pairwise consistent, we can reduce the time complexity to O(n) time. Since the remaining candidates are consistent to each other, for the overlapping candidates T_x and T_x + a, the overlap region is checked only once if T_x is order-isomorphic with the pattern P. Otherwise, for a mismatch position j, T_x+a should be checked from position j - a + 1 of T_x+a, because P[a:j-1] ≈ T_x[a:j-1] ≈ T_x+a[1:j-a]. Algorithm <ref> describes the procedure for the sweeping stage.[t] there are unchecked candidates to the right of T_x let T_x be the leftmost unchecked candidate there are no candidates overlapping with T_x T_x ≉P eliminate T_x let T_x+a be the leftmost candidate that overlaps with T_x T_x ≈ P start checking T_x+a from the location m - a + 1 let j be the mismatch position eliminate T_x start checking T_x+a from the location j-a The sweeping stage algorithm The sweeping stage can be completed in O(n) time. By Lemmas <ref>, <ref>, and <ref>, we summarize this section as follows. The duel-and-sweep algorithm solves 1d-OPPM Problem in O(n + mlog m) time. Moreover, the running time is O(n+m) under the natural assumption that the characters of P can be sorted in O(m) time. § EXPERIMENTIn order to compare the performance of proposed algorithm with the KMP-based algorithm, we conducted experiments on 1d-OPPM problem. We performed two sets of experiments. In the first experiment, the pattern size m is fixed to 10, while the text size n is changed from 100000 to 1000000. In the second experiment, the text size n is fixed to 1000000 while the pattern size m is changed from m 5 to 100. We measured the average of running time and the number of comparisons for 50 repetitions on each experiment. We used randomly generated texts and patterns with alphabet size |Σ|=1000. Experiments are executed on a machine with Intel Xeon CPU E5-2609 8 cores 2.40 GHz, 256 GB memory, and Debian Wheezy operating system.The results of our preliminary experiments are shown in Fig. <ref> and Fig. <ref>. We can see that our algorithm is better that KMP based algorithm in running time and number of comparison when the pattern size and text size are large. However, our algorithm is worse when the pattern size is small, less than 10. § TWO-DIMENSIONAL ORDER PRESERVING PATTERN MATCHINGIn this section, we will discuss how to perform two-dimensional order preserving pattern matching (2d-OPPM). Array indexing is used for two-dimensional strings, the horizontal coordinate x increases from left to right and the vertical coordinate y increases from top to bottom. [x,y] denotes an element ofat position (x,y) and [x:x + w - 1 , y:y + h - 1] denotes a substring ofof size w × h with top-left corner at the position (x, y).We say that two dimensional stringsandare order-isomorphic, written ≈, if [i_x,i_y] ≤[j_x,j_y] ⟺[i_x,i_y] ≤[j_x,j_y] for all 1 ≤ i_x, j_x ≤ w and 1 ≤ i_y, j_y ≤ h. For a simple presentation, we assume that both text and pattern are squares (w=h) in this paper, but we can generalize it straightforwardly. The two-dimensional order-preserving matching problem is defined as follows, Input: A textof size n × n and a patternof size m × m, Output: All occurrences of substrings ofthat are order-isomorphic with . Our approach is to reduce 2d-OPPM problem into 1d-OPPM problem, based on the following observation. For two-dimensional string , letbe a (one-dimensional) string which serializingby traversing it in the left-to-right/top-to-bottom order. We can easily verify the following lemma. ≈ if and only if ≈ for anyand . 2d-OPPM problem can be solved in O(n^2 m + m^2 logm). For a fixed 1 ≤ x ≤ n -m + 1, consider the substring [x:x+m-1, 1:n] and let S_x = [x:x+m-1, 1:n]. By Lemma <ref>,occurs inat position (x, y), i.e. ≈[x:x+m-1, y:y+m-1] if and only if ≈ S_x[m(y-1)+1, m(y-1)+m^2]. The positions m(y-1)+1 satisfying the latter condition can be found in O(nm + m^2logm) time by 1d-OPPM algorithms, which we showed in Section <ref> or KMP-based ones <cit.>, because |S_x| = nm and ||=m^2. Because we need the preprocess for the patternonly once, and execute the search in S_x for each x, the result follows.In the rest of this paper, we try a direct approach to two-dimensional strings based on the duel-and-sweep paradigm, inspired by the work <cit.>. A substring ofof size m × m will be referred as a candidate. _x,y denotes a candidate with the top-left corner at (x,y).§.§ Pattern preprocessingFor 0 ≤ a < m and -m < b < m, we say that a pair (i_x, i_y)(j_x, j_y) of locations is a witness pair for the offset (a, b) if either of the following holds: * [i_x, i_y] = [j]and [i_x+a, i_y+b] [j_x, j_y], * [i_x, i_y] > [j]and [i_x+a, i_y+b] ≤[j_x, j_y], * [i_x, i_y] < [j]and [i_x+a, i_y+b] ≥[j_x, j_y].The witness tablefor patternis a two-dimensional array of size m × (2m-1), where [a, b] is a witness pair for the offset (a, b).If the overlap regions are order-isomorphic whenis superimposed with offset (a, b), then no witness pair exists. We denote it as [a,b] = (m+1, m+1)(m+1, m+1).We show how to efficiently construct the witness table . Forand each 0 ≤ a < m, we define the Z-array a by a[i] = max_1 ≤ j ≤ |P_1| - i + 1{j | P_1[1:j]≈ P_2[i:i+j-1] } 1 ≤ i ≤ |P_1|,where P_1 = [1:m-a, 1:m], P_2 = [a + 1:m, 1:m], and |P_1| = |P_2| = m(m-a). For arbitrarily fixed a ≥ 0, we can compute the value of [a,b] in O(1) time and for each b, assuming that a is already computed. For an offset (a, b) with b ≥ 0, let us consider z_a,b = a[b · (m - a) + 1]. Case 1 z_a,b = (m-a)·(m-b): Note that the value is equal to the number of elements in the overlap region. Then [1:m-a,1:m-b] ≈[a+1:m,b+1:m], so that no witness pair exists for the offset (a,b). Case 2 z_a,b < (m-a)·(m-b): There exists a witness pair (i_x, i_y)(j_x, j_y), where (j_x, j_y) is the location of the element in , that corresponds to the (z_a,b + 1)-th element of P_1 = [1:m-a, 1:m]. By a simple calculation, we can obtain the values (j_x, j_y) in O(1) time. We can also compute (i_x, i_y) from (j_x, j_y) in O(1) time, similarly to the proof of Lemma <ref>, with the help of auxiliary arrays ,a and ,a.(Details are omitted.)Symmetrically, we can compute it for b<0. We can construct the witness tablein O(m^3) time. Assume that we sorted all elements of . For an arbitrarily fixed a, calculation of ,a and ,a takes O(m^2) time by using sorted . a can be constructed in O(m^2) time by Lemma <ref>. Furthermore, finding witness pairs for all offsets (a, b) takes O(m) time by Lemma <ref>. Since there are m such a's to consider,can be constructed in O(m^3) time.§.§ Dueling stage Similarly to Lemma <ref>, we can show the transitivity as follows. For any a , b, a', b' ≥ 0, let us consider three candidates _1 = _x,y, _2 = _x + a, y + b, and _3 = _x + a', y + b'. If _1 is consistent with _2 and _2 is consistent with _3, then _1 is consistent with _3.The dueling algorithm due to Amir <cit.> is also applicable to the problem.(<cit.>) The dueling stage can be done in O(n^2) time by using .§.§ Sweeping stage This is the hardest part for two-dimensional strings. We first consider two surviving candidates _x, y_1 and _x,y_2 in some column x, with y_1 < y_2. If we traverse [x:x+m-1, 1:n] from top-to-bottom/left-to-right manner we can reduce the problem to one-dimensional order-preserving problem. Thus performing the sweeping stage for some column x will take O(nm) time. Since there are n - m -1 such columns, the sweeping stage will take O(n^2m) time.Next, we propose a method that takes advantage of consistency relation in both horizontal and vertical directions. First, we construct m strings P_i = [1:m-i,1:m][m-i+1:m,1:m] for 0 ≤ i < m by serializingin different way.We then compute P_i and P_i for 0 ≤ i < m, thus we can compare the order-isomorphism of the pattern with the text in several different ways. P_i and P_i for 0 ≤ i < m can be computed in O(n^3) time by sortingonce and then calculated P_i and P_i by using the sorted . Fig. <ref> shows P_i for 0 ≤ i < m where m=5. We also do the same computation for bottom-to-top/left-to-right traversing direction.Let us consider two overlapping candidates _x_1, y_1 and _x_2, y_2, where x_1 < x_2 and y_1 < y_2.Suppose that _x_1, y_1 is order-isomorphic with the pattern and we need to check _x_2, y_2. Since _x_1, y_1 is consistent with _x_2, y_2, we need to check the order-isomorphishm of the region of _x_2, y_2 that is not an overlap region.We do this by using P_j, where j= x_2-x_1, without checking the overlap region. This idea is illustrated in Figure <ref> (a). The procedure for y_1 > y_2 is symmetrical.Next, consider three overlapping candidates _1 = _x_1,y_1, _2 = _x_2,y_2 and _3 = _x_3,y_3, such that x_1 ≤ x_2 ≤ x_3 and y_2 ≤ y_3. We assume that _1 and _2 are both order-isomorphic with the pattern. If y_1 ≤ y_2, we can use the method for two overlapping candidates that we described before to perform sweeping efficiently. However, if y_1 ≥ y_2, as showed in Fig. <ref> (b), we need to check the blue region twice since we do not know the order-isomorphism relation between the blue region with the overlap region of _2 and _3.By using the above method, we can reduce the number of comparisons for sweep stage. However, the time complexity remains the same. The sweeping stage can be completed in O(n^2m) time. By Lemmas <ref>, <ref>, and <ref>, we conclude this section as follows. The duel-and-sweep algorithm solves 2d-OPPM Problem in O(n^2 m + m^3) time. § DISCUSSION In the current status, the time complexity of duel-and-sweep algorithm for 2d-OPPM problem in Theorem <ref> is not better than straightforward reduction to 1d-OPPM problem explained in Theorem <ref>. We showed this result as a preliminary work on solving 2d-OPPM, and we hope the 2d-OPPM can be solved more efficiently by finding more sophisticated method based on some unknown combinatorial properties, as Cole <cit.> did for two dimensional parameterized matching problem.This is left for future work.abbrv | http://arxiv.org/abs/1705.09438v1 | {
"authors": [
"Davaajav Jargalsaikhan",
"Diptarama",
"Ryo Yoshinaka",
"Ayumi Shinohara"
],
"categories": [
"cs.DS"
],
"primary_category": "cs.DS",
"published": "20170526054054",
"title": "Duel and sweep algorithm for order-preserving pattern matching"
} |
Discriminative Metric Learning with Deep Forest Lev V. Utkin^1 and Mikhail A. Ryabinin^2 Department of Telematics Peter the Great St.Petersburg Polytechnic University St.Petersburg, Russia e-mail: ^[email protected], ^[email protected]=========================================================================================================================================================================================================== A Discriminative Deep Forest (DisDF) as a metric learning algorithm is proposed in the paper. It is based on the Deep Forest or gcForest proposed by Zhou and Feng and can be viewed as a gcForest modification. The case of the fully supervised learning is studied when the class labels of individual training examples are known. The main idea underlying the algorithm is to assign weights to decision trees in random forest in order to reduce distances between objects from the same class and to increase them between objects from different classes. The weights are training parameters. A specific objective function which combines Euclidean and Manhattan distances and simplifies the optimization problem for training the DisDF is proposed. The numerical experiments illustrate the proposed distance metric algorithm.Keywords: classification, random forest, decision tree, deep learning, metric learning, quadratic programming§ INTRODUCTION Real-world data usually has a high-dimensionality and a non-linear complex structure. In order to improve the performance of many classification algorithms especially those relying on distance computations (nearest neighbor classifiers, support vector machines, etc.) the distance metric learning methods are widely applied. Bellet et al. <cit.> selected three main groups of distance metric learning methods, which are defined by conditions of the available information about class labels of training examples. The first group consists of fully supervised algorithms for which we have a set of labeled training examples S={(𝐱_i,y_i ), i=1,...,n} such that every example has a label of a class y_i ∈𝒴, i.e., the class labels of individual training examples are known The second group consists of weakly supervised algorithms. This group is characterized by the lack of class labels for training every example, but there is a side information in the form of constraints corresponding to the semantic similarity or dissimilarity of pairs of training examples. This can be seen as having label information only at the pair level. The third group consists of semi-supervised algorithms for which there is a part of data that are labeled or belong to similarity constraints, but another part consists of fully unlabeled data.We study the first group of algorithms in this paper. The main goal of fully supervised distance metric learning is to use discriminative information to keep all the data samples in the same class close and those from different classes separated <cit.>. As indicated by Yang and Jin <cit.>, unlike most supervised learning algorithms where training examples are given class labels, the training examples of supervised distance metric learning is cast into pairwise constraints: the equivalence constraints where pairs of data points that belong to the same classes, and inequivalence constraints where pairs of data points belong to different classes.Metric learning approaches were reviewed in <cit.>. The basic idea underlying the metric learning solution is that the distance between similar objects should be smaller than the distance between different objects. If we have two observation vectors 𝐱_i∈ℝ^m and 𝐱_j∈ℝ^m from a training set, and the similarity of objects is defined by their belonging to the same class, then the distance d(𝐱_i,𝐱_j) between the vectors should be minimized if 𝐱_i and 𝐱_j belong to the same class, and it should be maximized if 𝐱_i and 𝐱_j are from different classes. Several review papers analyze various methods and algorithms of metric learning <cit.>. A powerful implementation of the metric learning dealing with non-linear data structures is the so-called Siamese neural network introduced by Bromley et al. <cit.> in order to solve signature verification as a problem of image matching. This network consists of two identical sub-networks joined at their outputs. The two sub-networks extract features from two input examples during training, while the joining neuron measures the distance between the two feature vectors. The Siamese architecture has been exploited in many applications, for example, in face verification <cit.>, in the one-shot learning in which predictions are made given only a single example of each new class <cit.>, in constructing an inertial gesture classification <cit.>, in deep learning <cit.>, in extracting speaker-specific information <cit.>, for face verification in the wild <cit.>. This is only a part of successful applications of Siamese neural networks. Many modifications of Siamese networks have been developed, including fully-convolutional Siamese networks <cit.>, Siamese networks combined with a gradient boosting classifier <cit.>, Siamese networks with the triangular similarity metric <cit.>.A new powerful method, which can be viewed as an alternative to deep neural networks, is the deep forest proposed by Zhou and Feng <cit.> and called the gcForest. It can be compared with a multi-layer neural network structure, but each layer in the gcForest contains many random forests instead of neurons. The gsForest can be regarded as an multi-layer ensemble of decision tree ensembles. Zhou and Feng <cit.> point out that their approach is highly competitive to deep neural networks. In contrast to deep neural networks which require great effort in hyperparameter tuning and large-scale training data, gcForest is much easier to train and can perfectly work when there are only small-scale training data. The deep forest solves the tasks of classification and regression. Therefore, by taking into account its advantages, it is important to modify it in order to develop a structure solving the metric learning task. We propose the so-called Discriminative Deep Forest (DisDF) which is a discriminative distance metric learning algorithm. It is based on the gcForest proposed by Zhou and Feng <cit.> and can be viewed as its modification. The main idea underlying the proposed DisDF is to assign weights to decision trees in random forest in order reduce distances between pairs of examples from the same class and to increase them between pairs of examples from different classes. We define the class distributions in the deep forest as the weighted sum of the tree class probabilities where the weights are determined by solving an optimization problem with the contrastive loss function as an objective function. The weights are viewed as training parameters. We also apply the greedy algorithm for training the DisDF, i.e., the weights are successively computed for every layer or level of the forest cascade. In order to efficiently find optimal weights, we propose to modify the standard contrastive loss in a way that makes the loss function to be convex with respect to the weights. This modification is carried out by combining Euclidean and Manhattan distances in the loss function. Moreover, we reduce the optimization problem for computing the weights to the standard convex quadratic optimization problem with linear constraints whose solution does not meet difficulties. For large-scale data, we apply the well-known Frank-Wolfe algorithm <cit.> which is very simple when the feasible set of weights is the unit simplex.The paper can be viewed as an extension of the results obtained by Utkin and Ryabinin <cit.> where the Siamese Deep Forest has been proposed. The main difference of the presented paper from <cit.> is that the case of the weakly supervised learning was used in the Siamese Deep Forest when there are no information about the class labels of individual training examples, but only information in the form of sets of semantically similar pairs is available. Now we study the case of the fully supervised learning when the class labels of individual training examples are known.The paper is organized as follows. Section 2 gives a very short introduction into the gcForest proposed by Zhou and Feng <cit.>. The idea to assign weights to trees in random forests, which allows us to construct the DisDF, is considered in Section 3 in detail. Algorithms for training and testing the DisDF are considered in Section 4. Section 5 provides algorithms for dealing with large-scale training data. Numerical experiments with real data illustrating the proposed DisDF are given in Section 6. Concluding remarks are provided in Section 7.§ DEEP FOREST According to <cit.>, the gcForest generates a deep forest ensemble, with a cascade structure. Representation learning in deep neural networks mostly relies on the layer-by-layer processing of raw features. The gcForest representational learning ability can be further enhanced by the so-called multi-grained scanning. Each level of cascade structure receives feature information processed by its preceding level, and outputs its processing result to the next level. Moreover, each cascade level is an ensemble of decision tree forests. We do not consider in detail the Multi-Grained Scanning where sliding windows are used to scan the raw features because this part of the deep forest is the same in the DisDF. However, the most interesting component of the gcForest from the DisDF construction point of view is the cascade forest.Given an instance, each forest produces an estimate of class distribution by counting the percentage of different classes of examples at the leaf node where the concerned instance falls into, and then averaging across all trees in the same forest. The class distribution forms a class vector, which is then concatenated with the original vector to be input to the next level of cascade. The usage of the class vector as a result of the random forest classification is very similar to the idea underlying the stacking method <cit.>. The stacking algorithm trains the first-level learners using the original training data set. Then it generates a new data set for training the second-level learner (meta-learner) such that the outputs of the first-level learners are regarded as input features for the second-level learner while the original labels are still regarded as labels of the new training data. In fact, the class vectors in the gcForest can be viewed as the meta-learners. In contrast to the stacking algorithm, the gcForest simultaneously uses the original vector and the class vectors (meta-learners) at the next level of cascade by means of their concatenation. This implies that the feature vector is enlarged and enlarged after every cascade level. The architecture of the cascade proposed by Zhou and Feng <cit.> is shown in Fig. <ref>. It can be seen from the figure that each level of the cascade consists of two different pairs of random forests which generate 3-dimensional class vectors concatenated each other and with the original input. After the last level, we have the feature representation of the input feature vector, which can be classified in order to get the final prediction. Zhou and Feng <cit.> propose to use different forests at every level in order to provide the diversity which is an important requirement for the random forest construction.§ WEIGHTED AVERAGES IN FORESTS The DisDF aims to provide large distances between pairs of vectors belonging to the same class and small distances between vectors from different classes.In order to achieve the above aim, we modify ideas provided by Xiong et al. <cit.> and Dong et al. <cit.>. Xiong et al. <cit.> considered an algorithm for solving the metric learning problem by means of the random forests. The proposed metric is able to implicitly adapt its distance function throughout the feature space. Dong et al. <cit.> proposed a random forest metric learning algorithm which combines semi-multiple metrics with random forests to better separate the desired targets and background in detecting and identifying target pixels based on specific spectral signatures in hyperspectral image processing. A common idea underlying the metric learning algorithms in <cit.> and <cit.> is to define a distance measure between a pair of training elements 𝐱_i and 𝐱_j for a combination of trees as average of some special functions of the training elements. For example, if a random forest is a combination of T decision trees {f_t(𝐱),t=1,...,T}, then the distance measure isd(𝐱_i,𝐱_j)=T^-1∑_t=1^Tf_t(ψ(𝐱 _i,𝐱_j)).Here ψ(𝐱_i,𝐱_j) is a function specifically defined in <cit.> and <cit.>.The idea of the distance measure (<ref>) produced by a random forest combined with the idea of probability distributions of classes for producing new augmented feature vectors after every level of the cascade forest proposed by Zhou and Feng <cit.> can be a basis for the modification of the gcForest which produce a new feature representation for efficient metric learning. According to <cit.>, each forest of a cascade level produces an estimate of the class probability distribution by counting the percentage of different classes of training examples at a leaf node where the concerned instance falls into, and then averaging across all trees in the same forest. In contrast to this approach for computing the class probability distribution, we propose to define the class distribution as a weighted sum of the tree class probabilities. The weights can be viewed as training parameters. They are optimized in order to reduce distances between examples of the same class and to increase them between examples from different classes.We apply the greedy algorithm for training the DisDF, i.e., we train separately every level starting from the first level such that every next level uses results of training obtained at the previous level.Let us introduce notations for indices corresponding to different deep forest components. The indices and their sets of values are shown in Table <ref>. One can see from Table <ref>, that there are Q levels of the deep forest, every level contains M_q forests such that every forest consists of T_k,q trees. It is supposed that all training examples are divided into C classes.Suppose we have trained trees in the DisDF. According to <cit.> , the class distribution forms a class vector which is then concatenated with the original vector to be input to the next level of cascade. Suppose an origin vector is 𝐱_i, and the p_i,c^(t,k,q) is the probability of class c for 𝐱_i produced by the t-th tree from the k-th forest at the cascade level q. Below we use the triple index (t,k,q) in order to indicate that the element belongs to the t-th tree from the k-th forest at the cascade level q. Following the results given in <cit.>, the element v_i,c^(k,q) of the class vector corresponding to class c and produced by the k-th forest in the gcForest is determined asv_i,c^(k,q)=T_k,q^-1∑_t=1^T_k,qp_i,c^(t,k,q).Denote the obtained class vector as 𝐯_i^(k,q)=(v_i,1 ^(k,q),...,v_i,C^(k,q)) and the concatenated M_q vectors 𝐯_i^(k,q) as 𝐯_i^(q). Then the concatenated vector 𝐱_i^(1) after the first level of the cascade is𝐱_i^(1)=(𝐱_i,𝐯_i^(1,1) ,....,𝐯_i^(M_1,1))=(𝐱_i ,𝐯_i^(k,1),k=1,...,M_1)=(𝐱 _i,𝐯_i^(1)).It is composed of the original vector 𝐱_i and M_1 class vectors obtained from M_1 forests at the first level. In the same way, we can write the concatenated vector 𝐱_i^(q) after the q-th level of the cascade as𝐱_i^(q) =(𝐱_i^(q-1),𝐯 _i^(1,q),....,𝐯_i^(M_q,q)) =(𝐱_i^(q-1),𝐯_i^(k,q), k=1,...,M_q )=(𝐱_i^(q-1),𝐯_i^(q)).Below we omit the index q in order to reduce the number of indices because all derivations will concern only level q, where q may be arbitrary from 1 to Q.The vector 𝐱_i in (<ref>) is derived in accordance with the gcForest algorithm <cit.> by using (<ref>). We propose to change the method for computing elements v_i,c^(k) of the class vector in the DisDF, namely, the averaging (<ref>) is replaced with the weighted sum of the form:v_i,c^(k)=∑_t=1^T_kp_i,c^(t,k)w^(t,k).Here w^(t,k) is a weight for combining the class probabilities of the t-th tree from the k-th forest. An illustration of the weighted averaging is shown in Fig. <ref>, where we partly modify a picture from <cit.> (the left part is copied from <cit.>) in order to show how elements of the class vector are derived as a simple weighted sum. One can see from Fig. <ref> that three-class distribution is estimated by counting the percentage of different classes of a new training example 𝐱_i at the leaf node where the concerned example 𝐱_i falls into. Then the class vector of 𝐱_i is computed as the weighted average. It is important to note that we weigh trees belonging to one of the forests, but we do not weigh classes, i.e., the weights do not depend on the class c. Moreover, the weights characterize trees, but not training elements. One can also see from Fig. <ref> that the augmented features v_i,c^(k), c=1,...,C, corresponding to the q-th forest are obtained as weighted sums, i.e., there holdv_i,1^(k) =0.4w^(1,k)+0.2w^(2,k)+1.0w^(3,k),v_i,2^(k) =0.4w^(1,k)+0.5w^(2,k)+0.0w^(3,k),v_i,3^(k) =0.2w^(1,k)+0.3w^(2,k)+0.0w^(3,k).The weights are restricted by the following obvious condition:∑_t=1^T_kw^(t,k)=1, w^(t,k)≥0, t=1,...,T_k.So, we have the weighted averages for every forest, and the weights are trained parameters which are optimized in order to decrease the distance between objects from the same class and to increase the distance between objects from different classes. Therefore, the next task is to develop an algorithm for training the DisDF, in particular, for computing the weights for every forest and for every cascade level.§ THE DISDF TRAINING AND TESTING In this section, we consider how to efficiently compute the weights in the DisDF. The main difficulty in solving this task is to choose a proper loss function which allows us to implement efficient computations. We overcome the difficulty by modifying the well-known contrastive loss function. The proposed modification gives us a convex loss function with respect to weights.Before considering the DisDF training, we introduce the following notation:P_ij^(t,k) =∑_c=1^C(p_i,c^(t,k)-p_j,c ^(t,k))^2,Q_ij^(t,k) =∑_c=1^C| p_i,c^(t,k)-p_j,c ^(t,k)| , π^(k,t) =∑_i,j(1-z_ij)P_ij^(t,k), 𝐏_ij^(k) =(P_ij^(t,k), t=1,...,T_k) , 𝐏_ij=(𝐏_ij^(k), k=1,...,M), 𝐐_ij^(k) =(Q_ij^(t,k), t=1,...,T_k) , 𝐐_ij=(𝐐_ij^(k), k=1,...,M), 𝐰^(k) =(w^(t,k), t=1,...,T_k), 𝐰 =(𝐰^(k), k=1,...,M), (𝐰^(k))^2 =((w^(t,k)) ^2, t=1,...,T_k),𝐰^2=((𝐰 ^(k))^2, k=1,...,M), π^(k) =(π^(t,k), t=1,...,T_k) , π=(π^(k), k=1,...,M).Vectors 𝐏_ij, 𝐐_ij, 𝐰, 𝐰^2, π have the same length ∑_k=1^MT_k, and they are produced as the concatenation of M vectors characterizing the forests, for example, the vector 𝐰 is the concatenation of vectors 𝐰^(k)=(w^(1,k),...,w^(T_k,k)), k=1,...,M. If a pair of examples (𝐱_i,𝐱_j) belongs to the same class, then we assume that z_ij=0, otherwise z_ij=1.We apply the greedy algorithm for training the DisDF, namely, we train separately every level starting from the first level such that every next level uses results of training at the previous level. The training process at every level consists of two steps. The first step aims to train all trees by applying all training examples. This step totally coincides with the training algorithm of the original gcForest proposed by Zhou and Feng <cit.>.The second step is to train the DisDF in order to get the weights w^(t,k), t=1,...,T_k. This can be done by minimizing the following objective function over M unit (probability) simplices in ℝ^T_k denoted as Δ_k, i.e., over non-negative vectors 𝐰^(k), k=1,...,M, that sum up to one:min_𝐰J_q(𝐰)=min_𝐰∑_i,j l(𝐱_i,𝐱_j,y_i,y_j,𝐰)+λ R(𝐰).Here l is the loss function, R(𝐰) is a regularization term, λ is a hyper-parameter which controls the strength of the regularization. We define the regularization term asR(𝐰)=‖𝐰‖ ^2.One of the most popular loss functions in metric learning is the contrastive loss which can be written in terms of the considered problem as follows:l(𝐱_i,𝐱_j,y_i,y_j,𝐰)=((1-z_ij )d(𝐱_i,𝐱_j)+z_ijmax( 0,τ-d(𝐱_i,𝐱_j))).Here τ is the tuning parameter. It can be seen from the above expression that the first term of the sum corresponds to the reduction of distances between points of the same class, and the second term increases the distances between points from different classes. We aim to find the minimum of the function with respect to 𝐰 in order to find an optimal vector of weights which fulfills the above properties of examples from the same and from different classes. Moreover, we aim to get the convex function J_q (𝐰) with respect to 𝐰 satisfying (<ref>) in order to simplify the solution of the optimization problem.Since the weights 𝐰 impact on values of augmented parts of vectors 𝐱_i, then we define the distances between these parts of vectors without taking into account the concatenated original vectors. Let us denote the distance between two vectors 𝐯_i^(k) and 𝐯 _j^(k) as d(𝐯_i^(k),𝐯_j^(k)). Then the k-th forest at level q produces the following Euclidean distance:d(𝐯_i^(k),𝐯_j^(k))=∑_c=1 ^C(v_i,c(k)-v_j,c(k))^2=∑_c=1^C∑_t=1^T_k(p_i,c^(t,k)-p_j,c ^(t,k))^2(w^(t,k))^2=∑_t=1^T_kP_ij^(t,k)(w^(t,k))^2.Taking into account all M forests at level q, we can write the total distance between 𝐯_i and 𝐯_j without original vectors as:d(𝐯_i,𝐯_j)=∑_k=1^M∑ _t=1^T_kP_ij^(t,k)(w^(t,k))^2=⟨𝐏_ij,𝐰^2⟩ .Here <·,·> is the dot product of two vectors. The function (<ref>) is convex with respect to w^(t,k). Unfortunately, the second term in (<ref>) as a function of 𝐰 is non-convex. In order to overcome this difficulty, we modify the contrastive loss and reformulate the distance metric in the second term as follows:d_1(𝐯_i,𝐯_j)=‖𝐯 _i-𝐯_j‖ _1=∑_k=1^M∑_t=1^T_k Q_ij^(t,k)w^(t,k)=⟨𝐐_ij,𝐰⟩ .The Manhattan distance is used instead of the Euclidean distance. We do not need to consider the absolute values of w^(t,k) because the weights are restricted by condition (<ref>) such that their values are non-negative. Moreover, we rewrite the second term in the following form:z_ij(max(0,τ-d_1(𝐯_i,𝐯 _j)))^2.The above function is convex in the interval [0,1] of w^(t,k). Then the objective function J_q(𝐰) as a sum of the convex functions is convex with respect to weights. Finally, we write the function J_q (𝐰) asJ_q(𝐰) =min_𝐰∑_i,j(1-z_ij)⟨𝐏_ij,𝐰^2⟩+∑_i,jz_ij(max(0,τ-⟨𝐐 _ij,𝐰⟩))^2+λ‖𝐰‖ ^2.Let us introduce new variablesα_ij=max(0,τ-⟨𝐐_ij,𝐰 ⟩).Then we can rewrite the optimization problem asJ_q(𝐰)=min_α_ij,𝐰(⟨π,𝐰^2⟩ +∑_i,jz_ijα_ij ^2+λ‖𝐰‖ ^2),subject to (<ref>) andα_ij≥τ-⟨𝐐_ij,𝐰⟩ ,α_ij≥0,∀ i,j.This is a standard quadratic optimization problem with linear constraints and with M· T_k variables w^(t,k) and n·(n-1)/2 variables α_ij.Let us prove that the problem can be decomposed into M quadratic optimization problems in accordance with forests, i.e., every optimization problem can be solved for every forest separately. Let us return to the problem (<ref>). The first term in the objective function can be rewritten as∑_i,j(1-z_ij)⟨𝐏_ij,𝐰^2⟩ =∑_k=1^M[∑_i,j(1-z_ij)∑_t=1^T_kP_ij ^(t,k)(w^(t,k))^2].Since constraints (<ref>) for w^(t,k_0) and w^(t,k_1) do not intersect each other by k_0≠ k_1, i.e., they consist of different weights, then we can separately consider M optimization problems for every k.Let us consider the second term in (<ref>) and rewrite it as follows:∑_i,jz_ij(max(0,τ-⟨𝐐 _ij,𝐰⟩))^2=∑_i,jz_ij(max(0,∑_k=1^M[τ_k -∑_t=1^T_kQ_ij^(t,k)(w^(t,k))]) )^2.Here ∑_k=1^Mτ_k=τ. In order to minimize J_q(𝐰), we need to maximize τ_k-∑_t=1^T_kQ_ij^(t,k)( w^(t,k)). But its maximizing does not depend on k because the corresponding constraints (<ref>) for w^(t,k_0) and w^(t,k_1) do not intersect each other. Since τ as well as τ _k are tuning parameters, then we can tune τ_k for every k=1,...,M, separately. Then we can write∑_i,jz_ij(max(0,[τ_k_0+A-∑_t=1 ^T_kQ_ij^(t,k)w^(t,k)]))^2.Here A is a term which does not depend on w^(t,k), but, of course, it depends on other weights from the vector 𝐰^(k). In fact, we have the parameter τ_k=τ_k_0+A for tuning. So, we can separately solve the problems for every forest. Then (<ref>)-(<ref>) can be rewritten for every forest as follows:J_q(𝐰^(k))=min_α_ij,𝐰^(k)( ⟨π^(k),(𝐰^(k)) ^2⟩ +∑_i,jz_ijα_ij^2+λ‖𝐰^(k)‖ ^2),subject to (<ref>) andα_ij≥τ-⟨𝐐_ij^(k),𝐰 ^(k)⟩ ,α_ij≥0,∀ i,j.In sum, we can write a general algorithm for training the DisDF (see Algorithm <ref>). Its complexity mainly depends on the number of levels. Having the trained DisDF, we can classify a new example 𝐱. By using the trained decision trees and the weights 𝐰, the vector 𝐱 is augmented at each level. Finally, we get the vector 𝐯 of augmented features after the Q-th level of the forest cascade corresponding to original example 𝐱.The class of the example 𝐱 is defined by the sum of the c-th elements of vectors 𝐯^(1,Q),...,𝐯^(M_Q,Q). The example 𝐱 belongs to the class c, if the sum of the c-th elements v_c^(1,Q)+...+v_c^(M_Q,Q) is maximal.§ LARGE-SCALE DATA The main difficulty in solving the problem (<ref>)-(<ref>) is that the number of variables as well as the number of constraints may be extremely large because we need to enumerate all pairs of training data. Therefore, we simplify the algorithm by using the well-known Frank-Wolfe algorithm <cit.>. It is represented as Algorithm <ref>.DenoteS_ij=τ-⟨𝐐_ij^(k),𝐰^(k)⟩ .The gradient of the function J_q(𝐰^(k)) with respect to the variable w^(k,t) is∇_w^(k,t)J_q(𝐰^(k)) =2w^(k,t)[λ +∑_i,j(1-z_ij)P_ij^(k,t)] -2∑_i,jz_ij·{[0, S_ij≤0,; S_ij^2· Q_ij^(k,t), S_ij>0. ].It should be noted that the linear problem in Algorithm <ref> can be solved by looking for the solution among T_k vertices of the unit simplex Δ_k. The vertices are of the form:𝐠=(0,...,0,1,0,...,0).Hence, we have to look for smallest value of f(𝐰_s) by t=1,...,T_k. In other words, we compute ∇_𝐰^(k) J_q(𝐰^(k)) for different t. Then 𝐠_s consists of T_k-1 zero elements and the unit element whose index t_0 coincides with the index of the smallest value of f(𝐰_s).So, we have obtained a very simple algorithm for solving the problem (<ref>)-(<ref>). Its simplicity is due to simple constraints for the weights w^(k,t). The modified Frank-Wolfe algorithm is represented as Algorithm <ref>.§ NUMERICAL EXPERIMENTS We compare the DisDF with the gcForest. The DisDF has the same cascade structure as the standard gcForest described in <cit.>. Each level (layer) of the cascade structure consists of 2 complete-random tree forests and 2 random forests. Three-fold cross-validation is used for the class vector generation. The number of cascade levels is automatically determined.We modify a software in Python implementing the gcForest and available at https://github.com/leopiney/deep-forest to implement the procedure for computing optimal weights and weighted averages v_i,c^(k). Accuracy measure A used in numerical experiments is the proportion of correctly classified cases on a sample of data. To evaluate the average accuracy, we perform a cross-validation with 100 repetitions, where in each run, we randomly select N training data and N_test=2N/3 test data.First, we compare the DisDF with the gcForest by using some public data sets from UCI Machine Learning Repository <cit.>: the Ecoli data set (336 instances, 8 features, 8 classes), the Parkinsons data set (197 instances, 23 features, 2 classes), the Ionosphere data set (351 instances, 34 features, 2 classes). A more detailed information about the data sets can be found from, respectively, the data resources. Different values for the regularization hyper-parameter λ have been tested, choosing those leading to the best results. In order to investigate how the number of decision trees impact on the classification accuracy, we study the DisDF as well as gcForest by different number of trees, namely, we take T_k=T=100, 400, 700, 1000.Results of numerical experiments for the Parkinsons data set are shown in Table <ref>. It contains the DisDF accuracy measures obtained for the gcForest (denoted as gcF) and the DisDF as functions of the number of trees T in every forest and the number N=50,80,100,120 of examples in the training set. It follows from Table <ref> that the accuracy of the DisDF exceeds the same measure of the gcForest in most cases. The difference is not significant by N=50 and 120. However, it is larger by N=100 and the small amount of trees T. Nevertheless, the largest difference between accuracy measures of the DisDF and the gcForest is observed by T=1000 and N=100.Results of numerical experiments for the Ecoli data set are shown in Table <ref>. This data set shows the largest difference between accuracy measures of the DisDF and gcForest by N=50. This implies that the proposed DisDF outperforms the gcForest by the very small amount of training data. It is interesting to note that the DisDF does not outperform the gcForest by N=120 and T=700. At the same time, the number of trees also significantly impact on the accuracy, namely, we can see from Table <ref> that the outperformance of the DisDF is observed by T=100 and 400.Numerical results for the Ionosphere data set are represented in Table <ref>. It follows from Table <ref> that the largest difference between accuracy measures of the DisDF and the gcForest is observed by T=100 and N=50. This again implies that the DisDF outperforms the gcForest by the very small amount of training data.By analyzing all results, we have to point out that, in contrast to comparative results, the highest accuracy measure can be obtained by the large number of training data and the large number of trees in every forest. If the first parameter (N) cannot be controlled, then the number of trees is a tuning parameter. It is interesting to note that the largest number of trees by some fixed values of N, for example, N=80 and 100, does not provide the largest accuracy measure. In particular, we can see from Tables <ref>-<ref> that the largest accuracy measures are achieved at T=400 and 700.It should be noted that the multi-grained scanning proposed in <cit.> was not applied to investigating the above data sets having relatively small numbers of features. The above numerical results have been obtained by using only the forest cascade structure.Another data set for comparison of the DisDF and gcForest is the well-known MNIST data set which is a commonly used large database of 28×28 pixel handwritten digit images <cit.>. It has a training set of 60,000 examples, and a test set of 10,000 examples. The digits are size-normalized and centered in a fixed-size image. The data set is available at http://yann.lecun.com/exdb/mnist/. In contrast to gcForest, we did not use the multi-grained scanning scheme for the DisDF implementation because its use provides worse results by the small amount of training data. Results of numerical experiments for the MNIST data set are shown in Table <ref>. Numerical experiments with the MNIST data set by the very small training data have shown that the use of the multi-grained scanning procedure may deteriorate the classification performance of the DisDF as well as the gcForest. Therefore, we did not use this procedure in numerical experiments with the MNIST. It can be seen from Table <ref> that the DisDF outperforms the gcForest in the most cases. The largest difference between accuracy measures of the DisDF and the gcForest is observed by T=100 and N=50.§ CONCLUSION A discriminative metric learning algorithm in the form of the DisDF has been presented in the paper. Two main contributions should be pointed out. First, we have introduced weights for trees which allow us to apply some new properties for the deep forest. The weights play a key role in the developing the DisDF. This role is similar to the role of weights of connections in neural networks which also have to be trained. This implies that we can control properties of the deep forest and its modifications by constructing the corresponding objective functions J(𝐰) which are similar to the loss or reconstruction functions in neural networks. This fact opens a way for developing new modifications of the deep forest which have certain properties. Moreover, we get an opportunity to consider the relationship between the deep forest and neural networks as it has been done by Richmond et al. <cit.> in exploring the relationship between stacked random forests and deep convolutional neural networks. Second, we have used a new objective function J_q(𝐰) which combines two different distance metrics: Euclidean and Manhattan distances. This combination allows us to get the convex objective function with respect to 𝐰 and to significantly simplify the optimization problem.It should be also noted that one of the implementations of the DisDF has been represented in the paper. It should be noted that other modifications of the DisDF can be also obtained. We can use more efficient modifications of the Frank-Wolfe algorithm, for example, algorithms proposed by Hazan and Luo <cit.> or Reddi et al. <cit.>. We can also consider non-linear functions of weights like activation functions in neural networks. Moreover, we can investigate imprecise statistical models <cit.> for restricting the set of weights, for example, we can reduce the unit simplex of weights in order to get robust classification models. These modifications can be viewed as directions for further research.§ ACKNOWLEDGEMENT The reported study was partially supported by RFBR, research project No. 17-01-00118. 10Bellet-etal-2013 A. Bellet, A. Habrard, and M. 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"authors": [
"Lev V. Utkin",
"Mikhail A. Ryabinin"
],
"categories": [
"stat.ML",
"cs.LG",
"68T10"
],
"primary_category": "stat.ML",
"published": "20170525122411",
"title": "Discriminative Metric Learning with Deep Forest"
} |
Department of Astronomy, Box 351580, University of Washington, Seattle, WA 98195, USA [email protected] L. Dodge Physics and Astronomy Department, University of Oklahoma, 440 W Brooks St., Norman, OK 73019, USA [email protected] Exoplanets & Stellar Astrophysics Laboratory, NASA Goddard Space Flight Center, Code 667, Greenbelt, MD, 20771, USA Department of Terrestrial Magnetism, Carnegie Institution of Washington, Washington DC 20015, USA Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA Adnet Systems, Inc., 6720B Rockledge Dr., Suite 504, Bethesda, MD, 20817 USA Department of Terrestrial Magnetism, Carnegie Institution of Washington, Washington DC 20015, USA We present coronagraphic long slit spectra of AU Mic's debris disk taken with the STIS instrument aboard the Hubble Space Telescope (HST). Our spectra are the first spatially resolved, scattered light spectra of the system's disk, which we detect at projected distances between approximately 10 and 45 AU. Our spectra cover a wavelength range between 5200 and 10200 Å. We find that the color of AU Mic's debris disk is bluest at small (12-35 AU) projected separations. These results both confirm and quantify the findings qualitatively noted by <cit.>, and are different than IR observations that suggested a uniform blue or gray color as a function of projected separation in this region of the disk. Unlike previous literature that reported the color of AU Mic's disk became increasingly more blue as a function of projected separation beyond ∼30 AU, we find the disk's optical color between 35-45 AU to be uniformly blue on the southeast side of the disk and decreasingly blue on the northwest side. We note that this apparent change in disk color at larger projected separations coincides with several fast, outward moving “features” that are passing through this region of the southeast side of the disk. We speculate that these phenomenon might be related, and that the fast moving features could be changing the localized distribution of sub-micron sized grains as they pass by, thereby reducing the blue color of the disk in the process. We encourage follow-up optical spectroscopic observations of the AU Mic to both confirm this result, and search for further modifications of the disk color caused by additional fast moving features propagating through the disk.§ INTRODUCTION AU Mic is a 23 ±3 Myr old <cit.>, nearby (9.9 pc), M1Ve star that is a member of the β Pic moving group <cit.>. The star is surrounded by a debris disk that extends out to 210 AU, which appears close to edge-on from our point of view <cit.>. While several early searches for planets in the system yielded null results <cit.>, <cit.> recently presented radial velocity data that suggested the presence of two Jovian-mass planets at semi-major axis separations of ∼0.3 and 3.5 AU.AU Mic exhibits numerous signatures of activity common for its youth. For example, <cit.> observed flares from AU Mic in both their FUSE and HST/STIS data. <cit.> monitored AU Mic for 28 continuous nights in 2005 with the CTIO 1-m telescope in the optical and found that the system exhibited a 4.847 day period, which they attributed to star spots, and underwent three flares, while <cit.> observed several flaring events from the system with HST/STIS that lasted between 10 s and 3 min. This is common for flares on M-dwarfs, which last on the order seconds to many hours and result in a significant increase in the stars' flux at NUV and blue optical wavelengths (see e.g. ). <cit.> found that the flares in the UV typically precede those in the X-ray wavelengths using the XMM-Newton EPIC and OM cameras. Chandra observations also detected a small flare from the system <cit.>, and simultaneous observations with XMM-Newton and the VLA showed that the largest X-ray flares also have a radio counterpart <cit.>. A high energy (10^36 ergs) flare was also observed in EUVE observations of the system, and interpreted to be accompanied by a coronal mass ejection <cit.>.AU Mic's debris disk has been spatially resolved across a wide range of wavelength regimes, including optical to near-IR scattered light <cit.>, optical polarized scattered light <cit.>, far-IR and sub-mm <cit.>, and mm <cit.> wavelengths, helping to trace the spatial distribution of a variety of grain size populations. The system appears to have a `birth ring' of material near 43 AU that forms micron sized particles from collisions between larger bodies (see e.g. ), as well as an extended halo comprised of approximately micron-sized grains <cit.>.Recent analysis of ALMA data suggest that there are millimeter sized grains at separations of ≤ 3 AU from the central star and an outer dust belt near the birth ring, the combination of which might be architecturally similar to the asteroid belt and Kuiper Belt in our own solar system <cit.>. Although, recent modeling of the ALMA data suggests that coronal heating may be responsible for the point source excess at millimeter wavelengths instead of an inner dust belt <cit.>. The inner (< 30 AU) disk appears to be largely devoid of small, micron-sized grains <cit.>, while analysis of the polarimetric properties of the disk led <cit.> to suggest that its grains were highly porous.There is a rich history of attempting to diagnose the optical and near-IR colors of AU Mic's debris disk. When comparing HST F606W and F435W coronagraphic imagery to each other, <cit.> found the disk's color to be gray. However, a comparison of the F435W and F814W data show that AU Mic's disk is blue relative to the central star, particularly at large radii (30 to 60 AU). <cit.> also qualitatively suggested that the disk looks increasingly blue inwards of 30 AU, but cautioned that PSF residuals may have biased that result. The increasingly blue color of the outer disk persists in the near-IR and when comparing <cit.>'s optical HST data to near-IR datasets <cit.>. Nevertheless, the increasingly blue trend at small projected separations has not been reproduced <cit.>. It's thought that the radiation pressure from the central M star, which is lower than that of more massive stars, is ineffective at clearing out grains, allowing small grains to persist longer than typically expected for a debris disk <cit.>. Instead, it is believed that the evolution of the grain populations within AU Mic's disk is more strongly affected by a combination of their porosity and the central star's wind, which may replace radiation pressure as the dominate mechanism for clearing out small grains. <cit.>. Local density enhancements and deficits have been discovered within scattered light images of the disk <cit.> that are consistent with ongoing or recent planet formation in the system.Moreover, recent analysis of multi-epoch imagery of the disk has revealed clear evidence of large scale features at projected separations of 10-60 AU on the southeast side of the disk that change location as a function of time <cit.>. These features are moving between 4 and 10 km s^ -1 and appear to be either in highly elliptical orbits, or unbound. While the fundamental mechanism responsible for producing this new observational phenomenon remains unclear, <cit.> did speculate that highly energetic episodic flares may be interacting with the disk material. Given the newly discovered dynamic nature of AU Mic's disk, one might expect that the disk's color could exhibit time variability if the fundamental mechanism driving the fast moving features also changed the local grain size distribution. In fact, it has been theoretically shown that coronal mass ejections, one of the possible causes of the fast moving features in AU Mic's disk, could remove infrared emitting dust from disks around active stars on timescales of days <cit.>.In this paper, we investigate the color of AU Mic's debris disk using the HST/STIS coronagraphic spectroscopy mode, which makes use of a long slit and occulting bar to produce optical spectra of the disk as a function of radius from the central star. Despite coronagraphic spectroscopy's relative power to measure the location dependent color of a disk at a higher spectral resolution than simple multi-band photometry, there is only one previously published dataset in which this technique has been used on a disk system (TW Hya; ). Therefore, in Sections 2 and 3 we describe our observing strategy and data reduction procedures in detail (see alsofor a comprehensive description of this technique). In Section 4 we present our spatially resolved disk color spectra. In Section 5 we describe the surface brightness and disk color profiles extracted from our dataset and compare them to datasets already in the literature. We discuss the implications of our findings in Section 6 and summarize our findings in Section 7.§ OBSERVATIONS The Hubble Space Telescope (HST) observed AU Mic and a point spread function (PSF) reference star, GJ 784, on 2012 July 29 as part of program GO-12512. All observations were taken using the G750L grating with the STIS instrument and cover the 5240-10270 Åwavelength range.Our observing strategy was similar to that of <cit.> and produces spatially resolved coronagraphic spectroscopy of AU Mic's disk by making use of the 086 fiducial occulting bar attached to the STIS long slits. Our AU Mic observations consist of a sequence of exposures that start with peak ups using a narrow slit (0.2 × 0.05ND) in both the wavelength (x) and spatial (y) directions of the detector to accurately position the star after the initial acquisition. These are followed by an unocculted spectrum of AU Mic, and a lamp flat using the 52"×02 slit.Next, a series of exposures using the 52"×02F2 slit are taken (hereafter “fiducial exposures", “fiducial data", or “fiducial spectra"). This is the same slit as the one used for the unocculted spectrum, but it places the 086 wide fiducial bar over the central star. We repeated this sequence of exposures for the PSF star GJ 784.During our first three HST orbits, we collected an unocculted point source spectrum and three fiducial spectra of AU Mic, as well as a lamp flat. The lamp flat, the unocculted AU Mic spectrum, and one fiducial spectrum were collected during the first orbit, and we obtained two additional fiducial spectra during the following two orbits. Therefore, the exposure time of our first fiducial spectrum of AU Mic (OBPZ05030) is significantly shorter (1520 s) than the following two spectra (OBPZ05040 and OBPZ05050 respectively; 2780 s). During our fourth and final orbit we obtained an unocculted point source spectrum and one fiducial spectrum of the PSF star, as well as an additional lamp flat. Table <ref> lists the start times, exposure times, and apertures used for our observations.In the case of AU Mic, the position angle of the slit, and therefore the roll angle of HST, was chosen so that it was aligned with the disk. For all of our AU Mic fiducial observations the slit was oriented at 307.1, which places it at very close to the same angle as the disk (Kalas et al. 2004 report the position angle of the disk as 124±2 for the south east side and 310±1 for the north west side). This causes the signal from the disk to be dispersed along the x-axis of the exposure so that each row of pixels represents a spectrum from a different spatial location of the disk (i.e. the x-axis is the wavelength direction and the y-axis is the spatial direction), as shown in Figure <ref>. Therefore, data on the +y (-y) side of the fiducial bar are from the north west (south east) side of the disk (hereafter NW and SE). In the spatial direction the pixel size is 0051. We note that no matter the roll angle of HST, the slit is always oriented away from the diffraction spikes <cit.>.lccccc 6 0ptObservation Log Obs IDTargetStart Time (UT)Exposure Time (s)ApertureCommentsOBPZ05010 AU Mic12:15:090.652 × 0.2Point source spectrumOBPZ05020 Lamp Flat12:17:212552 × 0.2OBPZ05030 AU Mic12:20:21152052 × 0.2F2Fiducial spectrumOBPZ05040 AU Mic13:32:30278052 × 0.2F2Fiducial spectrumOBPZ05050 AU Mic15:08:20278052 × 0.2F2Fiducial spectrumOBPZ06010 GJ 78417:30:270.452 × 0.2Point source spectrum, PSF starOBPZ06020 Lamp Flat17:32:392552 × 0.2OBPZ06030 GJ 78418:20:15167852 × 0.2F2Fiducial spectrum, PSF star All data were taken with the STIS CCD using the G750L grating. § DATA REDUCTIONAs described in Section 2, the AU Mic and PSF star exposures were obtained with the 52"×02F2 slit, which is an unsupported STIS observing mode. Therefore, we performed our own data reduction and calibration on both the point source and fiducial datasets following the prescription laid out by <cit.>, who used the same data collection technique on the TW Hya protoplanetary disk system. The general data reduction steps, which we discuss in detail below, include defringing, wavelength calibration, and flux calibration for both the point source and fiducial data. We also perform hot and cold pixel correction and PSF subtraction on our fiducial datasets. We explored processing a combined version of the AU Mic fiducial data, which we obtained by adding the raw exposures from all the orbits together, but we found that this reduced our ability to successfully defringe the data. Therefore, we processed the individual orbits of the AU Mic fiducial data separately. We began our data reduction by running the raw point source and fiducial exposures through theprepspec task, which subtracts bias and dark frames, divides by the pixel to pixel flat, subtracts overscan levels, and rejects cosmic rays.§.§ Defringing, Wavelength Calibration, and Flux Calibration Interference between reflections off the front and back surfaces of the STIS CCD causes fringing in all (point source and fiducial) our spectra at long wavelengths (see also Roberge et al. 2005). We defringed all of our data by using our contemporaneous lamp flats and thenormspflat, mkfringeflat, and defringetasks. This allows us to create and apply a flat field with fringes that have been shifted and scaled to match those in the point source and fiducial spectra. However, some nominal amount of fringing still remains, particularly above the fiducial bar on the NW side of the disk, and at wavelengths above 9,500 Å.Once our spectra were defringed, we used thewavecaltask to wavelength calibrate both the point source and fiducial spectra. We then flux calibrated the point source spectra with the x1d task and the fiducial spectra with the x2d task. This converts the units of each pixel from counts to erg s^ -1 cm^ -2 Å^ -1 arcsec^ -2 while also correcting for geometric distortions so that the wavelength and distance axes increase linearly. Figure <ref> displays the final reduced version of the AU Mic point source spectrum. For comparison, we also display the default calibration, which does not include defringing, for this spectrum. The defringing process significantly improves the visibility of intrinsic spectral features above 7,500 Å, while removing the interference pattern imprinted onto the data by the STIS CCD.§.§ Hot and Cold Pixel Correction We corrected our fiducial spectra for hot and cold pixels following a similar sigma-clipping procedure to that laid out by <cit.>. As part of that prescription pixels inside a clip box, whose size is determined by the shape of the spectra, are used to calculate both the spectrum and background noise, which are then combined in quadrature. This “local noise" is then used to determine if any individual pixel should be flagged for correction. Pixels that varied from the median flux for the whole clip box by more than three times the local noise were replaced with the median flux. In our case, the size of the clip boxes were 5 pixels in the spatial (y) direction and 30 pixels in the wavelength (x) direction. This size mitigated missing hot or cold pixels because it accounted for the large variations across the images in the spatial direction and the significant, but not as large, variations in the wavelength direction.For a given clip box, the spectrum noise was calculated by computing the standard deviation of the values for all pixels in the box. Similarly, the background noise is the standard deviation of a five by five box of pixels surrounding the pixel with the largest (or smallest) value inside the clip box. We note that we do not follow the background noise calculation presented in <cit.>, who used the standard deviation of the column of pixels containing the largest (or smallest) value. This is because many of our hot and cold pixels come in pairs (or more) directly next to each other, and the resulting background noise is therefore skewed toward not flagging pixels which require correction. For comparison, if we directly follow the procedure as laid out by <cit.> we would only correct 100 pixels, which is significantly below both the thousands of pixels <cit.> corrected in their data and below the number clearly present in our data. Alternatively, if we were only to use the spectrum noise to determine which pixels should be corrected, we would flag upwards of 3% of the image for correction. Our five by five background noise box produced a compromise between these two extreme approaches; approximately 1.5% of pixels were corrected in each individual fiducial dataset. Figure <ref> (top) displays a reduced AU Mic fiducial spectrum (OBPZ05040) after applying our hot and cold pixel corrections. Significantly fewer hot and cold pixels remain after correction, and hot and cold pixels that contained spectral data were not corrected at a rate higher than pixels located where the disk was not detected. In fact, the pixels in the disk spectra were less likely to be corrected than background pixels. The fiducial bar is clearly seen as a dark, horizontal band which passes through the vertical center of the image. Spectral features, particularly the Na1 5890 Åline, are visible as the dark vertical bands in our final image, as well as HST's Airy rings, which appear as slanted lines that angle away from the fiducial bar.§.§ PSF Subtraction We observed the PSF star GJ 784 immediately after AU Mic using the same exposure sequence. These data were reduced using the same reduction procedures we applied to the AU Mic spectra. GJ 784 was chosen because of its close spectral match (M0V) to AU Mic (M1V) allows us to isolate light reflected in AU Mic's disk from direct light from the central star. We correct for the brightness difference and slight spectral mismatch between AU Mic and GJ 784 by using our point source spectra. Each column of the fiducial PSF image was scaled by the ratio of the flux of the AU Mic to PSF point source spectra at that same wavelength. Once the fiducial PSF image was scaled, we subtracted it from the AU Mic fiducial images. In addition to removing the Airy rings visible in Figure <ref>, which are due to the telescope's PSF, this also removed the flux in the spectrum due to AU Mic's central star, leaving a spectrum containing only light reflected in the disk.Figure <ref> (bottom) displays the final PSF subtracted AU Mic fiducial image for observation OBPZ05040. Effects still present in the final dataset include some fringing at the longest wavelengths and PSF subtraction residuals near the fiducial bar. In general, these features are present within all three final fiducial images of AU Mic. The locations of the PSF residuals shift slightly in the wavelength direction and their intensity varies somewhat between images. We explored mitigating the presence of these features by scaling the brightness our final PSF fiducial spectrum by up to ±10% of the brightness difference between AU Mic and GJ 784, but found this to be ineffective in reducing the residuals at best and to produce more residuals at worst. We also explored shifting the final PSF image relative the the AU Mic fiducial spectra by up to several pixels in both the wavelength and distance directions. While this did not necessarily produce additional residuals, it did not appear to decrease their presence in our final datasets. This process primarily shifted the location of the dark bands seen near the fiducial bar in Figure <ref> to larger and smaller wavelengths. Therefore, the reduced AU Mic fiducial spectra were PSF subtracted with no realignment of the images. § EXTRACTION OF DISK COLOR SPECTRA§.§ Individual Orbits: Do flares affect our dataset?In order to explore the extent to which flares from AU Mic's central star might affect our disk color determination, we extracted spectra from the fiducial observations obtained during each individual orbit separately. Because flares from M dwarfs primarily affect flux at higher energies, we expect that they would appear as an increase in the scattered flux on the blue end of our spectra. We extracted spectra from every row of pixels for each of our three AU Mic fiducial exposures, which we corrected for slit loss using theheader keyword that was produced by the x2d task. This is is a wavelength averaged throughput value such that the value is correct for the center of the bandpass, but too high and too low at the ends. We then converted from surface brightness to flux units by multiplying by the pixel scale in the dispersion direction (0.051 arcseconds) and the slit width (0.2 arcseconds). Finally, in order to produce our `color spectra', we divided our extracted disk spectra by our unocculted stellar spectrum to remove the edge effects introduced during our slit loss correction and produce a color relative to the system's central star. We calculated error-weighted mean spectra using a sliding bin in the distance direction that was 5 pixels (approximately 2.5 AU or 025) wide to reduce our uncertainties. Figure <ref> shows our representative resulting color spectra, which we binned (20 pixels or 98 Å) in the wavelength direction. Our displayed uncertainties represent the combination of the statistical and systematic errors, which were added in quadrature. When comparing our fiducial spectra against themselves, their relative uncertainty is up to 2% of the total flux <cit.>. Because our data were taken in back to back orbits, residual fringing and the absolute flux calibration dominate the systematic uncertainties of our dataset, while factors like the instrument stability and variability in the time dependent photometric calibration are insignificant. Therefore, the relative photometric uncertainties of our dataset are more likely to be closer to 1%. However, out of an abundance of caution and desire to over, rather than under, estimate our uncertainties, the error bars displayed in Figure <ref> were calculated using the 2% systematic uncertainty value.All of the color spectra are largely consistent with each other at the same projected separation independent of the side of the disk and orbit during which the data were taken. In general, the color spectra are blue at small distances from the central star and become grayer as projected separation increases. Several strong features are clearly visible in the color spectra that were extracted from regions close to the fiducial bar. In particular, there is a strong minimum near ∼ 6500-7000 Å(black dotted line) at a projected separation of 10.1 AU (1012) on the NW side of the disk that appears in data from our first orbit (05030). This feature is consistent with a stellar titanium oxide band that remains present in our spectra and should not be considered when determining disk color. Additionally, there is a feature at ≈ 6500 Å(red dotted line) at a projected separation of 15.1 AU (1519) on the SE side of the disk that persists through all three of our orbits of data. Examination of many disk spectra extracted at projected separations between 10 and 17 AU (1012 - 11772) on both sides of the disk show that these minima behave like Airy rings; they move to longer wavelengths as the distance from the fiducial bar increases. We explored rescaling and shifting the alignment of our PSF, but found we could never completely remove these residuals. Therefore, we did not realign or rescale our PSF to produce our final color spectra. These features should not be considered significant in terms of determining disk color or the effects of flares on our color determination. However, because these PSF residuals do not affect the whole 15.1 AU (1519) spectrum, the disk to stellar flux outside of this feature can still be used to determine the disk's color. A comparison of how our disk spectra evolve with time reveals no evidence of flaring activity that might bias our color results.§.§ Combined Color Spectra We explored multiple ways to combine our three AU Mic fiducial datasets into one final fiducial spectrum. Ultimately, we found that combining these data after the reduction process produced the least amount of residual fringing. Therefore, each fiducial exposure underwent defringing, wavelength and flux calibration, hot and cold pixel correction, and PSF subtraction as described in Section 3 separately. We combined our individually extracted color spectra (see Section 4.1) into one final set of color spectra by calculating the error-weighted mean spectrum at each disk location. Figure <ref> displays our final combined disk color spectra at representative projected separations on both the NW and SE sides of the disk. During the combination process we did not leave out data from regions in our individual spectra that we believe to be affected by TiO lines or PSF residuals. Instead, we combined the data from all the orbits and then reevaluated whether these effects appear in the final spectra. Our displayed uncertainties are the same as Figure <ref> with the appropriate error propagation.The final combined color spectra behave similarly to the color spectra extracted from individual orbits. The general trend in the data suggests the disk is bluer at small projected separations from the central star and, as distance increases, the disk becomes less blue on both sides of the disk. The PSF residual in the color spectrum extracted at 15.1 AU (1519) on the SE side of the disk remains visible, but the effects from the TiO band at 10.1 AU (1012) on the NW side of the disk is no longer evident.We calculated the projected locations of the fast moving disk features B, C, and D, which were first detected by <cit.>, during our observations and extracted color spectra at those specific locations (15.10 AU, 26.67 AU, and 36.73 AU). We do not find any difference between the disk colors at these three specific projected disk locations and the disk colors from regions immediately surrounding these three locations. While our slit size was large enough to encompass some of these fast moving features, the apparent lack of color change associated with them may be partially due to our lack of vertical spatial resolution in the disk; spectra at the location of features B, C, and D includes light scattering off both the fast moving and other disk material. We did not attempt to determine the disk color for feature A (8.55 AU), because projected separations that close to the fiducial bar are significantly affected by PSF residuals, and feature E (50.32 AU), which is farther out in the disk than the maximum projected distance at which we have a spectral detection. § SURFACE BRIGHTNESS AND COLOR PROFILES Prior to dividing our extracted disk spectra by the stellar spectrum, we convolved our spectra with synthetic filters (Figure <ref>) to extract equivalent fluxes in the ACS F606W and F814W bands <cit.>, as well as ground-based R and I bands <cit.>. This allows us to produce a pseudo-midplane surface brightness profile in each of these filters and facilitates comparisons between our dataset and other disk color datasets available in the literature, albeit with several caveats. First, we cannot calculate a true midplane surface brightness profile for our dataset because our data contain no spatial information in the vertical direction of the disk. Second, our disk spectra stop more than 500 Åshort of the blue end of the F606W filter. Third, there is no published color information for AU Mic's disk derived in part from I band imagery. Additionally, while <cit.> has published F814W imagery of the system, no detailed surface brightness profiles were presented. Finally, while R band imagery of the system does exist <cit.>, it does not spatially overlap with the regions of the disk that we detect. All of our spectra are from disk locations which were under the occulting spot used by <cit.> in their discovery image. At regions outside of their occulting spot, the flux levels of our fiducial imagery are too low to detect the disk. Figure <ref> displays our surface brightness profiles for all four filters for our final combined (error-weighted mean) dataset. In general, our surface brightness profiles show no distinct or significant features other than a general decrease in brightness as projected distance increases. The NW side of the disk is brighter than the SE side by, on average, 0.04 magnitudes in the R band and 0.1 magnitudes in the I band. Additionally, our profiles agree well with those from archival F606W and R band datasets.We computed R minus I band color information for the disk relative to AU Mic's central star using the surface brightness profiles displayed in Figure <ref> and our unocculted AU Mic spectrum (Figure <ref>), which we convolved with the same filter response functions. Figure <ref> displays the resulting color profile for the NW and SE sides of the disk separately. The disk appears bluest from 10-17 AU on the NW and most of the SW side. The disk's color at larger projected separations is still modestly blue, albeit less so than at small projected separations. At projected separations larger than 40 AU, the disk color is consistent with being gray within 1σ uncertainties at many locations on the NW sides of the disk.On the SE side of the disk there is a sharp sudden graying of the disk around 14 AU (Figure <ref> open symbols). This is due to the PSF residual that can be seen at the same projected distances in our color spectra (Figure <ref>). Fast moving feature B <cit.> is located within the region of the our data affected by this PSF residual. Additionally, the locations of fast moving features A and E are outside of the regions of the disk that we detect. Sections of the disk immediately interior to features C and D appear to be less blue than regions immediately at and exterior to the fast moving features. It is difficult to discern the significance of this low level variability due to the size of our uncertainties. However, we note that our input spectra are binned using a moving box which is 5 rows of pixels wide in the original fiducial imagery. This has the effect of reducing the uncertainties on our disk color profile at the expense of reducing the amplitude of small changes in disk color such as those associated with the fast moving features.On the NW side of the disk, there appears to be a peak in the blueness between 13 and 14 AU. This feature does not coincide with any obviously visible PSF residual in either our reduced fiducial imagery or extracted color spectra. Interestingly, <cit.> reports a break in the ACS F606W surface brightness profile at a similar location in the disk (≈ 15 AU) which has not been observed at other wavelengths. Therefore, it is likely a real localized change in disk color. At projected distances immediately larger than this feature, the disk color appears to stay constant for approximately 15 AU before becoming increasingly less blue. The turnover to a less blue color occurs at approximately 30 to 35 AU on both sides of the disk, which is the same location where a second break in power law fits to surface brightness profiles have been reported (see e.g., F606W, J, H, and K' band imagery inand ).We calculated the error weighted mean R-I band color for several regions of the disk to determine the significance of its change with projected distance. We also performed a linear fit of our color data to characterize its variations across those same regions. We report the slope of this fit, which is positive when the color of the disk is becoming less blue with projected separation, zero when the disk is uniformly blue, and negative when the disk is becoming more blue with increasing projected separation. Between 12 and 17 AU, the color of the NW side of the disk is -0.214± 0.012, while the slope of our linear fit is 0.024± 0.008. This inner region of the disk is bluer than regions at immediately larger projected separations at the 3σ level; between 17 and 35 AU the error-weighted mean color of the disk is -0.154± 0.006 and -0.165± 0.007 on the NW and SE sides respectively. Across these same regions the slope of our linear fit is -0.004± 0.001 and 0.004± 0.001, respectively. This shows that the disk has not only become less blue in the 17-35 AU region when compared to the 12-17 AU region at a statistically significant level, but the variation in the color across these two regions is different at the 3σ level as well. The blueness of the disk continues to decline at a statistically significant level. The error-weighted colors of the farthest projected regions (35-45 AU) of the disk that we detect are -0.082± 0.011 for the NW and -0.096± 0.010 for the SE sides of the disk. This is less blue than the regions of the disk at projected separations immediately interior to 35-45 AU at a 3σ level as well. On the NW side, the slope of our linear fit between 35 and 45 AU is 0.021± 0.004, which is different at a 4σ level from the 17-35 AU region on the same side of the disk. However, on the SE side, the disk becomes less blue at essentially the same rate for both the 17-35 AU and 35-45 AU regions; the slope of our fit at projected separations between 35 and 45 AU on the SE side of the disk is 0.001± 0.003. This suggests that the trend of the disk becoming less blue with larger projected separations is real on both sides of the disk.The R bandpass contains a large red tail that overlaps with the I bandpass. This allows a significant amount of red light to be included in our R band photometry and potentially impairs our color determination efforts. In order to more accurately trace the color, and thus the grain size distribution and chemistry, of the disk we created a `Blue' synthetic filter that has 100% throughput between 5500 and 7000Åand 0% throughput everywhere else which we used to compute a second color profile for the system. Figure <ref> displays our resulting Blue-I color profile. The same general trends exist in these profiles as the R-I profiles. The disk appears bluest in the inner disk and less blue in the outer disk. There appears to be a peak in the blueness of the NW side of the disk at 15 AU and a second break in the behavior of the color profile between 30 and 35 AU on both sides of the disk. Small changes in disk color at the locations of fast moving features C and D remain visible, but their significance is still difficult to determine due to the size of our uncertainties. At every separation the disk appears bluer in the Blue-I color profile than the R-I. Beyond 40 AU the disk color no longer appears gray. This is due to the significant amount of red light `contamination' in the R bandpass due to its long red tail.§ DISCUSSION: IS THE COLOR OF AU MIC'S DISK CHANGING WITH TIME? The color of AU Mic's debris disk has been previously determined using a variety of high-contrast observations in optical to near-IR bandpasses. The optical (F435W-F814W; ), optical-near-IR (F606W-H; , ), and near-IR (J-H, K'-H ) colors of the disk become progressively bluer with projected separation at distances beyond ∼30 AU, except for the reported gray color within HST/ACS F435W-F606W observations <cit.>. At closer projected separations, the disk's color has been reported to be blue or gray and does not change with distance to the star <cit.>. The only possible exception to this is the qualitative report provided by <cit.> that the F435W-F814W color becomes progressively bluer at smaller projected separations; however, these authors cautioned that strong PSF subtraction residuals rendered the trend suspect.By contrast, our optical coronagraphic spectroscopy appears to show slightly different color behavior than most previous broad-band color determinations of the disk. Both our color spectra and our color profiles (Figures <ref>, <ref>, and <ref>) show that AU Mic's disk is bluest in regions closest to its central star (projected separations between 12 and 17 AU; see Section 5). Thus, our data both quantify and confirm the trend qualitatively reported by <cit.>. Moreover, these data seem to differ from mainly IR colors of the inner (10-30 AU) regions of the disk that find a more uniform blue or gray color with decreasing projected separation (see e.g. ). At more extended distances from the central star (17-35 AU), the disk still appears blue, albeit less so than that observed closer to the star at a 3σ statistical significance. Moreover, the color of the disk at these extended distances (35-45 AU) does not become progressively more blue, as generally reported in previous literature, but rather is less blue than regions at smaller projected separations from AU Mic at a 3σ significance level.Although our observations do not cover the full F606W bandpass and so do not allow us to make a complete comparison of our extracted colors with those quantified in the literature, our surface brightness data are generally consistent with archival imagery where comparisons can be made (Figure <ref>). We have also carefully checked each of our HST orbits to search for evidence of energetic flares, which as detailed in the Introduction are known to occur in AU Mic; however, we have found no clear evidence that the system experienced any significant flares during our observations. As far as we are aware, no other color dataset presented in the literature has been analyzed for such a signature outside of our own. We do note however that it would take a particularly strong flare to create such a signature.Therefore, we consider the possibility that AU Mic's scattered light disk could exhibit variability in its observed optical colors. Variability in spatially resolved scattered light imagery of younger disks has been reported in several cases, and has generally been interpreted to arise from variable illumination of the outer regions of these disks induced by structure in the inner disk regions <cit.>. AU Mic itself is also observed to have fast moving features in the SE side of its disk, whose origin is still unclear <cit.>. The color of a debris disk in scattered light can be influenced both by the grain size distribution present within the disk and by the detailed chemistry of these grains (e.g. ); hence, any purported color change must either be caused by a change in these grain properties or by a color change in illuminating source. Since we have already searched for and excluded the presence of large stellar flares during our observations and it seems unlikely that the chemistry of grains would change on the timescales involved, we suggest that the most likely cause of any time variable color change in AU Mic arises from a change in the grain size distribution.We speculate that the observed change in blue-optical colors of the disk that we observe at more extended projected separations (35-45 AU) could reflect a change in the grain size distribution that is scattering light. Specifically, the change in the disk's color from 30-45 AU from being “increasingly bluer with distance” <cit.> to a constant (SE side) and decreasing (NW side), small blue optical color could indicate a modest reduction in the relative number of sub-micron size grains being scattered at larger projected separations.Analogously, the bluer optical colors that we are now clearly quantifying for the first time at projected separations of 12-17 AU could indicate an enhancement of sub-micron size grains inside of the purported “birth ring” of the system. However, the dust seen at small projected separations could also be the same dust belt observed by <cit.> at 4due to projection effects. Therefore, it is possible that the bluer optical colors observed at small projected separations may also be due to either viewing the same dust belt at differing scattering angles (e.g. a difference in color due to forward scattered light versus light scattered at a 90 angle) or due to a scattering phase function with a slight wavelength dependence. While beyond the scope of this paper, detailed modeling of the scattered light from AU Mic's debris disk is needed to distinguish between the scenarios that may be responsible for the bluer color observed at small projected separations.Identifying the origin of the purported change in the population of sub-micron size grains at more extended projected separations (30-45 AU) will likely require detail modeling of the dynamics of the AU Mic disk, which is beyond the scope of this paper. We do note that the dynamical state of the AU Mic disk was fundamentally different during the original epoch (2004) of HST (F606W) observations, that have generally set the optical color baseline of the disk <cit.>. Specifically, during the 2004 observations, only the fast, outward moving feature “E” identified by <cit.> had just begun to enter this region of the disk; its extrapolated location in the disk during 2004 was ∼34 AU, at a projected separation just inside of the reported increase in blue color. Features “D” and “C” were at extrapolated locations of ∼21 and ∼16 AU respectively during 2004. By contrast, feature “E” had moved beyond the extent of our detection capabilities (∼50 AU) in our 2012 STIS coronagraphic spectroscopy data, as illustrated in Figures <ref> and <ref>.Moreover, a second fast, outward moving feature “D” (∼37 AU) had moved within the 30-45 AU range by the epoch of our 2012 STIS observations, and feature “C” (∼27 AU) was nearing this range. We speculate that there might be a causal correlation between these fast moving features propagating through the disk and the color changes that we are seeing in our scattered light data. Specifically, we speculate that as the fast moving features propagate outwards, they could preferentially elevate at least some small sub-micron size grains to higher latitudes above the disk midplane, and therefore reduce the previously noted increase in blue color with increasing projected separation. We note that the change in disk color revealed by our data seems to happen on both the NW and SE-side of the disk, whereas the fast, outward moving features have only been reported on the SE-side of the disk to-date. <cit.> recently suggested that the fast moving features are due to collisional chain reactions that form in an “avalanche zone", where AU Mic's birth ring intersects a secondary debris ring that was formed within the last several ten thousand years. They further suggest that this zone is at very small projected separations (∼3.5 AU) and produces sub-micron sized dust that is then blown outward by the central star's wind. Interestingly, this may explain the bluer color of the disk at small projected separations (<15AU) that we detected in our coronagraphic spectra, although perhaps only on the SE side of the disk where our data are strongly affected by a PSF residual. Coronal mass ejections might also be responsible for the clearing out of some small sub-micron sized grains from the disk (e.g. ) and recent work by <cit.> shows that the stellar wind may play a key role in the velocity profile of AU Mic's fast moving features. However, it is difficult to explain the bluer color of the disk at small projected separations on both the NW and SE sides while also explaining the time variability of the disk's color at larger projected separations, even with the combination of these scenarios. Regardless of the explanation, it is clear that AU Mic's disk is undergoing poorly understood complex physical phenomena that is changing the localized distribution of sub-micron sized grains on years long timescales. We encourage future observations to re-examine the optical colors of AU Mic's disk and to confirm the two new trends revealed by our data. We note that the fast outward moving features A,B,C,D, and E <cit.> would be located at projected separations of 12.6, 20.3, 32.9, 46.1, and 60.6 AU in 2017. If the propagation of these features through the disk influences the localized distribution of small grains as the features pass by, we would expect to see additional color changes in these regions of the disk in new coronagraphic spectroscopy data.§ CONCLUDING REMARKS We find that the color of AU Mic's debris disk is bluest at small (12-17 AU) projected separations.These results both confirm and quantify the speculative findings qualitatively noted by <cit.>, and are different than IR observations that suggested a uniform blue or grey color as a function of projected separation between 10-30 AU <cit.>. Unlike previous literature that reported the color of AU Mic's disk became increasingly more blue as a function of projected separation beyond ∼30 AU (e.g. <cit.> state that “The ratios of these [surface brightness] profiles indicate that the disk becomes increasingly blue at larger radii, at least for r=30-60 AU."), we find the disk's optical color between 35 and 45 AU to be uniformly blue on the SE side of the disk and decreasingly blue on the NW side. We note that this apparent change in disk color at larger projected separations coincides with several fast, outward moving “features” recently noted by <cit.> that are moving through this region of the disk. We speculate that these phenomenon might be related, and that the fast moving features could be changing the localized distribution of sub-micron sized grains as they pass by, thereby reducing the blue color of the disk in the process. We encourage follow-up optical spectroscopic observations of the AU Mic to both confirm this result, and search for further changes in the disk color caused by additional fast moving features propagating through the disk. This paper is based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with program GO-12512. Facilities: HST (STIS).[Ardila et al.(2007)]Ardila Ardila, D. R., Golimowski, D. A., Krist, J. E., et al. 2007, , 665, 512[Augereau & Beust(2006)]Augereau Augereau, J.-C., & Beust, H. 2006, , 455, 987 [Barrado y Navascués et al.(1999)]Barrado Barrado y Navascués, D., Stauffer, J. R., Song, I., & Caillault, J.-P. 1999, , 520, L123[Bessell(1990)]Bessell Bessell, M. 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A. 2001, , 562, L87 | http://arxiv.org/abs/1705.09291v2 | {
"authors": [
"Jamie R. Lomax",
"John P. Wisniewski",
"Aki Roberge",
"Jessica K. Donaldson",
"John H. Debes",
"Eliot M. Malumuth",
"Alycia J. Weinberger"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170525180000",
"title": "Optical Coronagraphic Spectroscopy of AU Mic: Evidence of Time Variable Colors?"
} |
Department of Physics,Faculty of Exact Sciences and Informatics, Hassiba Benbouali University of Chlef P.O. Box 151, 02000, Ouled Fares, Chlef, Algeria.We investigate effects ofthree-body contact interactions on a trapped dipolar Bose gas at finite temperature using the Hartree-Fock-Bogoliubov approximation. We analyze numerically the behavior of the transition temperature and the condensed fraction. Effects of the three-body interactions, anomalous pair correlations and temperature on the collective modes are discussed.03.75.Hh, 67.85.DeDipolar Bose gas with three-body interactions at finite temperature Abdelâali Boudjemâa===================================================================§ INTRODUCTION Recently, Bose-Einstein condensates (BEC) with dipole-dipole interactions (DDI) have received considerable attention both experimentally and theoretically<cit.>.Dipolar quantum gases, in stark contrast to dilute gases of isotropic interparticle interactions, offer fascinating prospects of exploring ultracold gasesand novel many-body quantum phases with atomic interactions that are long-range and spatially anisotropic.Impressionnant efforts has been directed towards the ground statesproperties and elementary excitations of such dipolar systems at both zero and finite temperatures <cit.>.Three-body interactions (TBI) are ubiquitous and play an important role in a wide variety of interesting physical phenomena,and yield a new physics and many surprises not encountered in systems dominated by the two-body interactions. Recently, many experimental and theoretical techniques have been proposed to observeand realize the TBI in ultracold Bose gas <cit.>.For instance, inelastic three-body processes, including observations of Efimov quantum states and atom loss from recombination have been also reported in Refs <cit.>. Few-body forces induce also nonconventional many-body effects such asquantum Hall problems <cit.>and the transition from the weak to strong-pairing Abelian phase <cit.>. In 2002, Bulgac <cit.> predictedthat both weakly interacting Bose and Fermi gases with attractive two-body and large repulsive TBI may form droplets. Dasgupta<cit.> showed that if the two-body interactions were attractive, the presence of the TBI leads toa nonreversible BCS-BEC crossover.Furthermore, many proposals dealing with effects of effective TBI in ultracold bosonic atoms in an optical lattice or a superlattice are reported in<cit.>. Moreover, the TBI in dilute Bose gases may give rise to considerably modify the collective excitations at both zero and finite temperatures <cit.>,the transition temperature, the condensate depletion and the stability of a BEC in a one (1D)- andtwo (2D)-dimensionaltrapping geometry <cit.>.However, little attention has been paid to effects of TBI on dipolar BECs.For instance, it has been argued that TBI play a crucial role in the stablization of the supersolid state in 2D dipolar bosons <cit.>and in the quantum droplet state in 3D BEC with strongDDI <cit.>.Our main aim here is to study effects of the TBI on weakly interacting dipolar Bose gases in a pencake trap at finite temperature. To this end, we employ the full Hartree-Fock-Bogoliubov (HFB) approximation. This approach which takes into account the pair anomalous correlations has been extensively utilizedto describe the properties of both homogenous and trapped BECs with contact interactions <cit.>.We will show in particular how the interplay between the DDI, TBI and temperature can enhance the density profiles, the condensed fraction and the collective modes of the system.The rest of the paper is organized as follows. In Sec.<ref>, we introduce the full HFB formalism for dipolar BECs with TBI. We discuss also the issues encountered in our model and present the resolution of these problems. Section <ref> is devotedto presenting and discussing our numerical results. Our conclusions are drawn in Sec.<ref>. § THREE-BODY MODEL FOR DIPOLAR BOSONSConsider a dipolar BEC with contact repulsivetwo-body interactions and TBI confined in a pancake-shaped trap with the dipole moments of the particles oriented perpendicular to the plane.It is straightforward to check that the condensate wavefunction Φ ( r)=⟨ψ̂( r)⟩, with ψ̂( r) beingthe Bose field operator, satisfies the generalized Gross-Pitaevskii (GP) equation <cit.>iħ∂Φ ( r,t)/∂ t ={ h^sp+ g_2 [n_c( r,t) +2 ñ( r,t) ]+ g_3/2 [n_c^2( r,t)+6n_c ( r,t)ñ( r,t)+ m̃^*( r,t) Φ^2( r,t)]}Φ ( r,t) + [ g_2 m̃ ( r,t) +3 g_3/2m̃( r,t)n_c ( r,t) ] Φ^*( r,t)+∫ d r' V_d( r- r')[ n ( r',t) Φ( r,t)+ ñ ( r, r',t)Φ( r',t) +m̃ ( r, r',t)ϕ^*( r',t)], where h^sp =-ħ^2 Δ/2m +U( r) is the single particle Hamiltonian, m is the particle mass,U( r)= m ω_ρ^2 (ρ^2+λ^2 z^2)/2, ρ^2=x^2+y^2, λ=ω_z/ω_ρ is the ratio between the trapping frequencies in the axial and radial directions. The two-body coupling constant is defined by g_2=4πħ^2 a/m with a beingthe s-wave scattering length which can be adjusted using a magnetic Feshbach resonance. The three-body coupling constant g_3 is in general a complex number with Im(g_3) describing the three-body recombination loss and Re(g_3)accounting for the three-body scattering parameter. In the present paper, we do not consider the three-body recombination terms i.e. Im(g_3)=0, so the system is stable which is consistent with recent experiments <cit.>. The DDI potential is V_d( r) = C_dd (1-3cos^2θ) / (4π r^3), where C_dd = M_0M^2 (= d^2/ϵ_0) is the magnetic (electric) dipolar interaction strength,and θ is the angle between the relative position of the particles r and the direction of the dipole. The condensed and noncondensed densities are defined, respectively as n_c( r)=|Φ( r)|^2, ñ ( r)= ⟨ψ̂̅̂^† ( r) ψ̂̅̂ ( r) ⟩ and n( r)=n_c( r)+ñ ( r) is the total density. The term ñ ( r, r') and m̃ ( r, r') are respectively, the normal and the anomalous one-body density matrices which account for the dipole exchange interaction between the condensate and noncondensate.Equation (<ref>) describes the coupled dynamics of the condensed and noncondensed components. For g_3=0, it recovers the generalized nonlocal finite-temperature GP equation with two-body interactions. For m̃ =0, Eq.(<ref>) reduces to the HFB-Popov equation <cit.> which is gapless theory. For m̃=ñ =0, it reduces to standard GP equation that describes dipolar Bose gases only at zero temperature.Upon linearizing Eq.(<ref>)around a static solution Φ_0, utilizing the parameterization Φ( r,t)=[Φ_0( r)+δΦ( r,t) ] e^-iμ t/ħ,where δΦ = ∑_k [u_k ( r) e^-i ε_k t/ħ+ v_k( r) e^i ε_k t/ħ],and ε_k is the Bogoliubov excitations energy. The quasi-particle amplitudes u_k( r), v_k( r) satisfy the generalized nonlocal Bogoliubov-de-Gennes (BdG) equations <cit.>: ε_k u_k ( r)= L̂ u_k ( r)+ M̂ v_k ( r) + ∫ d r' V_d( r- r') n ( r, r') u_k ( r')+ ∫ dr'V_d( r- r') m̅( r, r') v_k ( r'),-ε_k v_k ( r)= L̂ v_k ( r)+ M̂ u_k ( r) + ∫ d r' V_d( r- r') n ( r, r') v_k ( r')+ ∫ dr'V_d( r- r') m̅( r, r')u_k ( r'), whereL̂=h^sp+ 2g_2n ( r)+ 3g_3 [n_c^2 ( r) +4 n_c ( r) ñ ( r)+m̃^* ( r) Φ^2( r)+m̃ ( r) Φ^*2( r)]/2+ ∫ d r' V_d( r- r') n ( r')-μ,M̂=g_2 [Φ_0^2( r)+ m̃ ( r)]+g_3[ n_c^2( r)+3Φ_0^2( r) ñ ( r)+3Φ_0^2( r) m̃ ( r) ], n ( r, r')= Φ_0^*( r') Φ_0( r)+ ñ ( r, r') and m̅( r, r')= Φ_0( r') Φ_0( r) +m̃( r, r'). Equations (<ref>) and (<ref>) describe the collective excitations of the system. The normal and the anomalous one-body density matricescan be obtainedemploying the transformationψ̂̅̂=∑_k [u_k ( r) b̂_k+ v_k^*( r) b̂_k^†] ñ ( r, r') = ∑_k {[u_k^*( r') u_k ( r)+v_k( r')v_k^*( r) ] N_k( r)+v_k( r')v_k^*( r)}, m̃ ( r, r') = -∑_k {[u_k( r') v^*_k ( r)+u_k( r)v_k^*( r') ] N_k( r)+u_k( r')v_k^*( r)},where N_k=⟨b̂_k^†b̂_k⟩=[exp(ε_k/T)-1]^-1 are occupation numbers for the excitations. The noncondensed and anomalous densities can simply be obtained by setting, respectively ñ ( r)=ñ ( r, r) and m̃ ( r)=m̃ ( r, r) in Eqs.(<ref>) and (<ref>).From now on we assume that ñ ( r, r')=m̃ ( r, r')=0 for r≠ r' <cit.>. It is worth stressing that the omission of the long-range exchange term does not preclude the stability of the system <cit.>.As is well known, the full HFB theory sustains some hindrances notably the appearence ofan unphysical gap in the excitation spectrum and the divergence of the anomlaous density. In fact, this violation of the conservation laws in the HFB theory is due to the inclusion of the anomalous density which in general leads to a double counting of the interaction effects. The common way to circumvent this problem is to neglect m̃ in the above equations, which restores the symmetry and hence leads to a gapless theory, but this is nothing else than the Popov approximation. To go consistently beyond the Popov theory, one should renormalize the coupling constanttaking into account many-body corrections for scattering between the condensed atoms on one hand and the condensed and thermal atoms on the other. Following the procedure outlined in Refs <cit.> we obtaing_2 |Φ|^2Φ+g_2m̃Φ^*+3g_3 /2 n_cm̃Φ^* =g_2 [1+m̃(1+3g_3n_c/g_2) /Φ ^2] |Φ|^2Φ= g_R |Φ|^2Φ. This spatially dependent effective interaction, g_R is somehow equivalent to the many body T-matrix <cit.>. A detailed derivation of g_R, including the term g_3, will be given elsewhere.It is easy to check that if one substitutes (<ref>) in the HFB equations, we therefore, reinstate the gaplessness of the spectrum and the convergence of the anomalous density. § NUMERICAL RESULTS For numerical purposes, it is useful to set theEqs.(<ref>)-(<ref>) into a dimensionless from. We introduce the following dimensionless parameters: the relative strength ϵ_dd=C_dd/3g_2 (ϵ_dd=0.16 for Cr atoms)which describes the interplay between the DDI and short-range interactions, and g̅_3 =g_3 n_c/g_2 describes the ratio between the two-body interactions and TBI. Throughout the paper, we express lengths and energies in terms of the transverse harmonic oscillator length l_0=√(ħ/m ω_ρ) and the trap energy ħω_ρ, respectively. Figure.<ref> shows that the noncondensed and the anomalous densities increase with the TBI which leads to reduce the condensed density. A careful observation of the same figure reveals that the m̃ is larger than ñ at low temperature which is in fact natural since the anomlaous densityitself arises and grows with interactions<cit.>. When the temperature approaches to the transition, one can expect that m̃vanishes similar to the case of a BEC with a pure contact interaction <cit.>.In Fig.<ref> we compare our prediction for the condensed fraction N_c/N with the HFB-Popov theoretical treatment and the noninteracting gas. As is clearly seen, our results diverge from those of the previous approximations due to the effects of the TBI. This means that both the condensed fraction and the transition temperature decrease with increasing the TBI. Before leaving this section, let us unveil the role of the TBI on the collective excitations. According to Fig.<ref>, we can observe that our results deviate from the HFB-Popov (which have also been found in <cit.>)at higher temperatures for m=0 and 2 excitations.The reason of such a downward shift which enlarges as the temperature approaches T_c, is the inclusion of both anomalous pair correlations and TBI. A similar behavior holds in the case of BEC with short-range interactions (see e.g. <cit.>). Fig.<ref> depicts also that both the full HFBand the HFB-Popov produce a small shift from 1 for the Kohn mode ω/ω_ρ=1 at higher temperatures. One possibility to fix this problem might be the inclusion of the dynamics of the noncondensed and the anomalous components. A suitable formalism to explore such a dynamics is the time-dependent HFB theory <cit.>.§ CONCLUSIONIn conclusion, we have deeply investigated the properties of dipolar Bose gas confined by a cylindrically symmetric harmonic trapping potential in the presence of TBI at finite temperature. The numerical simulation of the full HFB model emphasized that the condensed fraction and the transition temperatureare reduced by the TBI. Effects of the TBI and temperature on the collective modes of the system are notably highlighted. We found that the full HFB approach in the presence of the TBI reproduces the HFB-Popov results of <cit.> only at low temperature while both approaches diverge each other when the temperature is close to T_c. One can expect that the same behavior persists in the case of a density-oscillating ground states known as a biconcave state predicted in Ref <cit.>. § ACKNOWLEDGMENTSWe are grateful to Dmitry Petrov and Axel Pelster for the careful reading of the manuscript and helpful comments.28Baranov See for review:M. A. Baranov, Physics Reports 464, 71 (2008). PfauSee for review: T. Lahaye et al., Rep. Prog. Phys. 72, 126401 (2009). Carr See for review: L.D. Carr, D. DeMille, R.V. Krems, and J. Ye, New J. Phys.11,055049 (2009). 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"authors": [
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Experimental status of the nuclear spin scissors modeP. Schuck Received/ Accepted======================================================empty emptyRecovering surgical scene structure in laparoscope surgery is crucial step for surgical guidance and augmented reality applications. In this paper, a quasi dense reconstruction algorithm of surgical scene is proposed. This is based on a state-of-the-art SLAM system, and is exploiting the initial exploration phase that is typically performed by the surgeon at the beginning of the surgery. We show how to convert the sparse SLAM map to a quasi dense scene reconstruction, using pairs of keyframe images and correlation-based featureless patch matching. We have validated the approach with a live porcine experiment using Computed Tomography as ground truth, yielding a Root Mean Squared Error of 4.9mm.§ INTRODUCTIONThere is much ongoing research to develop Augmented Reality (AR) surgical guidance systems for improving laparoscopic surgery. The idea is to allow hidden structures such as organ vessels and tumors to be visualized in real-time, by registering imaging data from different modalities such as CT or MRI <cit.>. An important research objective is to register the second modality with the laparoscopic images automatically without the use of artificial tracking markers or external tracking equipment. Feature-based solutions were proposed <cit.>, however they fail in the case of textureless tissues where feature detection is extremely unreliable. Feature tracking <cit.> is very fragile at textureless regions, and can suffer from drift and drop-off due to occlusion, sudden camera motion and motion blur. Furthermore feature-based solutions do not produce a dense 3D scene reconstruction, which is necessary for accurate inter-modal registration <cit.>.Several computer vision based solutions have been proposed for 3D reconstruction of laparoscopic scenes. Structure-from-Motion (SfM) <cit.> is a well-established approach, however it requires offline batch processing and is not suitable for real-time application. Shape from Shading <cit.> exploits the shading effect to deduce the scene's 3D structure, however it is weakly constrained problem and require multiple depth cues in a hybrid method, such as SfM/SfS as concluded by Collins et al. <cit.>. Stereo-vision based solutions were also proposed <cit.> for dense scene reconstruction, however they are not applicable to monocular laparoscopes, which are by far the most common type.Recently, SLAM (Simultaneous Localization And Mapping) systems have emerged as an excellent approach to reconstruct laparoscopic scenes and to compute the laparoscope's 3D pose in real-time. Typically, in the laparoscopic surgery an initial exploration is performed in order to explore the abdominal cavity. During exploration the surgeon does not manipulate the environment with tools, so the scene can be assumed to be rigid. This has been developed for both stereo<cit.> and monocular laparoscopes <cit.>. ORB-SLAM <cit.> in particular shows remarkable tracking performance but at the expense of low map density <cit.>. This is because it only reconstructs the 3D positions of sparse ORB features that have been matched in two or more keyframes.We propose a SLAM based quasi dense reconstruction algorithm which is able to reconstruct the surgical environment using only a monocular endoscope and no extra tracking equipment. It works by densifying a sparse reconstructed map computed during exploration phase by e.g. ORB-SLAM, using pairs of keyframe images and correlation-based featureless patch matching. Only a small number of relevant keyframe pairs are selected, therefore keeping computation time low. Keyframe pairs are selected using their respective baseline in the covisibilty graph, and are treated as a stereo pair. Densification is then done in three main steps: initial feature based densification where we do a 3D reconstruction of unmatched features (cf. Sec. <ref>), depth propagation where we propagate the reconstruction to featureless regions (cf. Sec. <ref>) and finally reconstruction post-processing, where outliers are removed and the reconstruction is smoothed (cf. Sec. <ref>).Semi-dense SLAM has been proposed in <cit.> limiting the dense reconstruction to highly textured image areas. In contrast we densify regions with low contrast and without planarity assumption in low gradient regions as proposed in <cit.>. The advantage of densifying the sparse map after exploration phase is to maintain the per-frame computation time during SLAM, which is important to achieve reliable and robust tracking before and after the dense mapis estimated. Furthermore, unlike <cit.> we use densification based on Normalized Cross Correlation (NCC), which handles significant illumination changes that are common in surgical scenes <cit.>. The resulting quasi dense reconstruction can be used to accurately register pre-operative data to laparoscope's images, which is a vital component for AR surgical guidance.§ QUASI DENSE 3D SCENE RECONSTRUCTION§.§ Method overview We outline the proposed method in Figure <ref>.We assume the laparoscope is pre-calibrated with fixed intrinsic using <cit.> immediately before exploration. During both SLAM and reconstruction densification we pre-process each frame to handle particular challenges of laparocopic image data (cf. Sec. <ref>). During exploration, SLAM is run until the end of this phase, which typically lasts no more than a minute. We use ORB-SLAM but any good feature-based SLAM approach could be used. The SLAM process is denoted by the top loop in Figure <ref>. This outputs a set of keyframes, their respective camera poses, a set of features detected in each keyframe and sparse 3D map. Next the three stages: feature based densification, depth propagation and reconstruction post-processing are run. Once finished, the laparoscope's pose can be tracked in the incoming laparoscopic frames in real-time using the sparse ORB-SLAM map.§.§ Frame pre-processing We first detect and eliminate specular reflections to avoid introducing false features to the SLAM system. This is done by converting the RGB frame to HSV and thresholding the saturation component. All detected features in these areas are ignored. Most feature detectors (including ORB) work on monochrome frames. We compute these by converting the RGB frame to monochrome using the average of the green and blue channels. This is because they give the highest contrast for human tissue <cit.>. §.§ Building the keyframe neighborhood graphWe construct a neighborhood graph G that connects pairs of SLAM keyframes. This is a sparse graph with typically O(g) edges, where g is the number of keyframes. Sparseness is necessary to keep processing time low. Each edge (i,j) in G corresponds to a stereo pair, and we use this for both feature-based densification and depth propagation. This is constructed as follows. For each keyframe i we compute the stereo baseline α with respect to all neighbor keyframes as a ratio between sparse map median depth and the distance between the two keyframes. We add an edge (i,j) if α falls within the rangeα_0 ≤α≤α_1. The lower-bound α_0 ensures there is sufficient baseline with which to reliably reconstruct points in 3D. The upper-bound α_1 ensures that the keyframes are not too far from each other. We give the default values of α_0 and α_1 (and all other parameters defaults) in Table <ref>. §.§ Feature based densification We process each keyframe pair (i,j)∈ G as follows. We have two types of features detected in i: matched f and unmatched features f'. The matched features are those that have been already matched by the SLAM system in another keyframes, have been triangulated, and have been inserted into the map. Our goal is to reconstruct each feature in f'. The process works by a cross-correlation search guided by epipolar geometry as illustrated in Fig. <ref>. This works by searching exhaustively over an epipolar line segment lwith a margin of 10 pixels using NCC matching with a N X N window. We then triangulate the matches and filter them for removing outliers according to three criteria: Matches with NCC score less than a threshold τ are eliminated. Matches with negative depths are eliminated. Matches with a parallax angle lower than γ are eliminated. The triangulated 3D position of each matched feature that passes the filters is then inserted into the map.We bound the length of the epipolar segment l to keep computation cost low using median depth of all visible points in keyframe j. Two extreme points on the back-projected ray are used to bound l which are P_min and P_max. The two points depths are computed by averaging and doubling median depth of visible map points, respectively.§.§ Featureless depth propagation After feature-based densification, we further densify the map at featureless regions through a depth propagation algorithm. The process works on each keyframe pair (i,j)∈ G as follows. First we take all points that were matched in keyframes i and j, and use their depths as seed depths which are then propagated to neighboring pixels. We then continue to propagate depth around seeds on best-first basis by popping a seeds queue, as proposed in <cit.>. New matches are added to the queue as the algorithm iterates until no more matches can be popped.Consider a seed point with a 2D position m in keyframe i and m' in keyframe j, with N(m) and N(m') spatial neighbored pixels, respectively in a 6 × 6 window. These seed matches are used to control the smoothness of the disparity estimation of all N(m) and N(m') pixels. For each neighbored pixel u ∈ N(m) a NCC is used to find a corresponding match u' in a 6 × 6 window centered in the corresponding spatial location in keyframe j, that has higher NCC score than τ and satisfy the smoothness constrain defined in eq. (1). We use β to control the smoothness of the disparity estimation. NCC is used as a similarity measure during propagation step. N(m,m') = { (u,u'), ||(u-u')-(m-m')|| ⩽β} §.§ Outlier removal and denoising Because depth propagation operates on keyframe pairs, there will be some disagreement due to noise across different keyframe pairs, typically at very low-textured regions. We deal with this by a robust averaging and merging. First we detect any remaining outliers in the quasi dense map using point neighborhood statistics <cit.>. This works by eliminating points if they are unusually far apart from the nearest κ points, according to a threshold ρ multiplied by standard deviation of all points distances. We then remove noise using Moving Least Square (MLS) <cit.>. This works by fitting a local plane to each point using all η nearest neighbors. The denoised point is then computed by orthogonally projected onto the fitted plane.§ EXPERIMENTAL RESULTS §.§ Experimental designWe evaluated our approach with a live in-vivo porcine experiment using a CT scan as ground-truth. In this experiment we reconstructed a porcine liver surrounding abdominal viscera, abdominal wall and diaphragm (cf. Fig. <ref>(e,f)) from 10 seconds exploratory video. Example frames are shown in Fig. <ref>(a). A CT was then acquired during 10 second expiration breath hold and is manually segmented by an expert to generate a 3D volume with 0.876mm x 0.876mm x 0.799mm voxel size and 749, 318 vertices, Fig. <ref>(a).Fig. <ref> shows the results of each step in reconstruction algorithm. Fig. <ref>(b) shows the sparse ORB-SLAM map from the initial exploration, where map points are shown in red and keyframes poses are the blue rectangles. Fig. <ref>(c), shows the newly added points, in blue after the feature based densification step. Fig. <ref>(d) shows the reconstruction after the featureless densification stage. Fig. <ref>(e,f) shows the reconstruction from different view points after outlier removal and denoising with normal ORB map point highlighted in red and the newly added points from feature based densification stage are highlighted in blue. Note that there are holes in the reconstruction due to specular reflections and regions of extremely homogeneous texture. More details can be appreciated at our video <cit.>. Another video <cit.> shows the reconstruction results of one liver sequence available at Hamlyn Centre Laparoscopic/Endoscopic Video Datasets <cit.>.§.§ Implementation details and computation timeThe system is implemented in C++ with OpenCV and PCL libraries and executed on desktop PC with Intel Core i7 CPU @ 2.6 GHz and 4GB RAM. The average tracking time is 23ms with 1600x900 image resolution. Average time required for: feature based densification was 5ms per keyframe, depth propagation was 25ms per keyframe (time for matching and triangulating points), MLS denoising was 1min due to computing normal for each points and polynomial fitting. The total number of reconstructed points was 348, 068 point. After obtaining dense scene reconstruction, the endoscope pose is then tracked in 25ms on average because only the sparse ORB-SLAM map is considered. §.§ Registering the map to CT and measuring accuracyAs with any SLAM system, our reconstruction is up to a similarity transform (i.e. an arbitrary scale and rigid coordinate transform). To evaluate accuracy we aligned the reconstruction to the CT model (cf. Fig. <ref>(a)) using a best-fitting similarity transform. This was found by manually selecting 3 landmarks to roughly estimate the scale and initial orientation by Horn's algorithm <cit.>. Then Iterative Closest Point (ICP) was run until convergence to refine the initial alignment. Fig. <ref> (b,c) shows the alignment between CT model in yellow and reconstructed dense map in white. Accuracy was measured by the euclidean distance of each map point to its closest point on the CT model's surface. The Root Mean Squared Error (RMSE) was used to evaluate the overall error, which was 4.9mm. Fig. <ref>(a,b) shows distances distributions of total number of map points besides the accumulative histogram. It can be seen that 85% of points with lower distances than 6.7mm. Thus, thresholding errors lower than 6.7mm as inliers (85%) and the rest as outliers(15%), (cf. Fig. <ref>(d)) reduces the RMSE error to 2.8mm. Those 15% outliers were abdominal wall points and it is quite hard to correctly reconstruct them due to non-rigid deformation by the breathing cycle which prohibits correct re-projection <cit.>.§ CONCLUSIONSWe have presented a simple approach for quasi-dense 3D reconstruction of laparoscopic scenes, that robustly densifies a sparse SLAM reconstruction. Because densification is embedded in SLAM, we keep all the benefits of state-of-the-art feature-based SLAM system such a ORB-SLAM, including fast tracking, mapping and automatic relocalization. Our preliminary results on an in-vivo porcine dataset are very promising, with a RMSE of 4.9mm. Future directions include improving reconstruction accuracy by including multiple views in the reconstruction process of the same point. Reducing dense reconstruction computation time, especially the denoising stage. Additionally, we aim to use the reconstruction for automatic registration with a CT model.-12cm § ACKNOWLEDGMENTThis work is part of a project of the Investissements d'Avenir program ("Investing in the Future") called 3D-Surg, funded by BPIfrance. It is also partially funded by the Spanish government DPI2015-67275-P and Aragonese DGA T04-FSE.Puerto-Souza G. Puerto-Souza, J. A. Cadeddu, and G. Mariottini, Toward long-term and accurate augmented-reality for monocular endoscopic videos. IEEE Trans. Bio. Eng., vol. 61(10), pp. 2609-20, 2014. Sylvain S. Bernhardt, S. A. Nicolau, L. Soler, C. Doignon, The status of augmented reality in laparoscopic surgery as of 2016. Medical Image Analysis, vol.37, pp. 66-90, 2017. Collins2017 T. Collins, P. Chauvet, C. Debize, D. Pizarro, A. Bartoli, M. Canis, N. Bourdel,A System for Augmented Reality Guided Laparoscopic Tumour Resection with Quantitative Ex-vivo User Evaluation, CARE 2016. Haouchine N. Haouchine, J. Dequidt, M. Berger, and S. Cotin, Monocular 3d reconstruction and augmentation of elastic surfaces with self-occlusion handling. IEEE Trans. Vis. Comput. Graph, vol 21(12), pp. 1363-1376, 2015. KLT C. Tomasi and T. Kanade, Detection and Tracking of Point Features. Carnegie Mellon University Technical Report CMU-CS-91-132, 1991. Sun D. Sun, J. Liu, C.-A. Linte, H. Duan, R.-A. Robb, Surface Reconstruction from Tracked Endoscopic Video Using the Structure from Motion Approach. MIAR 2013, pp. 127-135, 2013 Hu M. Hu, G. Penney, M. Figl, P. Edwards, F. Bello, R. Casula, D. Rueckert, D. Hawkes, Reconstruction of a 3D surface from video that is robust to missing data and outliers: application to minimally invasive surgery using stereo and mono endoscopes. Med. Image Anal, vol. 16(3), pp. 597-611, 2012. Collins2012 T. Collins, A. Bartoli, Towards Live Monocular 3D Laparoscopy using Shading and Specularity. IPCAI 2012, pp 11-21, 2012 LinB. Lin, A. Johnson, X. Qian, J. Sanchez, Y. Sun, SimultaneousTracking,3D Reconstruction andDeformingPointDetectionforStereoscopeGuidedSurgery. MICCAI, pp. 35-44, 2013. Stoyanov2010 D. Stoyanov, M.V. Scarzanella, P. Pratt,G.Z. Yang, Real-time stereo reconstruction in robotically assisted minimally invasive surgery, MICCAI, pp. 275–282, 2010 Mountney1 P. Mountney, D. Stoyanov, A.J. Davison, G.Z. Yang, Simultaneous stereoscope localizationandsoft-tissuemappingforminimalinvasivesurgery. MICCAI 2006, Part I. LNCS, vol. 4190, pp. 347-354, 2006 Mountney2 P. Mountney, G.-Z. Yang, Motion compensated SLAM for image guided surgery. MICCAI, pp. 496-504, 2010 Grasa O. G. Grasa, E. Bernal, S. Casado, I. Gil, J.M.M Montiel, VisualSLAMfor handheld monocular endoscope. IEEE Trans. on Med. Imag., vol. 33(1), pp. 135-146, 2014. NaderN. Mahmoud, I. Cirauqui, A. Hostettler, C. Doignon, L. Soler, J Marescaux, J. M. M. Montiel, ORBSLAM-based Endoscope Tracking and 3D Reconstruction, CARE, pp. 72-83, 2017 Raul R. Mur-Artal, J. M. M. Montiel, J. D. Tardos, ORB-SLAM: A Versatile and Ac- curate Monocular SLAM System. IEEE Trans. on Robotics, vol. 31(5), pp. 1147-1163, 2015 Engel J. Engel, T. Schops, and D. Cremers, LSD-SLAM: Large-scale direct monocular SLAM.ECCV, pp. 834-849, 2014 Mur-Artal R. Mur-Artal and J.D. Tardós, Probabilistic Semi-Dense Mapping from Highly Accurate Feature-Based Monocular SLAM. RSS 2015. Alejo A. Concha and J. Civera, DPPTAM: Dense piecewise planar tracking and mapping from a monocular sequence. IROS 2015, pp. 5686-5693. 2015. Zhang Z. Zhang. A Flexible New Technique for Camera Calibration. IEEE Trans. Pattern Anal, vol. 22(11), pp. 1330-1334, 2000. Tromberg B. J. Tromberg, N. Shah, R. Lanning, A. Cerussi, J. Espinoza, T. Pham, L. Svaasand, J. Butler, Non-Invasive In Vivo Characterization of Breast Tumors Using Photon Migration Spectroscopy. Neoplasia, vol. 2(1-2), pp.26-40, 2000 Rusu R. B. Rusu, Z. C. Marton, N. Blodow, M. Dolha, and M. Beetz, Towards 3D Point cloud based object maps for household environments. Robotics and Autonomous Systems, vol. 56(11), pp. 927–941, 2008 Alexa M. Alexa, J. Behr, D. Cohen-Or, S. Fleishman, D. Levin, C. T. Silva, Computing and Rendering Point Set Surfaces. IEEE Trans. on Vis. and Comp. Graph., vol. 9(1), 2008 video1_myliver Youtube. (2017, May 25). SLAM based Quasi Dense Reconstruction For Minimally Invasive Surgery Scenes (Private Dataset)) [Video file]. Retrieved from https://www.youtube.com/watch?v=HQmVRSNFVu0&feature=youtu.be video2_HamlynLiver Youtube. (2017, May 25). SLAM based Quasi Dense Reconstruction For Minimally Invasive Surgery Scenes (Hamlyn Dataset) [Video file]. Retrieved from https://www.youtube.com/watch?v=oG54CBzqVh0&feature=youtu.be HamlynDataset London, I.C., Hamlyn centre laparoscopic / endoscopic video datasets (2017). URL http://hamlyn.doc.ic.ac.uk/vision/. [Accessed 24 May 2017] Horn B.K. Horn, Closed-form solution of absolute orientation using unit quaternions. J OPT SOC AM A, vol. 4(4), pp. 629-642, 1987 Marcinczak J.M. Marcinczak, R.R. Grigat, Total Variation Based 3D Reconstruction from Monocular Laparoscopic Sequences.ABD-MICCAI, pp 239-247, 2014. | http://arxiv.org/abs/1705.09107v1 | {
"authors": [
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"J. M. M. Montiel"
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"published": "20170525094434",
"title": "SLAM based Quasi Dense Reconstruction For Minimally Invasive Surgery Scenes"
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equationsection=-17mmḍ e• ( [Antonov problem reviewed]Radial orbit instability in systems of highly eccentric orbits: Antonov problem reviewedE. V. Polyachenko and I. G. Shukhman]E. V. Polyachenko,^1E-mail: [email protected], I. G. Shukhman,^2E-mail: [email protected] ^1Institute of Astronomy, Russian Academy of Sciences, 48 Pyatnitskya St., Moscow 119017, Russia ^2Institute of Solar-Terrestrial Physics, Russian Academy of Sciences,Siberian Branch, P.O. Box 291, Irkutsk 664033, Russia[ [ December 30, 2023 ===================== Stationary stellar systems with radially elongated orbits are subject to radial orbit instability – an important phenomenon that structures galaxies. Antonov (1973) presented a formal proof of the instability for spherical systems in the limit of purely radial orbits. However, such spheres have highly inhomogeneous density distributions with singularity ∼ 1/r^2, resulting in an inconsistency in the proof. The proof can be refined, if one considers an orbital distribution close to purely radial, but not entirely radial, which allows to avoid the central singularity. For this purpose we employ non-singular analogs of generalised polytropes elaborated recently in our work in order to derive and solve new integral equations adopted for calculation of unstable eigenmodes in systems with nearly radial orbits. In addition, we establish a link between our and Antonov's approaches and uncover the meaning of infinite entities in the purely radial case. Maximum growth rates tend to infinity as the system becomes more and more radially anisotropic. The instability takes place both for even and odd spherical harmonics, with all unstable modes developing rapidly, i.e. having eigenfrequencies comparable to or greater than typical orbital frequencies. This invalidates orbital approximation in the case of systems with all orbits very close to purely radial.Galaxy: model, galaxies: kinematics and dynamics. § INTRODUCTION The radial orbit instability (ROI), first mentioned in a preprint by <cit.>, plays an important role in the evolution of initially spherically symmetric and axisymmetric systems leading to bar-like perturbations. It has been widely studied both analytically <cit.> and numerically <cit.>. There are two basic candidates for a physical mechanism of ROI:an analog of Jeans instability in the anisotropic media andan orbital approach based on tendency of any pair of orbits to align under their mutual gravity. Discussion on these topics can be found in <cit.>.A distinct approach is suggested by <cit.>, who give an example of dissipation-induced ROI. A comprehensive modern review on ROI can be found in <cit.>, who also suggest a new symplectic method for exploring stability of equilibrium gravitating systems.<cit.> presents a first formal proof of ROI for purely radial motion using the Lyapunov method. However, his proof is doubtful: the Lyapunov function is ill-defined due to a divergence of its time derivative at the lower limit of integration. Although the main conclusion of the paper is correct, a rigorous examination of the purely radial case is still needed.The goal of this paper is to reconsider the Antonov problem by applying our technique of an eigenvalue problem in the form of integral equations.For this purpose, we shall use a general family of modelsF(E,L)= H(L_T-L)/L_T^2F_0(E) ,where H(x) is the Heaviside function. We retain an arbitrary form for F_0(E) whenever possible, otherwise we admit a polytropic lawF_0(E) = N(L_T,q)/4π^3(-2E)^q.Here E=1/2(v_r^2+v_⊥^2)+Φ_0(r)≤ 0 and L=rv_⊥ are the energy and absolute value of the angular momentum, respectively; Φ_0(r) is the unperturbed gravitational potential. The additive constant in Φ_0 is chosen so that the potential vanishes at the outer radius of sphere R; the normalization constant N(L_T,q) is chosen so that the total mass of the system is M. In the calculations below we shall assume that M=R=G=1. Equilibrium properties of family (<ref>) with polytropic dependence from energy (<ref>), calledsoftened polytrope models, arespecially built to consider the limit of purely radial motion and studied in our paper <cit.>. Stability properties of some series (fixed q) are studied in <cit.>.The polytropic law includes a series of mono-energetic models, in which all stars have zero total energy, at the limit q→ -1<cit.>. A limit L_T → 0 in this series gives a well-known <cit.> modelwhich was employed in the Antonov's work. The model is particularly usefulin our case, since it provides the simplest eigenvalue equations, yet preserving all features of interest.As is already said, our proof is based on analysis and solution of characteristic equations for eigenmodes – spherical harmonics and corresponding complex frequencies ω, such that ones with the positive imaginary parts give unstable solutions. In Section 2 we derive the equation for a model withpurely radial orbits, using delta-function expansion technique <cit.>. The unperturbed distribution function (DF) of the purely radial system is proportional to the Dirac delta-function of the angular momentum (<ref>), while the perturbed DF is a linear combination of the delta-function and its derivatives (<ref>). The linearised kinetic equation and Poisson equation provide matrix equations (<ref>) and (<ref>), for even and odd spherical harmonics, respectively. Both of them contain infinite entities p_k defined by (<ref>) which are a manifestation of the central singularity.In Section 3 we use the integral equation technique for the two-parametric family of models (<ref>) withnearly radial orbits <cit.>. Since this family includes the purely radial model of Section 2, we can get a link between different parts of the integral equations obtained in Sections 2 and 3, as the control parameterL_T in the DF approaches zero. In particular, we infer the meaning of the infinite entities, eqs. (<ref>, <ref>).Then, this finding helps us in Section 4 to reduce further the legitimate integral equations of Section 3 for nearly radial orbits to fairly compact limiting integral equations (L_T ≪ 1), for even and odd spherical harmonics, (<ref>) and (<ref>), respectively. They allow to prove existence of the aperiodic even unstable spherical solutions, and absence of the odd unstable spherical solutions. The analytical results are accompanied in Section 5 with numerical eigenmodes' calculation for series q=-1 and q=-1/2.In Section 6, we show how the orbital approach breaks down in spherical systems with orbits very close to radial. Comparison of our numerical results with qualitative results by <cit.> shows that orbital approach is satisfactory for the systems with orbits of moderate eccentricity only.Lastly, Section 7 contains a summary and conclusion. Appendix A is devoted to Antonov's `proof' of the existence of ROI (in our terms and notations) with the help of Lyapunov function, as a reminder and demonstration of difficulties appearing in the investigation of the systems with pure radial systems. Appendix B clarifies the sense of diverging coefficients which appear in the equations for purely radial models with the help of limiting procedure from models with finite dispersion over the angular momentum (L_T 0).§ PURE RADIAL MOTION: Δ-FUNCTION EXPANSION Purely radial orbits possess zero angular momentum, L=0. Thus, for systems with purely radial motion, we demandF(E,L)=δ(L^2)F_0(E)=π/r^2 δ(v_θ) δ(v_φ) F_0(E) ,where δ(x) is the Dirac delta-function. The analysis for instability prescribes the following ansatz for the perturbed DF:f_1(t,r,θ,φ,v_r, v_θ,v_φ) = A(t,r,θ,φ, v_r) δ(v_θ) δ(v_φ) + B(t,r,θ,φ, v_r) δ'(v_θ) δ(v_φ)+ C(t,r,θ,φ, v_r) δ(v_θ) δ'(v_φ) ,where δ' denotes a derivative of the delta-function. From the linearized kinetic equationf_1/ t+v_rf_1/ r+v_θ/rf_1/θ+ v_φ/r sinθf_1/φ + (v_θ^2+v_φ^2/r-dΦ_0/dr)f_1/ v_r- (v_r v_θ/r-θ v_φ^2/r)f_1/ v_θ-(v_r v_φ/r+θ v_θ v_φ/r)f_1/ v_φ= Φ_1/ rF/ v_r+1/r Φ_1/θF/ fv_θ+ 1/r sinθ Φ_1/φF/ v_φ ,relations between decomposition coefficients A, B, and C can be obtained:A/ t+D̂A+2 v_r/r A-1/rB/θ-θ/r B -1/rsinθC/φ=π/r^2 F_0'(E) v_r Φ_1/ r ,B/ t+D̂B+3 v_r/r B=π/r^3 F_0(E) Φ_1/θ , C/ t+D̂C+3 v_r/r C=π/r^3 sinθ F_0(E) Φ_1/φHere D̂ is a differential operator,D̂=v_r / r-dΦ_0/dr / v_r ,and Φ_1 denotes a perturbed potential depending on the polar angle θ only through Legendre polynomials P_l,Φ_1 ≡χ(t,r) P_l(cosθ) ,since the eigenmode spectrum does not depend on the azimuthal number m<cit.>.Substitution to (<ref>–<ref>) gives C=0, while A ∝ P_l(cosθ), and B ∝P̣_l(cosθ)/ θ̣. It is convenient to introduce new functions A and B independent of the angles:A(t,r,θ,φ,v_r)= A(t,r,v_r)/r^2 P_l(cosθ) ,B(t,r,θ,φ,v_r)= B(t,r,v_r)/r^3 d P_l(cosθ)/dθ .The perturbed densityρ_1(t,r,θ,φ) ≡Π(t,r) P_l(cosθ)is an integral from A over the radial velocity,Π(t,r) = 1/r^2∫ A(t,r,v_r)dv_r .The eqs. (<ref>–<ref>) and the Poisson equation for the new functions take the form:A/ t+D̂ A+l (l+1)/r^2B=π v_r d χ/dr F_0'(E) ,B/ t+D̂ B=π χ(r) F_0(E) ,χ(r)=-4π G/2l+1∫ dr'∫ dv_r'A(r',v_r')F_l(r,r') ,with F_l(r,r')=r_<^l/r_>^l+1, r_<= min(r,r') and r_>= max(r,r').Below, we shall explore the system of eqs. (<ref>–<ref>) in terms of action–angle variables. Radial orbits can be treated as highly eccentric ellipses with vanishingly small minor axis, see Fig. <ref>. The stellar position is fixed by four variables, three of which determine the orbit length and orientation (e.g., energy E, angles θ and φ), and the last one – radial angle variable w– sets the position along the orbit,w=Ω(E)∫_0^rdr'/v_r(E,r') ,where Ω(E) is the frequency of radial oscillations, v_r is the radial velocity:v_r=±√(2[E-Φ_0(r)]) . During full revolution, star's angular variable changes in the range -2π≤ w ≤ 2π. Therefore, the most general functions of w have a period of 4π, and their Fourier expansions should read{χ(t,w,E),A(t,w,E),B(t,w,E) }= = ∑_n=-∞^∞{Φ_n/2(t,E), A_n/2(t,E), B_n/2(t,E)} e^inw/2with n running over all integers. Thus, variables (t, r, v_r) are changed to (t, E, w).As the star travels from the upper part of the orbit 0<w< 2π to the lower part -2π < w < 0, the polar and azimuthal angles change discontinuously:θ→π - θ ,φ→π + φ .This results in additional factor in case of odd spherical harmonics, so further analysis should be done separately for even and odd cases.§.§ Equations for even spherical harmonics In action–angle variables, the evolutionary equations for perturbations in purely radial systems areA/ t+Ω_1A/ w+l (l+1)/r^2(E,w)B= π Ω_1 χ(t, E,w)/ w F_0'(E), B/ t+Ω_1B/ w=π χ(t,E,w) F_0(E),χ(t,E,w)=-4π G/2l+1∫dE'/Ω_1(E') ×∫ dw'A(t, E',w')F_l[r(E,w),r'(E',w')] . Radius of a star as a function of angle w obeys the following symmetry conditions:r(2π-w) = r(-w) = r(w) ,thus coefficients{Φ_n/2(t,E), A_n/2(t,E), B_n/2(t,E)}= 1/4π∫_-2π^2π{χ(t,r),A(t,r,v_r),B(t,r,v_r) } e^-inw/2 dwvanish for odd n, and equal to{Φ_k(t,E), A_k(t,E), B_k(t,E)}= 1/π∫_0^π{χ(t,r),A(r,v_r),B(r,v_r) } cos(kw) dw ,otherwise (k=n/2). In this case, periodicity changes to 2π due to symmetry of potential and density perturbations with respect to transformation w → -w.Assuming that perturbations are ∝exp(-iω t), one can obtain from (<ref>)–(<ref>)-i (ω-k Ω) A_k+l (l+1)∑_k'=-∞^∞p_k-k'B_k'=iπ k Φ_k F_0'(E) ,-i (ω-k Ω) B_k=π Φ_k F_0(E)andΦ_k(E)=-2G/2l+1∫dE'/Ω(E')∑_k'=-∞^∞ K_k k'^ even (E,E') A_k'(E') ,where k and k' are integers,p_k(E)=1/2π∮dw/r^2(E,w) e^-ikw=1/2π∮cos (kw)/r^2(E,w) dwandK_k k'^ even(E,E')=4 ∫_0^πdw∫_0^πdw' cos(kw) cos(k'w')F_l(r,r') .Eq. (<ref>) emphasises an issue arising in systems with purely radial orbits – the integrals diverge in the centre (w → 0, |2π|). Thus, these expressions for p_k require an interpretation. Note that a similar difficulty appeared in <cit.>, but then no adequate attention has been paid. For example, ϖ(E)≡ p_0(E) is 1/r^2, averaged along the orbit:p_0=⟨1/r^2⟩≡1/2π∮dw/r^2=Ω/π∫_0^r_ max(E)dr/r^2 √(2E-2Φ_0(r)) ,and diverges evidently at r=0, since singularity of Φ_0 is weaker than 1/r^2. We plan to tackle the issue employing a family of models with nearly radial orbits and study the system of interest by considering more and more radially anisotropic systems (see Section 3).With (<ref>) and (<ref>), one can exclude A_k and B_k from the equation in favour of Φ_k:Φ_k(E)=-2 π G/2l+1∫dE'/Ω(E')∑_k'K_k k'^ even(E,E')/ω-n' Ω(E')×[l (l+1) F_0(E') ∑_m p_k'-m(E') Φ_m(E')/ω-m Ω(E').. - Ω(E') k' Φ_k'(E') dF_0(E')dE'] . For some F_0(E) (e.g., polytropes (<ref>) with q<0) the integral from the term including dF_0/dE' diverges. In this case one should use the Lagrangian form, which is obtained formally by integration by parts and omission of the surface term:Φ_n(E)=-2 π G/2l+1∫ dE' F_0(E')×∑_n'{l (l+1)/Ω(E')K_n n'^ even(E,E')/ω-n' Ω(E')∑_m p_n'-m(E') Φ_m(E')/ω-m Ω(E').. + d/dE'[n' Φ_n'(E')K_n n'^ even(E,E')/ω-n' Ω(E')]}(see <cit.> for details). For further analysis, it is convenient to use an alternative form of the last equation. Using the identity provided m n' 1ω-n' Ω·1ω-m Ω=1/Ω (n'-m)(1/ω-n' Ω-1/ω-m Ω),one can haveΦ_n(E)=-2 π G l (l+1)/2l+1∫dE' F_0(E')/Ω(E') ×∑_n' K_n n'^ even(E,E') {∑_m n'p_n'-m(E') Φ_m(E')/Ω(E') (n'-m)×[1/ω-n' Ω(E') - 1/ω-m Ω(E')]+Φ_n'(E') p_0(E')/[ω-n' Ω(E')]^2}-2 π G/2l+1∫ dE' F_0(E')∑_n'd/dE'[n' Φ_n'(E')K_n n'^ even(E,E')/ω-n' Ω(E')] .In the final equation, we have separated the last term which retains even in the case of radial oscillations l=0.To find out the meaning of integrals (<ref>) in expressions for p_k, a specific model is not important, since p_k(E) depends on the orbit, but not on the orbit distribution over the phase space. For simplicity we consider a monoenergetic model corresponding to q→ -1 limit in (<ref>), which leads to algebraic equations:Φ_k=Q_L^ pure radial+Q_E^ pure radial ≡ -1/4π^2 l (l+1)/2l+1∑_k' K_k k'^ even(0,0) [∑_m k'p_k'-m Φ_m/Ω (k'-m)×(1/ω-k' Ω - 1/ω-m Ω)+Φ_k' p_0/(ω-k' Ω)^2]-1/4π^2 Ω/2l+1[d/dE'∑_k' k' Φ_k'(E')K_k k'^ even(0,E')/ω-k' Ω(E')]_E'=0 ,where Φ_k now denotes Φ_k(E=0).§.§ Equations for odd spherical harmonics The jump of the polar angle (<ref>) gives rise to additional factorσ(w) = [sin(12 w)] ,in case of the odd spherical harmonics, i.e.Φ_1 =χ(t,r) σ(w) P_l(cosθ) ,ρ_1(t,r,θ,φ) =Π(t,r) σ(w) P_l(cosθ) .The functions to be expandedχ̅(t,w) ≡χ(t, r) σ(w) ,A̅(t,w,E) ≡ A(t,w,E) σ(w)are antisymmetric, i.e.χ̅(t,-w) = -χ̅(t,w) ,A̅(t,-w,E) = -A̅(t,w,E) .Now the expansion coefficients for even n vanish, while for odd n one has (n=2k+1):{Φ_k+1/2(t,E),A_k+1/2(t,E)}=1/iπ∫_0^π{χ(t,r), A(t,r,v_r)} sin[(k+12) w] dw. From the eqs. similar to (<ref>) and (<ref>), it follows that expansion coefficients B_n/2(t,E) also vanish for even n. For a new set of variables {Φ̅_k, A̅_k,B̅_k}≡{Φ_k+1/2, A_k+1/2,B_k+1/2}, one obtains equations for the odd spherical harmonics l:-i [ω-(k+12) Ω] A̅_k+l (l+1) ∑_k'=-∞^∞p_k-k'B̅_k'=iπ (k+1/2) Φ̅_k F_0'(E) ,-i [ω-(k+12) Ω] B̅_k=π Φ̅_k F_0(E)andΦ̅_k(E)=-2G/2l+1∫dE'/Ω(E')∑_k'=-∞^∞ K_k k'^ odd(E,E') A̅_k'(E') ,where k and k' are integers; p_k(E) is given by (<ref>), i.e. the same as for the even l;K_k k'^ odd(E,E') =4 ∫_0^πdw∫_0^πdw' sin[(k+12) w] sin[(k'+12) w']F_l(r,r') .Eliminating A̅_k(E) and B̅_k(E) in favour of Φ̅_k(E), one obtains the equations similar to (<ref>) for even l:Φ̅_k=-1/4π^2 l (l+1)/2l+1∑_n' K_k k'^ odd(0,0) {∑_m k'p_k'-m Φ̅_m/Ω (k'-m)×[1/ω-(k'+12) Ω-1/ω-(m+12) Ω]+Φ̅_k' p_0/[ω-(k'+12) Ω)^2}-1/4π^2 Ω/2l+1[d/dE'∑_k' (k'+12) Φ_k'(E')K_k k'^ odd(0,E')/ω-(k'+12) Ω(E')]_E'=0 . § NEARLY RADIAL ORBITS: INTEGRAL EQUATIONS The integral equations (<ref>) and (<ref>) are of no use, since they contain infinite coefficients p_k. In this section we consider nearly radial series of `dispersed' Agekyan models (q=-1) with a control parameter L_T which includes the purely radial (Agekyan) model of the previous section as a limiting case L_T → 0. We shall see, that the dispersed models allow for a well-defined integral equations, and their solutions indeed give infinitely large growth rates in the limit of the purely radial case.The integral equations for the nearly radial models in the Lagrangian form are <cit.>:ϕ_ l_1, l_2(E,L)=-4π G/2l+1∑_l_1'=-∞^∞∑_l_2'=-l^l D_l^l_2'∫ dE'∫ dL' F(E',L')×[ E' Ω_l_1'l_2'(E',L') + l_2'L' ] L'/Ω_1(E',L') ×ϕ_ l_1' l_2'(E',L') Π_l_1, l_2; l_1', l_2'(E,L;E',L')/ω-Ω_l_1'l_2'(E',L') ,where Ω_l_1'l_2'(E',L')≡ l_1' Ω_1(E',L') + l_2' Ω_2(E',L'); differentiation operator / E' acts both on Ω_l_1'l_2'(E',L') and the last row; coefficients D_l^k vanish for odd |l-k| andD_l^k= 12^2 l (l+k)!(l-k)![(1/2 (l-k))! (1/2 (l+k))!|]^2otherwise. For the given models, the right-hand side can be written as the sum of two terms:ϕ_l_1l_2(0,L_T) = Q_L+Q_E ≡-K̅(L_T)/2π^2(2l+1) L_T∑_l_1'=-∞^∞∑_l_2'=-l^l (l_2'D_l^l_2') ×ϕ_l_1'l_2'(0,L_T) Π_l_1,l_2;l_1'l_2'(0,L_T;0,L_T)/Ω_1(0,L_T)[ω-Ω_l_1'l_2'(0,L_T)]-K(L_T)/2π^2 (2l+1) L_T^2∑_l_1'=-∞^∞∑_l_2'=-l^l D_l^l_2'[/ E'∫_0^L_T L'dL' Ω_l_1'l_2'(E',L')/Ω_1(E',L') ×ϕ_l_1'l_2'(E',L') Π_l_1,l_2;l_1'l_2'(0,L;E',L')/ω-Ω_l_1'l_2'(E',L') ]_E'=0 .Due to orbit symmetry, r(w) = r(-w), the kernel functions Π_l_1, l_2; l_1', l_2' and unknown expansion coefficients of the potential ϕ_l_1 l_2 can be expressed in integral forms with integration reduced from [-π,π] to [0,π]:Π_l_1, l_2; l_1', l_2'(E,L;E',L') =∮ dw cosΘ_l_1 l_2(w)∮ dw'cosΘ_l_1' l_2'(w') F_l(r,r')=4∫_0^π dw cosΘ_l_1 l_2(w)∫_0^π dw'cosΘ_l_1' l_2'(w') F_l(r,r') ;ϕ_l_1 l_2(E,L)=1/π∫_0^πcosΘ_l_1 l_2(E,L;w) χ[r(E,L,w)] dw .The angleΘ_l_1l_2(E,L,w) isΘ_l_1 l_2(E,L;w)= (l_1+l_2 Ω_2/Ω_1) w -l_2δφ(E,L;w) ,withδφ(E,L,w) =L∫_r_ min(E, L)^r(E,L,w)dx/x^2 √(|[2E+2Ψ(x)]-L^2/x^2)=L/Ω_1∫_0^w dw'/r^2(w')denoting the azimuthal change of the particle coordinate as it passes from the pericentre to the current radius r; Ψ(r) ≡ - Φ_0(r). At the apocentre, δφ(E,L;π)=(Ω_2/Ω_1) π.Further, we shall expand the functions entering eq. (<ref>) considering L_T as a small parameter. For nearly radial orbits, the precession velocityΩ_ pr≡Ω_2-1/2 Ω_1is small compared to frequencies Ω_1,2. So, it can be separated out in the linear combinationΩ_l_1l_2=l_1Ω_1+l_2Ω_2= (l_1+12 l_2) Ω_1+l_2 Ω_ pr .The angle Θ_l_1 l_2(E,L;w) can be written as a sumΘ_l_1 l_2(E,L;w)= [(l_1+12 l_2) w-12 l_2π]+l_2 β ,where the expression in the square brackets retains in the limit L→ 0, whileβ=Ω_ pr/Ω_1 (w-π)+L/Ω_1∫_w^πdw'/r^2(w') =Ω_ pr/Ω_1 (w-π) +L∫_r^r_ maxdr'/r'^2 √(|[2E+2Ψ(r')]-L^2/r'^2)vanishes.In Appendix B, we give details of the expansion of eq. (<ref>) in L_T and β for the even spherical harmonics l. It should be compared with eq. (<ref>) for the systems with purely radial orbits. The equations coincide entirely, if p_0(E) is substituted by the limiting ratio Ω_ pr(L_T)/L_T, i.e.p_0 ≡1/2π∮dw/r^2(w)→lim_L_T→0Ω_ pr(L_T)/L_T ,and coefficients p_k(E) for k 0 are understood as the limitsp_k(E)≡1/2π∮cos(kw)/r^2(w) dw →lim_L_T→ 01/L_T[L_T/π∫_0^πcos(kw)/r^2 dw-Ω_1(L_T)/2].Given that (L_T/π)∫_0^π dw/r^2=Ω_2, one can havep_k-p_0 = 2/π∫_0^πsin^2 (1/2 k w)/r^2(w) dw ,where the right-hand side converges in the usual sense. Note that expansion of (<ref>), not given here, and comparison with (<ref>) for the odd harmonics lead to the same results (<ref>) and (<ref>).The obtained relation between p_k and the limiting value of ϖ≡Ω_pr(L_T)/L_T implies that p_k are infinitely large. Indeed, in purely radial systems the density isnecessarily singular, at least not weaker than 1/r^2<cit.>. Thus the potential and gravitational force are also singular and linear law of the precession rate Ω_pr(L) is no longer valid. In particular, for the softened polytropes all purely radial models (q ≤ 1/2) give ϖ→∞ in the limit L_T → 0<cit.>. Besides, for dispersed Agekyan model, we found numerically that Ω_ pr(L_T)≈ 0.316 (L_T)^0.26, i.e., ϖ≈ 0.316/(L_T)^0.74 (see below Sect. 5.1). § LIMITING INTEGRAL EQUATIONS Eqs. (<ref>) and (<ref>) for p_k(E) show that infinitely large coefficients occur in (<ref>)–(<ref>) and (<ref>)–(<ref>) as L_T goes to zero. This enables us to obtain a simplified counterparts of the stability equations. We shall start from the equations in the form (<ref>)–(<ref>), and assume everywhere that L_T≪ 1. The right hand side in (<ref>) should be omitted since it does not contain p_0. Then, one should neglect the difference between p_k and p_0=ϖ≡Ω_pr(L_T)/L_T ≫ 1 since p_k/p_0 -1 =O(1/p_0) ≪ 1. The expansion 1/r^2=∑ p_k exp (ikw) then turns into1/r^2≈ 2π p_0 ∑_n δ( w-2 π n) ,so that (<ref>) and (<ref>) turn intoA/ t+ΩA/ w=-2π ϖ l (l+1) δ(w)B≡ R ,B/ t+ΩB/ w=π χ(w) F_0(E) .Now changing / t to -iω and solving the equations, taking into account symmetry of functions R(w) and χ(w), one obtains for A:A(E,w) = e^ i ν w/Ω(E) [1/1-e^ 2iπ ν∫_-π^πdwR(w) e^-i ν w. -.∫_w^π dw' R(w') e^-i ν w'] ,orA=-iπl (l+1) ϖ/Ω sin(π ν)B(0) exp [i ν (w- πw)] ,where ν≡ω/Ω is the dimensionless frequency. For B the solution isB(E,w)=π F_0(E)/Ω(E) e^ i ν w[1/1-e^2iπ ν∫_-π^πdw χ(w) e^-i ν w. .-∫_w^π dw'χ(w') e^-i ν w'].In particular, B(E, 0) isB(E, 0)=i π F_0(E)/Ω sin(π ν)Φ_ω(E) ,whereΦ_ω(E)≡∫_0^πχ cos[ν (w-π)] dw .Multiplying the Poisson equation (<ref>) bycos[ν (w-π)] and integrating from 0 to π, one obtains an integral equation for even l:Φ_ω(E)=-8π^3 G l (l+1)/2l+1∫dE'/(Ω')^3 sin^2(π ν')F_0(E') ×ϖ(E',L_T)K_ω(E,E') Φ_ω(E') ,where Ω' and ν' denote Ω(E') and ω/Ω', and the kernel isK_ω(E,E')=∫_0^πcos[ν (w-π)] dw∫_0^πcos[ν' (w'-π)] dw'F_l(r,r') . The analogous equation of odd l has the form:Φ_ω(E)=-8π^3 G l (l+1)/2l+1∫dE'/(Ω')^3 cos^2(π ν')F_0(E')×ϖ(E',L_T)K_ω(E,E') Φ_ω(E') . We shall refer further to eqs. (<ref>) and (<ref>) aslimiting integral equations. Note that they lack the advantage of the linear eigenvalue problem, since frequency ω enters into the kernel function and into the argument of sine and cosine in denominators. However, they retain their forms during the change ω→ -ω, so both equations should depend on ω^2.We need to emphasise that ϖ depends on L_T, and that it is assumed that L_T ≪ 1 and ϖ(E,L_T) ≫ 1. This is the only variable dependent on L_T, in all other places L_T → 0 limit leads to finite quantities, so there we assume L_T=0.Relative simplicity of the limiting integral equations (<ref>) and (<ref>) allows us to demonstrate analyticallyexistence ofaperiodic unstable solutions (ω = iγ with γ>0) for even l and their absence for odd l. Introducing σ(E) ≡γ/Ω(E) one obtains equivalent equations for γ: for even l,Φ_γ(E)=8π^3 G l (l+1)/2l+1∫F_0(E') ϖ(E',L_T) dE'/(Ω')^3 sinh^2(π σ')× K_γ(E,E') Φ_γ(E')andfor odd l,Φ_γ(E)=-8π^3 G l (l+1)/2l+1∫F_0(E') ϖ(E',L_T) dE'/(Ω')^3 cosh^2(π σ')× K_γ(E,E') Φ_γ(E') .In both equationsK_γ(E,E')=∫_0^πcosh[σ (w-π)] dw×∫_0^πcosh[(σ' (w'-π)] dw'F_l(r,r') .Redefinition of the eigenfunctionΨ_γ(E) = Φ_γ(E)/sinh(πσ) √(F_0(E) ϖ(E)/Ω^3(E))allows one to symmetrize the integral equations. Using an integral representation for F_l(r,r') through Bessel functions <cit.>,F_l(r,r')=(2l+1)∫_0^∞ dk J_l+1/2(kr)/√(kr) J_l+1/2(kr')/√(kr') ,it can be proven that the kernel of the symmetrized equation for even l, Q_γ^ even ispositive. So the eigenvalue problem (<ref>) can be rewritten as∫ dE'Q_γ^ even(E,E') Ψ_γ(E')=Λ_n(γ) Ψ_γ(E) .Here Λ_n(γ)(n=0,1,2,...),are a set of positive eigenvalues of the linear problem depending on γ as a parameter. The eigenvalues Λ_n can be ordered so that Λ_0>Λ_1>Λ_2>..., and larger n correspond to eigenfunctions with larger number of nodes (n=0 eigenfunction has the largest scale). The needed values of γ satisfyΛ_n(γ)=1 . In the limit γ→ 0 frequencies σ and σ' are vanishingly small, and the kernelQ_γ^ even(E,E')γ→ 0≈8π G l (l+1)/2l+1 √(F_0(E) ϖ(E)/Ω^3(E))×√(F_0(E') ϖ(E')/Ω^3(E'))1/σ σ'∫_0^π dw∫_0^π dw'F_l(r,r')≫ 1is large, so many Λ_n are greater than 1. On the other hand, for γ≫ 1 sinh(σ π)≈12 e^σ π , cosh[σ (w-π)]≈12 e^σ (π-w) ,and the kernel takes a form:Q_γ^ even(E,E')γ→∞≈8π^3 G l (l+1)/2l+1 √(F_0(E) ϖ(E)/Ω^3(E))×√(F_0(E') ϖ(E')/Ω^3(E'))∫_0^π e^-σ w dw∫_0^π e^-σ' w' dw'F_l(r,r') .Due to rapidly decreasing exponents the kernel is small, and thus Λ_n are small. When γ is changing from zero to infinity, many Λ_n cross the unity value. Since for a given γ, the eigenfunction with the largest scale has the largest eigenvalue, Λ_0 will be the first to cross unity as γ increases, so γ_0 > γ_1 > γ_2 > ....An equation analogous to (<ref>) for odd l has a negative kernel, Q_γ^ odd < 0. It means that all Λ_nare negative for any γ, and no aperiodic solution is possible.§ NUMERICAL RESULTS§.§ Aperiodic modes in the dispersed Agekyan model (q=-1) For the dispersed Agekyan modelF_0(E)=Ω8π^3 δ(E)which corresponds to q=-1 in (<ref>) the integral equation (<ref>) is reduced to an algebraic one,l (l+1)/2l+1 ϖ(0,L_T)/Ω^2K_γ(0,0)/sinh^2(π σ)=1 ,where σ = γ/Ω,K_γ(0,0)=∫_0^πcosh[σ (w-π)] dw×∫_0^πcosh[(σ (w'-π)] dw'F_l(r,r') .It is known <cit.> that this equation has only one even aperiodic solution γ, which is large when ϖ is large. Thus, keeping in the hyperbolic functions the leading exponents only, one obtains the characteristic equation for even l aperiodic modes,l (l+1)/2l+1 ϖ(0,L_T)/Ω^2 × ∫_0^πdw∫_0^π dw'exp[-γ/Ω (w+w')]F_l(r,r')=1 . For this model, the function ϖ(L_T) can be approximated by the power lawϖ(L_T)≃0.316/L_T^ 0.74 ,obtained numerically in the range -3< L_T<-1. With this approximation formula, one can find solutions at arbitrary small L_T. Fig. <ref> shows the dependence of the growth rate γ of the unstable aperiodic solution for l=2. As expected, γ is large as L_T → 0and it scales approximately as γ∼ϖ for very small L_T, and γ∼ϖ^1/2 for L_T ∼ 0.1.Recall that the dynamic frequency Ω=2.16 is of the order unity, thus the obtained growth rates obey the inequality |γ| > Ω for L_T < 10^-2. §.§ Oscillatory modes in the Agekyan model This approximation formula (<ref>) allows us to calculate oscillatory unstable solutions in the form of even and odd spherical harmonics using-l (l+1)/2l+1 ϖ(L_T)/Ω^2K_ω(0,0)/sin^2(π ν)=1for even l, and-l (l+1)/2l+1 ϖ(L_T)/Ω^2K_ω(0,0)/cos^2(π ν)=1for odd l, where ν = ω/Ω andK_ω(0,0)=∫_0^π dw cos[ν (w-π)] ×∫_0^π dw' cos[(ν (w'-π)]F_l(r,r') . The results for the first three harmonics in a wide range of frequencies and L_T = -2 ... -6 are presented in panels of Fig. <ref>. As expected, the aperiodic solutions are absent for odd modes. The growth rates of aperiodic solutions (for l=2) rapidly increase as L_T → 0, so for most values of L_T the apriodic solutions are outside the (middle) panel.Growth rates of the oscillatory solutions show weak dependence on ω, especially for the smallest L_T when γ = ω≈ 1. Real parts of frequencies obey approximately ω /Ω = n-1/4 in the limit L_T → 0, but these limiting values approach from different sides (in case of even l– from the right, and in case of odd l– from the left). In all cases the oscillatory solutions obey |ω| ≳Ω. §.§ Series q=-1/2 In this section we study a series of models with nontrivial dependence of the DF on the energy. For q=-1/2 the potential can be obtained in an analytical form for arbitrary L_T<cit.> and this explains our choice of q.In the purely radial limit Φ = ln r, Ω = √(2π) e^-E. If L_T is small but finite, there is a small radius r_1 =O(L_T) which separates two intervals. From r_1 to approximately 1, the potential is close to Φ = ln r, but in the interval [0, r_1] it behaves like -sinkr/r, with k ∝ L_T^-1. However, for energies in the range [E_c, 0], where E_c=E_c(L_T) is the energy of the particle on the circular orbit with angular momentum L_T (see Fig. <ref>), the pericentre distance is not less than r_1, i.e. one can use the potential for the purely radial case to calculate the precession rate. The azimuth change g(α) during the pericentre passage isg (α) = π +12 πμ(1 +12 μ ln 2μ) +O (μ^3ln^2μ) ,where α≡ L/L_ circ(E), L_ circ(E) = e^E-1/2, μ = [ln(1/α)]^-1<cit.> and thus the precession rate of the particle with energy E and angular momenta L isΩ_pr(E,L)≈e^-E/√(8π) μ(1 +12 μ ln 2μ) ,whereμ=[ ln(1/α)]^-1=[ln (e^E-1/2/L)]^-1≫ 1 .However, this formula is valid for nearly radial orbits E>E_c only and fails for the circular orbit E=E_c where Ω_pr=0.293/L_T. So, instead of(<ref>), we shall calculate the precession rate numerically, using Φ = ln r for the potential, andΩ_pr(E,L_T)= g(α_T)-π/2h(α_T) e^-E ,whereg(α)=2α/√(e)∫_x_ min^x_ maxdxx √(-2x^2 ln x-α^2/e)andh(α)=∫_x_ min^x_ maxx dx√(-2x^2 ln x-α^2/e) .With new variables z=e^E, z_c=√(e) L_T and ν=ω/Ω_1(z,L_T), where Ω_1(z,L_T)=π/z h(z_c/z) ,the limiting integral equations (<ref>) and (<ref>) becomeΦ(z)=-√(1/2π^3) l(l+1)/(2l+1)∫_z_c^1z'^2 ϖ(z',L_T) dz'/√(ln(1/z'^2))×K_ω(z,z')/sin^2(πν') Φ(z')for even l, andΦ(z)=-√(1/2π^3) l(l+1)/(2l+1)∫_z_c^1z'^2 ϖ(z',L_T) dz'/√(ln(1/z'^2)) × K_ω(z,z')/cos^2(πν') Φ(z') ,for odd l, whereϖ(z,L_T)=Ω_ pr(z,L_T)/L_T ,and the kernel is given by eq. (<ref>).The instability growth rates of the aperiodic modes ω = iγ (l=2) are presented in Fig. <ref>. Contrary to the Agekyan model (Fig. <ref>), in this case we have many aperiodic solutions. Results of our calculations of oscillatory eigenmodes for l=1...3 spherical harmonics are presented in Fig. <ref>. Panels of the figure show both real and imaginary parts of ω of the first two modes versus the control parameter L_T. The difference between two successive real parts is ≈ 2.3, and the overall behaviour resembles one of the oscillatory modes in the dispersed Agekyan model (see Fig. <ref>).§ THE ORBITAL APPROACH IN SYSTEMS WITH NEARLY RADIAL STELLAR ORBITS In this section we shall analyse validity of the orbital approach in studying systems with purely radial and nearly radial orbits. Recall that the orbital approach turns from consideration of a particle trajectory to precessing motion of the closed orbital wires. An angle between two successive apocentres of the particle on the radial orbit in the scale free potentials Φ∝ r^s isδφ= {[π ,s ≥ 0 ,; 2π/2 + s ,s < 0 ]. <cit.>. The potentials of the softened polytropes in the limit of purely radial motion L_T → 0 diverge in the centre as <cit.> Φ∝ln^p (1/r)and p = (1/2 - q)^-1 ,i.e. weaker than any negative power s, so δφ = π, and according to (<ref>), Ω_1(E,L=0) = 2Ω_2(E,L=0). The precession rate of the nearly radial orbits L ≪ 1,Ω_pr = Ω_2 - 1/2Ω_1is slow, Ω_pr≪Ω_1,2.Now consider the motion of a particle on a nearly radial orbit in presence of a weak non-rotating slowly growing bar potentialH = H_0 + ϵΦ_b(,t).Switching to the action–angle variables in the orbital plane, = (, ), the small perturbation due to the bar can be written as the Fourier series over radial angle w_1 of the unperturbed orbit:Φ_b(, t) = ^γ t_lΦ_l() ^i l w_1 + i m w_2 , m=2.The phases l w_1 + m w_2 = (lΩ_1 + mΩ_2)t vary quickly for all l except l=-1. Thus, omitting quickly oscillating terms one obtains an `averaged' hamiltonianH = H_0 + ϵΦ_-1() ^i m w_2 + γ t ,which possesses an adiabatic invariant J ≡ I_2 + I_1/2 and `slow' angle variable w_2 =w_2-1/2 w_1<cit.>. The equations of motion areJ̇= -H/ w_1 = 0 ,İ_2 = -H/w_2 = -i m ϵΦ_-1() ^i m w_2 + γ t ,ẇ_1 = Ω_1 + ϵ. Φ_-1/ J|_I_2^i m w_2 + γ tand ẇ_2 = Ω_pr + ϵ. Φ_-1/ I_2|_J^i m w_2 + γ t .In particular, eq. (<ref>) gives the change of the angular momentum perpendicular to the orbital plane, I_2, and eq. (<ref>) describes the apsidal precession. The requirement of adiabaticity impliesγ≪Ω_1. <cit.> obtained an expression for the growth rate in the monoenergetic model (q=-1) with pure radial orbits in the framework of spoke approximation, when the orbital wires turn into spokes. In our notations the growth rate isγ^2≡-ω^2=l(l+1) Ω^2ϖ/π^2∫_0^∞ dk I_l^2(k) ,where Ω is the radial frequency,I_l(k)=∫_0^1dr ρ_ lin(r) J_l+1/2(kr)/√(kr) , ρ_ lin(r) =1/|v_r|= 1/√(2Ψ(r)) is a linear density of the spoke; ϖ≡[dΩ_ pr/dL]_L=0. However, as we argued in Sections 3 and 5, the last parameter grows infinitely, as we turn to more and more radially anisotropic systems.It is interesting to note that if weformally assume the scaled growth rate of the mode σ=γ/Ω to be small in Eq. (<ref>) for monoenergetic model (q=-1), and use identity (<ref>) for F_l(r,r'), we obtain exactly the same expression for the growth rate (<ref>) found by <cit.> in the spoke approximation. This fact justifies the spoke approximation for systems with sufficiently small ϖ (moderately elongated orbits), but not for the very eccentric orbits! Note also that the growth rate γ for q=-1 series scales as ϖ^1/2 for not too small L_T in agreement with (<ref>), but for very small L_T grows even faster than ϖ.Hence stability study of the spherical systems with nearly radial or purely radial orbits cannot be made in the framework of the orbital approach (and the spoke approximation in particular), since γ grows with ϖ and condition (<ref>) fails. § SUMMARY AND CONCLUSIONS Using a new technique based on integral eigenvalue equations, we reconsider here a well-known work on radial orbit instability by Antonov (1973) in which spherical models with purely radial motion are studied. The Antonov problem cannot be correctly solved in the purely radial models due to singularity in the centre. Thus series of models with parameter L_T controlling orbit eccentricity including purely radial model (corresponding to L_T=0) should be used.The derived integral equations involve an only large quantity ϖ≡Ω_pr(L_T)/L_T in case of small L_T tending to infinity as L_T goes to zero. This quantity coincides with the Lynden-Bell derivative [Ω_pr/L]_L=0 playing a crucial role in theory of radial orbit instability <cit.>.We investigated stability of the spherically symmetric models with respect to perturbations ∝χ(r) P_l(cosθ) and obtained numerical solutions for two series of softened polytropic models F(E,q) ∝ H(L_T-L) (-2 E)^q (H(x) is the Heaviside function) allowing the purely radial limit <cit.>.The first one, q=-1, is a dispersed Agekyan model. The instability exists both for even and odd spherical harmonics l, for which multiple oscillatory modes with ω≈ (n-1/4) Ω are found, where n=1,2, ...; Ω≈ 2.16 is the radial frequency of particles (in units G=M=R=1). The modes growth rates γ≡ω≈ 1. Besides, we found aperiodic modes ω=0 for even spherical harmonics with growth rates tending to infinity as L_T → 0.The second series q=-1/2 provides an analytic potential for any value of parameter L_T, and relatively simple formulae for the radial frequency and the precession rate for nearly radial models. As with the previous series, we found multiple oscillatory modes for even and odd spherical harmonics. A characteristic feature of this model is multiple aperiodic modes with growth rates increasing as L_T → 0. We conclude that in all cases (both series, aperiodic and oscillatory modes, even and odd l) |ω| values are of the order of or larger than Ω.There are several interpretations for the physical mechanism of radial orbit instability (ROI). One relates ROI to the well-known Jeans instability in anisotropic medium for which insufficient velocity dispersion perpendicular to the radial direction cannot resist gravitational clusterization <cit.>. Another one is connected to precession dynamics of eccentric orbits that attract to each other, provided [Ω_pr/L]_L=0>0<cit.>. This point of view can be justified only for the so-called `slow modes' which satisfy `slow' integral equation in which only one resonance term ∝ [2Ω_pr - ω]^-1 is retained <cit.>. In turn, this implies (i) even l only, and `slowness' of the mode, i.e. |ω| should be much less than the dynamical frequencies, e.g. Ω<cit.>. As we saw, none of the solutions obtained in this work satisfy any of these requirements, and we must conclude that the orbital interpretation is limited.Using the energy approach, <cit.> argue that instability in sufficiently anisotropic systems can be induced by dissipation inevitably present in the real stellar systems. The energy approach claims that if the second order variation of energy due to the perturbation, H^(2),is negative, then the system may be unstable. If, in addition, a small dissipation takes place, the system is guaranteed to be unstable, with the growth rate proportional to the dissipation. Note, however, that the energy approach makes no conclusions for systems without dissipation in the case of negative sign of H^(2). In other words, it is of little help for highly anisotropic spherical systems subject to very strong collisionless (i.e., non-dissipative) radial orbit instability, which is apparently more important than the instability potentially induced by dissipation.Similar to <cit.>, we consider here non-radial perturbations independent of parity l, and an instability mechanism independent of the suggestion of slowness. However, strong singularity of central density inherent to the system with purely radial orbits <cit.> leads to singularity of the potential, consequently infinite ϖ and the growth rates γ for even aperiodic modes. We suppose that this instability is manifestation of Jeans instability modified due to periodic radial motion of stars along their orbit.It is worth recalling, in this context, the argument against our interpretation of ROI, raised for the first time by <cit.>. Accordingto the virial theorem, the growth rate of Jeans instability is of the same order as the inverse crossing time, ∼(Gρ)^1/2. Since radially anisotropic systems are also strongly radially inhomogeneous, he claims that “unstable mode would scarcely begin to grow before the particles contributed to it had moved away from their initial positions, to regions of very different density and velocity dispersion”. Thegrowth rates obtained in our calculations, however, are large compared to the inverse crossing time, which prevent particles from being escaped before the instability takes over the system. 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The Poisson equation (<ref>) then can be reduced to an algebraic equation:χ(r)=-4π G/l^2 r^2 Π(r) ,Π(r)=1/r^2∫ A dv_r .Now we introduce new variables A and B:B=π F_0 B ,A=-l^2π F_0 A ,and the systems (<ref>) and (<ref>) can be rewritten asB/ t+D̂ B=(2π)^2 G ∫ F_0(E) Adv_randA/ t+D̂ A=B/r^2 .Following <cit.>, we construct the Lyapunov functionL=∫∫ dr dv_rF_0(E)A B=∫ dw ∫F_0(E) dE/ν(E) A B .Differentiating over time, one obtains:dL/dt=∫∫ dr dv_r F_0(E) ( A/ tB+ B/ t A) .Now rewriting (<ref>)B/ t+D̂ B=(2π)^2 G ×∫ dr' δ(r'-r) ∫ F_0(E') A(r',v_r')dv_r' ,we havedL/dt=∫∫ dr dv_rF_0(E) {(-D̂ A+B/r^2)B.+.[-D̂ B+(2π)^2 G∫ dr' δ(r'-r)∫ F_0(E') A(r',v_r')dv_r']A}.Since B (D̂ A)+A (D̂ B) is a full derivative over w,B (D̂ A)+A (D̂ B) =D̂ (AB)=ν(E)(AB)/ w ,the corresponding integral∫ dr∫ dv_r (...) = ∫ν^-1(E) dE ∫ dw (...)vanishes and we obtaindL/dt=∫∫ dr dv_rF_0(E) B^2/r^2+(2π)^2 G ∫∫ dr dr' δ(r'-r)×∫ F_0(E') A(r',v_r')dv_r' ∫ F_0(E) A(r,v_r) dv_r ,or finallydL/dt=∫∫ dr dv_rF_0(E) B^2/r^2 + (2π)^2 G∫ dr [∫ F_0(E) A(r,v_r) dv_r]^2.The last equation is the full analog of the Antonov's expression for dF/dt (but expressedin our variables).[ Note that in the cited paper by <cit.> this expression (following eq. (7)) contains a misprint: the second term in the r.h.s. of the expression for dF/dt should read: 2π G ∫ r^2 dr [∫ dE_0ρ_E_0 (ξ_++ξ_-)]^2.]The proof is based on the evident positiveness of both terms in (<ref>). However, as we already noted in the main text, the first term diverges at r=0, so rigorously speaking such a proof of the radial orbit instability is invalid. § INTEGRAL EQUATIONS IN THE LIMIT OF SMALL L_T (EVEN L). CLARIFYING A SENSE OF DIVERGING COEFFICIENTS P_K In this appendix we restrict ourselves to the relatively compact derivation for the case of dispersed Agekyan model (q=-1), although generalisation to arbitrary F(E) is possible. Besides, we shall consider even spherical harmonics l only, but the desired relations used for interpretation of diverging integrals in the delta function technique are universal and valid for odd l as well. So we assumeF_0(E)=K(L_T)/8π^3 δ(E) ,wherelim_L_T→ 0K(L_T)=Ω≡Ω_1(0,0)≈ 2.16 ,and starting from the integral equation in the Lagrange form,ϕ_ l_1, l_2(E,L) =-4π G/2l+1∑_l_1'=-∞^∞∑_l_2'=-l^l D_l^l_2'∫ dE' ∫ dL'× F(E',L')[ E' Ω_l_1'l_2'(E',L')+l_2'L' ]×L'/Ω_1(E',L') ϕ_ l_1' l_2'(E',L')Π_l_1, l_2; l_1', l_2'(E,L;E',L')/ω-Ω_l_1'l_2'(E',L') .Here we denoteϕ_l_1 l_2(E,L)=1/π∫_0^πcosΘ_l_1 l_2(E,L;w) χ[r(E,L,w)] dw ,where Θ_l_1l_2(E,L,w) is an angle,Θ_l_1 l_2(E,L;w)= (l_1+l_2 Ω_2/Ω_1) w -l_2δφ (E,L;w)andδφ(E,L,w) =L/Ω_1∫_0^w dw'/r^2(w')= L∫_r_ min(E, L)^r(E,L,w)dx/x^2 √(|[2E+2Ψ(x)]-L^2/x^2)is the azimuthal angle as the particle travels from pericentre to current radius r, Ψ is the relative potential, Ψ(r) ≡ - Φ_0(r). In particular, in the apocentre (w=π) this angle is δφ(E,L;π)=(Ω_2/Ω_1) π. The kernel functions areΠ_l_1, l_2; l_1', l_2'(E,L;E',L')=∮ dw cosΘ_l_1 l_2(w) ×∮ dw'cosΘ_l_1' l_2'(w') F_l(r,r')=4∫_0^π dw cosΘ_l_1 l_2(w)∫_0^π dw'cosΘ_l_1' l_2'(w') F_l(r,r') .Note that the symmetry of the radial function r(2π- w) = r(w) allows one to reduce integration in eqs (<ref>) and (<ref>) over full range of the angle variable to the interval [0,π].The r.h.s. of (<ref>) can be divided into two partsϕ_l_1l_2(0,L_T)=Q_E+Q_L ,whereQ_E=-K(L_T)/2π^2 (2l+1) L_T^2∑_l_1'=-∞^∞∑_l_2'=-l^l D_l^l_2' ×[/ E'∫_0^L_T L'dL' Ω_l_1'l_2'(E',L')/Ω_1(E',L'). ×. ϕ_l_1'l_2'(E',L') Π_l_1,l_2;l_1'l_2'(0,L;E',L')/ω-Ω_l_1'l_2'(E',L') ]_E'=0andQ_L=- K(L_T)/2π^2(2l+1) L_T∑_l_1'=-∞^∞∑_l_2'=-l^l (l_2'D_l^l_2') ×ϕ_l_1'l_2'(0,L_T) Π_l_1,l_2;l_1'l_2'(0,L_T;0,L_T)/Ω_1(0,L_T)[ω-Ω_l_1'l_2'(0,L_T)] . Now we shall expand the integral equation entities on the small parameter L_T. The linear combination of frequencies can be rewritten through the precession rate Ω_ pr,Ω_l_1l_2=l_1Ω_1+l_2Ω_2 = (l_1+12 l_2) Ω_1+l_2 (Ω_2-12 Ω_1) = (l_1+12 l_2) Ω_1+l_2 Ω_ pr .For Θ_l_1 l_2(E,L;w) one can writeΘ_l_1 l_2(E,L;w)=[(l_1+12 l_2) w- 12 l_2π]+l_2 β ,whereβ=Ω_ pr/Ω_1 (w-π)+L/Ω_1∫_w^πdw'/r^2(w') ,orβ=Ω_ pr/Ω_1 (w-π)+ L∫_r^r_ maxdr'/r'^2 √(|[2E+2Ψ(r')]-L^2/r'^2) .Here terms proportional to Ω_ pr and L are considered to be small and vanishing as L approaches zero. To be clear, we assume the lower limit in the integral in(<ref>)(L/Ω_1)∫_w^πdw'/r^2(w') is not too close to zero, otherwise this integral becomes of the order unity, since for w=0 it equals to π (Ω_2/Ω_1)≈1/2 π. However, the range of w where the integral becomes ∼ 1 is very small for L → 0, and we shall see below that this bring no difficulties in further integrations. In (<ref>) we take into account that the angle δφ changes from zero to ≈π/2 in the centre, and then remains almost constant in the remaining part of the orbit. Angle β is the remaining part of angle δφ gained from r to r_ max. Thus β is small as long as r≫ r_ min in (<ref>), and contribution to δφ gained near the centre is taken into account by the term -1/2 l_2 π in the square brackets in (<ref>).For the even l, values of l_2 in the integral equation are even, so the sum l_1+12 l_2 is an integer. Introducing new indicesn=l_1+12 l_2 , n'=l_1'+12 l_2'one can switch in expressions for Q_1 and Q_2 from double summation over l_1' and l_2' to summation over n' and l_2',Θ_l_1l_2→Θ_n l_2=(n w-12 l_2π)+l_2 β ,Ω_l_1l_2→ n Ω_1+l_2Ω_ pr . For ϕ_l_1l_2 one obtains, providing Ω_ pr and α are small,ϕ_l_1', l_2'(E',L')=[Φ_n'(E')- l_2' (δΦ)_n'(E')] (-1)^l_2'/2 ,whereΦ_n'(E')=1/π ∫_0^πcos (n' w') χ(r') dw'and(δΦ)_n'(E')=1/π ∫_0^πsin (n' w') β(E',w') χ(r') dw' .Similarly, for the kernel functionsΠ_l_1 l_2; l_1'l_2'(E,0;E',L')=[ K_n,n'(E,E')- l_2' (δ K)_nn'(E,E')] (-1)^l_2/2+l_2'/2,K_n,n'(E,E')=4∫_0^π dw cos(nw) ×∫_0^π dw' cos(n'w')F_l(r,r')and(δ K)_nn'(E,E')=4∫_0^π dw cos(nw) ×∫_0^π dw' sin(n'w') β(w')F_l(r,r') .Since β(w') in (<ref>) and (<ref>) is multiplied by sin(n'w'), which vanishes at w=0, the uncertainty in β at w≈ 0 does not lead to any difficulties.Now it is easy to relate eqs. (<ref>) and (<ref>). In the leading order over L, ϕ_l_1l_2 coincides with (-1)^l_2/2Φ_n and the kernel functions Π_l_1l_2;l_1'l_2' coincide with K^ even_nn'· (-1)^l_2/2+l_2'/2 of eq. (<ref>). Using the identities∑_l_2=-l^l D_l^l_2=1 , ∑_l_2=-l^l l_2 D_l^l_2=0 and ∑_l_2=-l^l l_2^2 D_l^l_2=l(l+1)/2one can show that Q_E turns into the last term containing the energy derivative. The remaining term, Q_L, vanishes in the leading order O(1/L_T),Q_L=(-1)^l_2/2 [- K̅(L_T)/2π^2(2l+1) L_T∑_n'=-∞^∞∑_l_2'=-l^l (l_2' D_l^l_2') ×Φ_n'(E') K_n,n'(E,E')/Ω(ω-n'Ω)] = 0because of the second identity in (<ref>). To proceed further, we have to expand Q_L to the next order O(L_T^0) and compare it with the first square bracket in (<ref>).Small additional terms (δΦ)_n' and (δ K)_nn' can be expanded over functions of the leading order. According to (<ref>)χ(r)=∑_k Φ_k(E) e^ikw ,and from (<ref>) one has(δΦ)_n(E)=∑_k β_nk(E) Φ_k(E) ,whereβ_nk(E,L)= 1/π∫_0^πsin(n w)cos(kw) β(E,L,w) dw .Similarly, for (δ K)_nn' one obtains(δ K)_nn'(E,E')=∑_k K_nk(E,E') β_n'k(E',L') .Summarising, for L=0, L' ≪ 1 one obtains:ϕ_l_1'l_2'(E',L')→ [Φ_n'(E')-l_2'∑_mβ_n'm(E',L') Φ_m(E')] (-1)^l_2'/2,Π_l_1l_2;l_1'l_2'(E,L=0;E',L')→[ K_nn'(E,E')l_2'∑_m K_nm(E,E') β_n'm(E',L'). - . l_2'∑_m K_nm(E,E') β_n'm(E',L')] (-1)^l_2/2+l_2'/21/ω-Ω_l_1'l_2'→1/ω-n' Ω-l_2' Ω_ pr ≈1/ω-n' Ω+l_2' Ω_ pr/(ω-n' Ω)^2. Using these expressions in (<ref>), we obtain in the order O(L_T^0):Q_L≈-(-1)^l_2/2/2π^2(2l+1) L_T∑_n'=-∞^∞∑_l_2'=-l^l (l_2'^2 D_l^l_2')×{-1/ω-n'Ω[ K_nn'∑_m β_n'm Φ_m+Φ_n'∑_mβ_n'mK_nm] . +. Ω_ prK_nn'Φ_n'/(ω-n'Ω)^2}.Summing up over l_2' with the help of (<ref>) allows us to reduce (<ref>) toQ_L ≡Q̅_L(-1)^l_2/2≈(-1)^l_2/2/4π^2 l (l+1)/2l+1 1/L_T ×∑_n'=-∞^∞[∑_m β_n'm ( K_nn' Φ_m+ K_nm Φ_n')/ω-n'Ω-Ω_ prK_nn'Φ_n'/(ω-n'Ω)^2].The expression Q̅_L should be compared with the first terminr.h.s. of Eq.(<ref>), which can be rewritten as Q_L^ pureradial=-1/4π^2 l (l+1)/2l+1 ×∑_n' [∑_m n'p_n'-m ( K_n n' Φ_m+ K_n mΦ_n')/Ω (n'-m) 1/ω-n' Ω+K_n n'Φ_n' p_0/(ω-n' Ω)^2].In particular, comparison of the underbraced terms in (<ref>) and (<ref>) gives that p_0 should be associated with lim _L_T→ 0Ω_ pr(L_T)/L_T, i.e.p_0≡1/2π∮dw/r^2(w)→lim_L_T→ 0Ω_ pr(L_T)/L_T .Next, from (<ref>) and (<ref>)β_nm=-1/2 Ω {1/n+m[L/π∫_0^πcos(n+m)w/r^2 dw-Ω/2] +1/n-m[L/π∫_0^πcos(n-m)w/r^2 dw-Ω/2]}for m± n andβ_n,± n=-1/4 Ω n [L_T/π∫_0^πcos(2nw)/r^2 dw-Ω/2]for m=± n. Then, introducingP_k(E,L_T)=1/L_T [L_T/π∫_0^πcos (kw)/r^2 dw-Ω(L_T)/2] ,one can have(β_nm)_m± n=-L_T/2Ω (P_n+m/n+m+P_n-m/n-m) ,β_n,± n=-L_T/4 Ω n P_2n .Now it is not difficult to show that Q̅_L completely coincides with Q_L^ pure radial if one associate p_k as limiting values of P_k(L_T):p_k=1/π∫_0^πcos(kw) dw/r^2(E,w)→lim_L_T→ 0P_k(E,L_T).In particular, for k=0 one obtains (<ref>)p_0≡1/2π∮dw/r^2(w)→lim_L_T→ 01/L_T[L_T/π∫_0^πdw/r^2-Ω_1(L_T)/2] =lim_L_T→ 0Ω_2(L_T)-1/2 Ω_1(L_T)/L_T≡lim_L_T→ 0Ω_ pr(L_T)/L_T. From definition of p_k,p_k=p_0- 2/π∫_0^πsin^2 (1/2 kw)/r^2(w) dw ,with the integral converging in the usual sense, thus p_k 0 - p_0 =O(1). | http://arxiv.org/abs/1705.09150v1 | {
"authors": [
"E. V. Polyachenko",
"I. G. Shukhman"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170525124526",
"title": "Radial orbit instability in systems of highly eccentric orbits: Antonov problem reviewed"
} |
[email protected] [email protected]@shnu.edu.cnShanghai United Center for Astrophysics(SUCA), Shanghai Normal University, 100 Guilin Road, Shanghai 200234, China It is known that the Cardassian universe is successful in describing the accelerated expansion of the universe, but its dynamical equations are hard to get from the action principle. In this paper, we establish the connection between the Cardassian universe and f(T, 𝒯) gravity, where T is the torsion scalar and 𝒯 is the trace of the matter energy-momentum tensor. For dust matter, we find that the modified Friedmann equations from f(T, 𝒯) gravity can correspond to those of Cardassian models, and thus, a possible origin of Cardassian universe is given. We obtain the original Cardassian model, the modified polytropic Cardassian model, and the exponential Cardassian model from the Lagrangians of f(T,𝒯) theory. Furthermore, by adding an additional term to the corresponding Lagrangians, we give three generalized Cardassian models from f(T,𝒯) theory. Using the observation data of type Ia supernovae, cosmic microwave background radiation, and baryon acoustic oscillations, we get the fitting results of the cosmological parameters and give constraints of model parameters for all of these models.Action functional of the Cardassian universe Xin-zhou Li December 30, 2023 ============================================ § INTRODUCTIONThe Cardassian universe <cit.> has been known to describe the accelerating expansion of the universe with remarkable agreement with observations, whereas it lacked a solid theoretical foundation until now. In Cardassian models, the Friedmann equation is modified by the introduction of an additional nonlinear term of energy density while without the introduction of a cosmological constant or any dynamic dark energy component. In these models, the universe can be flat and yet consist of only matter and radiation and still be compatible with observations. Matter can be sufficient to provide a flat geometry. The possible origin for Cardassian models is from the consideration of braneworld scenarios, where our observable universe is a three dimensional membrane embedded in extra dimensions<cit.>. The modified Friedmann equation may result from the existence of extra dimensions, but it is difficult to find a simple higher dimensional theory, i. e., a higher-dimensional momentum tensor that produces the Cardassian cosmology<cit.>. Inspired by the study on the correspondence between thermodynamic behavior and gravitational equations, a couple of us have studied the thermodynamic origin of the Cardassian universe<cit.>. However, it is still hard to get the dynamic equations of this model from the action principle. To explain the accelerated expansion of the universe, besides adding Cardassian terms or unknown fields such as quintessence <cit.> and phantom <cit.>, there is another kind of theory known as modified gravity, which uses an alternative gravity theory instead of Einstein's theory, such as f(R) theory <cit.>, MOND cosmology <cit.>, Poincaré gauge theory <cit.>, and de Sitter gauge theory <cit.>. On the other hand, Einstein constructed the "Teleparallel Equivalence of General Relativity" (TEGR) which is equivalent to the general relativity (GR) from the Einstein-Hilbert action<cit.>. In TEGR, the curvatureless Weitzenböck connection takes the place of the torsionless Levi-Civita one, and the vierbein is used as the fundamental field instead of the metric. In the Lagrangian of TEGR, the torsion scalar T, from contractions of the torsion tensor, takes the place of the curvature scalar R. The simplest approach in TEGR to modify gravity is f(T) theory <cit.>, whose important advantage is that the field equations are second order and not fourth order as in f(R) theory. Recently, we established two concrete f(T) models that do not change the successful aspects of the Lambda cold dark matter scenario under the error band of fitting values, as describing the evolution history of the universe including the radiation-dominated era, the matter-dominated era, and the present accelerating expansion <cit.>. We also considered the spherical collapse and virialization in f(T) gravities<cit.>. Furthermore, extensions of f(T,𝒯) theory<cit.> where 𝒯 is the trace of the matter energy-momentum tensor 𝒯_μν were constructed, whose cosmological implications are rich and varied.Recently, it was shown<cit.> that modified gravity models may lead to a Cardassian-like expansion. In this paper, we try to find the relation between Cardassian models and f(T,𝒯) theory. Under the reconsidered scheme of f(T,𝒯) theory, we obtainthe original Cardassian model, the modified polytropic Cardassian model, and the exponential Cardassian model through the action principle and thus give a possible origin of the Cardassian universe. Furthermore, by adding an additional term to the corresponding Lagrangians, we give three generalized Cardassian models from f(T,𝒯) theory. Using the observation data of type Ia supernovae(SNeIa), cosmic microwave background radiation(CMB), and baryon acoustic oscillations(BAO), we get the fitting results of the cosmological parameters and give constraints of model parameters for all of these models.The paper is organized as follows: in Sec. II, with the discussion of the self-consistent form of the Lagrangian of barotropic perfect fluid, we give a new derivation of f(T,𝒯) theory. In Sec. III, the actions of the three Cardassian models from f(T,𝒯) theory are given explicitly and generalized Cardassian models from f(T,𝒯) theory are further given. We also examine the observational constraints of each model in this section.Finally, Sec. IV is devoted to the conclusion and discussion. We use the signature convention (+,-,-,-) in this paper.§ F(T,𝒯) THEORY WITH BAROTROPIC PERFECT FLUID §.§ The Lagrangian of barotropic perfect fluidThere exist two types of Lagrangian ℒ_m for the perfect fluid in modified gravity theories, so we have to define one or the other of these two. Harko has pointed out that ℒ_m=ϵ(ρ) is a more reasonable choice <cit.> in modified gravity theories, where ϵ(ρ) is the total energy density of the fluid and ρ is the rest mass density.In the work of Brown<cit.>, it is shown that the on shell perfect-fluid Lagrangian in GR can be ℒ_m=ρ or ℒ_m=-p, where ρ is the rest mass density and p is the pressure. Both Lagrangians lead to the same perfect fluid stress-energy tensor concordant with the laws of thermodynamics and hence, the same equations of motion. In past years, some authors have adopted some specific form of ℒ_m=-p from the work of Brown for their alternative theories of gravity <cit.>. However, according to Refs. <cit.>, we have to reconsider how to take the form of ℒ_m for the perfect fluid in modified gravity theories, including f(T,𝒯) theory.The usual form of the stress tensor of a barotropic perfect fluid is𝒯^μν=-[ϵ(ρ)+p(ρ)]u^μu^ν+p(ρ)g^μνwhere ϵ(ρ) and p(ρ) are the total energy density and the pressure of the fluid, respectively, which both depend on the rest mass density ρ. On the other hand, if the Lagrangian of a barotropic perfect fluid ℒ_m does not depend on the derivatives of the metric, the usual definition of the stress-energy tensor 𝒯^μν𝒯^μν=-ℒ_m g_μν+ 2∂ℒ_m/∂ g^μνwhere ℒ_m can be assumed to depend on ρ only. Considering the conservation of the matter current ∇_σ (ρ u^σ)=0, one can prove that<cit.>δρ=1/2ρ(g_μν-u_μu_ν)δ g^μνwhere the four velocity of the fluid u^α satisfies the conditions u^αu_α=1. Substituting these results into Eq.(<ref>), one can obtain<cit.>𝒯^μν=-ρdℒ_m/dρu^μu^ν- (ℒ_m-ρdℒ_m/dρ )g^μν.From a comparison of Eq.(<ref>) and Eq.(<ref>), we haveℒ_m=ϵ(ρ)=ρ [c^2+∫ p(ρ)/ρ^2dρ].anddϵ(ρ)/dρ=ϵ(ρ)+p(ρ)/ρ,where c is the speed of light, and the unit c=1 is taken hereinafter. In other words, ℒ_m=ϵ(ρ) is a direct and reasonable generalization from ℒ_m=ρ in GR to f(T,𝒯) theory, because Brown's argument becomes invalid in modified gravity theories. When compared with it, ℒ_m=-p is only a direct employment from GR.Furthermore, we can verify the conservation of the total energy . Actually, one can easily obtain the divergence of the energy density current∇_σ( ϵ u^σ)=( 1+∫p/ρ^2dρ+p/ρ)∇_σ( ρ u^σ)-p∇_σ u^σ.Under the conservation of matter current ∇_σ( ρ u^σ)=0, Eq.(<ref>) is the conservation of the total energy. For example, under the Friedmann-Walker-Robertson (FRW) metric it becomesϵ̇+3H( ϵ+p )=0,which is the usual form of energy conservation in cosmology. §.§ The field equations in f(T,𝒯) Theory We can find a set of smooth basis vector fields ê_(μ) in different patches of the manifold ℳ and make sure things are well-behaved on the overlaps as usual, where Greek indices run over the coordinates of spacetime. The set of vectors e_A, comprising an orthonormal basis, is known as a tetrad or vierbein, where Latin indices run over the tangent space T_p at each point p in ℳ. A natural basis of T_p is given by ê_(A)=∂ / ∂ x^A. Any vector can be expressed aslinear combinations of the basis vector, so we have ê_(A)=e_A ^μê_(μ)where the components e_A^μ form a 4× 4 invertible matrix. We will also refer to e_A^μ as the vierbein inaccordance with the usual practice of blurring the distinction between objects and their components. The vectors ê_(μ) in terms ofê_(A) areê_(μ)=e^A _μê_(A)where the inverse vierbeins e^A _μ satisfye^A_μe_B^=δ_B^A,e_A^μe^A_ν=δ_ν^μ.Therefore, the metric is obtained from e^A_μg_μν=η_ABe^A_μe^B_ν,or equivalentlyη_AB=g_μνe_A^μe_B^ν,and the root of the metric determinant is given by |e|=√(-g)=(e^A_μ).In TEGR, one uses the standard Weitzenböck's connection defined asΓ^α_μν=e_A^α∂_ν e^A_μ=-e^A_μ∂_ν e_A^α,and the covariant derivative D_μ satisfies the equationD_μ e^A_ν=∂_μ e^A_ν-Γ^α_νμe^A_α=0.Then the components of the torsion and contorsion tensors are given byT^α_μν = Γ^α_νμ-Γ^α_μν=e_A^α(∂ _μ e^A_ν-∂_ν e^A_μ),K^μν_α = -1/2 (T^μν_α-T^νμ_α-T_α^μν).By introducing another tensorS_α^μν=1/2(K^μν_α+δ ^μ _α T^βν_β-δ ^ν_α T^βμ_β),we can define the torsion scalar asT≡ T^α_μνS_α^μν. The action for f(T,𝒯) gravity takes the following form <cit.>S=1/16π G∫ e f(T,𝒯)d^4 x+∫ e ℒ_m d^4 x,where f(T, 𝒯) is an arbitrary function of the torsion scalar T and the trace 𝒯 of the matter stress-energy tensor. On the variation with respect to the vierbein that leads to the field equations, a question that should be noted is how to deal with the variation of the trace of the energy-momentum tensor δ𝒯. This question has been met in theories with 𝒯 included in the action, including f(R, 𝒯) theory <cit.> and f(T, 𝒯) theory <cit.>. With the discussion in the last subsection, we can reexamine this question now.From Eq. (<ref>) and Eq.(<ref>), the variation of ϵ isδϵ=-1/2(ϵ+p)(g^αβ-u^αu^β)δ g_αβ.Using (<ref>), (<ref>), and (<ref>), one can express the variation of 𝒯 asδ𝒯 = δ(3p-ϵ)= (3d p/dρρ/ϵ+p-1)δϵ= (1-3d p/dρρ/ϵ+p)(𝒯^α_β+ϵδ^α_β)e_A^βδ e_α^A.The field equations then read asf e_A^α+4/e f_T ∂_β(e S_σ^αβe_A ^σ)+4S_σ^αβe_A^σ∂_βf_T+ 4f_TS_ρ^ασT^ρ_σβe_A^β+f_𝒯 (1-3d p/dρρ/ϵ+p)ϵ e_A^α= (f_𝒯(3d p/dρρ/ϵ+p-1)+16π G)𝒯^α_β e_A^βwhere f_T and f_𝒯 denote derivatives with respect to torsion scalar T and the trace of T^μν, respectively.In contrast to f(T,𝒯) theory in previous papers<cit.>, this is the new derivation of f(T, 𝒯) theory with δ𝒯 reconsidered, since we have taken ℒ_m=ϵ(ρ) but not ℒ_m=-p. The crucial difference lies in the different choice of the matter Lagrangian ℒ_m. The derivation of the field equations in the references mentioned above depends on the assumption that ℒ_m=-p. And the same assumption is used in works on f(R,𝒯) gravity (see <cit.>). However, from the discussion in Sec. IIA and also in Refs. <cit.>, ℒ_m=ϵ(ρ) would be a more reasonable choice. This is what leads to the difference between the field equations (<ref>) that we got and the ones in the literature.Since f(T) theories are known to violate local Lorentz invariance<cit.>, particular choices of tetrad are important to get viable models in f(T) cosmology, as has been noticed in Ref. <cit.>. For a flat FRW metric in Cartesian coordinates,ds^2=dt^2-a(t)^2(dx^i)^2where a(t) is the scale factor, the diagonal tetrad e^A_μ=diag(1,a,a,a) is a good choice to get viable models<cit.>. The torsion scalar T=-6H^2, where H=ȧ/a is the Hubble parameter. Then the equations of motion (<ref>) give rise to the modified Friedmann equationsf_T H^2=-4/3π Gϵ-1/12fand4Ḣf_T=[f_𝒯 (3∂ p/∂ϵ-1 )+16π G](ϵ+p)-3H ḟ_T,which are consequently different from those in previous references for f(T, 𝒯) theory. It is easy to confirm the energy conservation (<ref>) from Eq. (<ref>) and Eq.(<ref>). § CARDASSIAN UNIVERSE FROM F(T,𝒯) THEORY§.§ The action of Cardassian models from f(T,𝒯) theoryIn Ref. <cit.>, we studied the cosmology of gravity with the Lagrangian in the forms of ℒ∝ -T+α√(-T)+f(T,ℒ_m) and ℒ∝ -T+β T^-1+f(T,ℒ_m). In the first form, the square root term is easy to prove as null, so α is actually a free parameter, and hence the correction of this term will not affect the local gravity tests. Similar to Ref. <cit.>, here we choosef(T, 𝒯)=-T+α√(-T)+g(𝒯).For dust matter, the pressure is p=0. Then from (<ref>) we have ϵ(ρ)=ρ, and Eq. (<ref>) reduces toH^2=8π G/3ρ+1/6g(𝒯),where 𝒯=-ρ for dust matter, and ρ∝ a^-3 from Eq.(<ref>). It is obvious that Eq. (<ref>) is the very equation for Cardassian models and it is easy to find the forms of f(T, 𝒯) corresponding to specific Cardassian models. Here, we examine three Cardassian models. The units 8π G=1 is used hereinafter. For the original Cardassian model (OC)<cit.>,H^2=ρ/3[1+(ρ/ρ_c)^n-1]where ρ_c is the critical energy density at which the two terms of Eq.(<ref>) are equal, we haveg(𝒯)=2ρ(ρ/ρ_c)^n-1=2/ρ_c^n-1( -𝒯)^n.For the modified polytropic Cardassian model (MPC)<cit.>H^2=ρ/3[1+(ρ/ρ_c)^q(n-1)]^1/q,we haveg(𝒯)= 2ρ[[1+(ρ/ρ_c)^q(n-1)]^1/q-1]= 2𝒯[1-[1+(𝒯/ρ_c)^q(n-1)]^1/q],and for the exponential Cardassian model (EC)<cit.>H^2=ρ/3exp[(ρ/ρ_c)^-n],we haveg(𝒯)= 2ρ[exp[(ρ/ρ_c)^-n]-1]= 2𝒯[1-exp[(-𝒯/ρ_c)^-n]].Therefore, we claim that we find the possible origin of Cardassian models from f(T,𝒯) theory. §.§ f(T,𝒯)-generalized Cardassian models Alternatively, inspired by the Lagrangian with the term β T^-1 considered in Ref. <cit.>, if we replace the α√(-T) term in Eq.(<ref>) with-3λ^2H_0^4/T, we can obtain the f(T,𝒯)-generalized Cardassian models. For generalized OC (Model I), the modified FRW equation readsE^2 -λ^2/4E^-2 = Ω_0(1+z)^3 + Ω_x (1+z)^3n.Here E(z) = H(z)/H_0, H_0 is the Hubble parameter, Ω_0≡ρ_0/3H_0^2 = Ω_m0 + Ω_b0, where Ω_m0 and Ω_b0 correspond to dark matter and baryons respectively, andΩ_x = 1- λ^2/4- Ω_0. For the generalized MPC (Model II), the modified FRW equation readsE^2 -λ^2/4E^-2 = {Ω_0^q(1+z)^3q+ [ (Ω_x + Ω_0)^q -Ω_0^q ] (1+z)^3qn}^1/q,and for generalized EC (Model III), the modified FRW equation readsE^2 -λ^2/4E^-2 = Ω_0(1+z)^3 exp[ (1+z)^-3nln(Ω_x + Ω_0/Ω_0)]. In all the cases, the modified FRW equations can be expressed unifiably asE^2 = 1/2[ ϕ(z) + √(ϕ^2(z) + λ^2)]where ϕ(z) is the right hand side of Eqs.(<ref>), (<ref>), or (<ref>). §.§Observational Constraints In this subsection, using the observational data of SNeIa, CMB, and BAO, we give constraints and the best fit parameters of each model. For SNeIa data, we use the joint light-curve analysis(JLA)sample, which contains 740spectroscopically confirmed type Ia supernovae with high quality light curves. The distance estimator in this analysis assumes that supernovae with identical colors, shapes, and galactic environments have, on average, the same intrinsic luminosity for all redshifts. This hypothesis is quantified by a linear model, yielding a standardized distance modulus<cit.>μ_obs = m_B - (M_B - A · s + B · C + P ·Δ_M) ,where m_B is the observed peak magnitude in rest-frame B band, and M_B, s, C are the absolute magnitude, stretch, and color measures, which are specific to the light-curve fitter employed, and P(M_* >10^10 M_⊙) is the probability that the supernova occurred in a high-stellar-mass host galaxy. The stretch, color, and host-mass coefficients (A, B, Δ_M, respectively) are nuisance parameters that should be constrained along with other cosmological parameters.The CMB temperature power spectrum is sensitive to the matter density, and it also precisely measures the angular diameter distance θ_* at the last-scattering surface. We use the Planck measurement of the CMB temperature fluctuations and the WMAP measurement of the large-scale fluctuations of the CMB polarization. This CMB data are often denoted by "Planck + WP". The geometrical constraints inferred from this data set are the present values of baryon density Ω_b0h^2, dark matter Ω_m0h^2, and θ_* <cit.>, where h is given byH_0= 100 h km s^-1 Mpc^-1.The BAO measurement provides a standard ruler to probe the angular diameter distance versus the redshift relation by performing a spherical average of their scale measurements, see Ref. <cit.>. We use the measurement of the BAO scale from Refs. <cit.>.In Table <ref>, we present thebest-fit parameters by using the data of CMB+BAO+JLA, andalso quote their 1σ bounds from the approximate Fisher information matrix. We also examine the constraints on parameters from the 1σ to the 3σ confidence levels for each model and Figs. 1-3 are the illustrations of the constraints on Ω_m0 and n for Models I, II, and III, respectively.§ CONCLUSION AND DISCUSSION Using the result of the Lagrangian of a barotropic fluid given in Ref.<cit.>, we rederive f(T,𝒯) gravity, obtaining the modified Friedmann equations. We find the connection between f(T,𝒯) gravity and the Cardassin universe. For dust matter, the modified Friedmann equations from f(T,𝒯) theory can correspond to those of the Cardassian models, and thus, a possible origin of the Cardassian universe is given. We present the Lagrangians of the original Cardassian model, the modified polytropic Cardassian model, and the exponential Cardassian model from f(T,𝒯) theory. Furthermore, we get generalized Cardassian models by adding an additional term to the corresponding Lagrangians of f(T,𝒯) theory that lead to the three Cardassian models mentioned above. Using the data of CMB+BAO+JLA, we get the fitting results of the cosmological parameters and give constraints of model parameters for all these models.As one of the candidates for explaining the acceleration of the universe, Cardassian models have advantages in that the universe can be flat and yet consist of only matter and radiation, bothof which satisfy the conservation laws. However, there is not a satisfactory answer in the literature for the origin of the Cardassian models. In our new derivation of f(T,𝒯) theory, the usual energy conservation still holds, which is necessary for Cardassian models. 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Soc.427, 2132 (2012).Anderson:2012sa L. Anderson, et al.,Mon. Not. Roy. Astron. Soc.427, 3435 (2012). | http://arxiv.org/abs/1705.09490v1 | {
"authors": [
"Xiang-hua Zhai",
"Rui-hui Lin",
"Chao-jun Feng",
"Xin-zhou Li"
],
"categories": [
"gr-qc",
"astro-ph.CO",
"hep-th"
],
"primary_category": "gr-qc",
"published": "20170526091354",
"title": "Action functional of the Cardassian universe"
} |
Direct Multitype Cardiac Indices Estimation via Joint Representation and Regression Learning Wufeng Xue, Ali Islam, Mousumi Bhaduri, and Shuo Li* Copyright (c) 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]. W. Xue, A. Islam, M. Bhaduri and S. Li are with the Department of Medical Imaging, Western University, London, ON N6A 3K7, Canada. W. Xue and S. Li are also with the Digital Imaging Group of London, London, ON N6A 3K7, Canada. * Corresponding author. (E-mail: [email protected]) ======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================= It can be difficult to tell whether a trained generative model has learned to generate novel examples or has simply memorized a specific set of outputs.In published work, it is common to attempt to address this visually, for example by displaying a generated example and its nearest neighbor(s) in the training set (in, for example, the L^2 metric).As any generative model induces a probability density on its output domain, we propose studying this density directly.We first study the geometry of the latent representation and generator, relate this to the output density, and then develop techniques to compute and inspect the output density.As an application, we demonstrate that "memorization" tends to a density made of delta functions concentrated on the memorized examples.We note that without first understanding the geometry, the measurement would be essentially impossible to make.§ INTRODUCTIONVariational Auto-Encoder (VAE) and Generative Adversarial Network (GAN) models have enjoyed considerable success recently in generating natural-looking images.However, in many cases it can be difficult to tell when a trained generator has actually learned to generate new examples; it is entirely possible for a VAE to simply memorize a training set.GAN training provides only indirect access to a training set, so direct memorization is less of an issue.However, it is still possible for any generative model to concentrate its probability mass on a small set of outputs, and the intrinsic dimension of the output is unclear.To identify memorization, experimenters often provide visual evidence: e.g. some generated examples may be shown alongside their nearest neighbors in a training set (e.g. [1], [2]).If the neighbors differ from the generated example, memorization is declared to be unlikely.Alternatively, outputs along a path in latent space may be plotted; if the output changes smoothly this suggests generalization, whereas sudden changes suggests memorization.Instead, we propose studying the induced probability density on output space.To do this, we must change variables and transform a density on the latent space to a density on the output space.This computation requires some care, as the data lie on low-dimensional submanifolds of the input and output space, and the standard formulas will become degenerate and fail.In what follows we first establish the local geometry of the situation, which allows us to obtain a formula for the output density.We then introduce ways to measure the degree to which a generator has memorized, and show experimental results.This enables us to characterize "memorizing" as learning a probability density on output space which concentrates its mass on a finite number of points (in the limit, the learned measure tends to a collection of delta functions).In contrast, generalization implies a density which smoothly interpolates points, assigning mass to large regions of output space. § MAPPING LATENT SPACE TO OUTPUT SPACEConsider a trained generative model, where a learned (but now fixed) generator function f maps a space of random variables Z ⊆^m to an output space X ⊆^n.We assume our generator mapping f:^m →^n is differentiable.(This may be false, particularly when f is a neural network with non-differentiable nonlinearities, but f will still be piecewise smooth.)We further assume that f^-1 is so difficult to compute that it is effectively unavailable.A main difficulty is that the latent space ^m may be large relative to the intrinsic dimension of the learned representation .m is typically chosen to be "large enough" for the problem at hand, and may be larger than necessary.That is, the learned latent representation may have dimension l < m.Assuming the typical case where m << n, we observe that f can only map the latent space onto a submanifold of the output space with dimension at most l ≤ m < n.Thus we see that ⊂^m is a latent (sub)manifold of dimension l and f() = ⊂^n is our output (sub)manifold of dimension ≤ l.In particular, as a map ^m →^n, we see that f is degenerate as its range lies on a low-dimensional submanifold. §.§ Tangent Spaces, Singular Vectors, and the Volume Element While global understanding of f:^m →^n is not possible in general, the local behavior can be understood by computing the Jacobian matrix J_f and considering the linearized mapf(z + h v) = f(z) + h J_f v + o(h)for h ∈ small.In particular, the rank of J_f tells us about the intrisic dimension of the manifold near a point (see [3]).Further, the singular value decomposition (SVD) allows us to writeJ_f = U Σ V^Twhere V, U are orthogonal matrices whose columns (the "singular vectors") span ^m and ^n respectively, and Σ is a diagonal matrix (the "singular values").It follows that the right and left singular vectors corresponding to non-zero singular values form a basis for the tangent spaces toand([3], [4]).The singular vectors v_i of V with degenerate (i.e. 0) singular values correspond to subspaces which get collapsed (via projection) onto the tangent space before the linearized mapping f.In other words, if J_f has l non-zero singular values, f locally maps ^m onto an l-manifold in ^n.Moving in directions v_i ∈^m corresponding to large singular values σ_i will cause greater change (in the L^2 distance) in the output than directions with smaller σ_i.(See section <ref> for experimental analysis of the intrinsic dimension.)Assuming we have l < m < n non-zero singular values at some point Z, then (as mentioned above) f is degenerate and it does not make sense to talk about a volume element.However, we can consider the restriction of f as a map → between l-manifolds.This restriction of f will still be a diffeomorphism fromonto its image, and we can talk about a volume element here.In particular, the change-of-variable formula will give dVol_ = ( ∏_σ_i0σ_i ) dVol_That is, volumes onanddiffer by the product of the nonzero singular values at corresponding points X = f(Z). § THE DENSITY ON OUTPUT SPACEThe random vectors Z = (z_1, z_2, ... z_m) are typically drawn from distributions that are easy to sample, e.g. each z_k may be an independent normal or uniform random variable.Whatever the distribution of p(Z) is, in conjunction with f it induces a density p on outputs X = f(Z); in the case m = n the induced density would have (by change-of-variable) the well-known formp(X) = p(Z) dVol()/dVol() = p(Z)/|J_f|where J_f is the Jacobian matrix of f (implicitly at Z), and |·| denotes the determinant (recall the determinant describes the volume element and is the product of eigenvalues of J_f).However, we have m < n and cannot use this formula directly.If J_f had rank m we could replace the denominator with √(|J_f^TJ_f|), but as discussed above, we find ourselves in a still more degenerate case.However, using equation <ref> we can compute the induced density as p(X) = p(Z) dVol()/dVol() = p(Z)/∏_σ_i0σ_iwhere {σ_i }_i=1^l are the singular values of J_f at Z.Note that we can also restrict to even lower-dimensional problems by discarding more singular values and vectors; in practice one typically sets a threshold below which any singular values are considered to be zero.§ MEASURING MEMORIZATION We introduce two measurements: the first is based on the density p(X) on lines (in latent space) joining two outputs, and considers how much the density drops in-between sample points.The second measure is based on the local rate of decay of the density about individual points, and provides a local measure of the density's concentration.It may help to consider figure <ref>:if p(X) has memorized a few examples, it implies f must map large regions ofto small regions of .If we plot p(X) as a function of distance (in ), we can examine both the drop in density between output samples and the rate at which the density decays around a given example or examples.Particularly for considering the decay rate, it is important that we consider p(X) as opposed to p(f(Z)) – that is, we should use distances on , not distances in .In cases of memorization, large regions incorrespond to nearly constant output.This implies the density p will appear to be spread over "large" neighborhoods when viewed as a function on , while simultaneously appearing obviously concentrated in(see again figure <ref>). §.§ The Density Along LinesWhen latent representations z_1, z_2 for outputs x_1, x_2 are known, one can construct a path joining the sample points in latent space.For example, the linear pathγ_z(t) ≡ (1 - t)z_1 + t z_2, t ∈ [0, 1]corresponds immediately to a path in output space γ_x(t) ≡ f(γ_z(t))However, we wish to detect point masses in output space, which means we should measure distances there as well.This unfortunately means we need to choose a metric on .For a VAE, the standard reconstruction loss obtained from assuming a Gaussian distribution on output space leads to a square loss on output space.This means the L^2 metric is in some sense natural in this case, and it is what we will use.[The L^2 metric is typically a poor metric for comparing, say, images.However in our case a metric which made related objects seem close, independent of their visual details, might actually be a disadvantage.Using a metric in which all objects of a certain type were at distance zero from each other, we could not tell if the generator had memorized or not.]We can then reparametrize to obtain γ_x(s) by constructing s(t) ass(t) = ∫_0^t ∂γ_x/∂ t (η) _L^2 dηand then inverting the mapping (numerically) to obtain t(s).Finally, we can plot p(s).In practice we compute the integral s(t) by summing L^2 distances between sample points along a discrete curve as described in procedure <ref>.In summary: Remarks: * When looking along paths joining two endpoints, it is possible for the path γ_z to come near the latent representation of a third point.In this case, the density may noticeably rise in the middle, or it may simply cause a falsely high density in between the endpoints.* To avoid the false readings above, we suggest either * Using the local measure described next, or* Computing only along paths joining nearest neighbors.Indeed, perhaps the most convincing evidence of memorization comes from computingpaths joining two nearby instances of a single class and seeing essentially no probability mass in the middle.§.§ Local MeasuresRather than consider paths between sample points, one might wish for a local measure of concentration.One alternative to interpolating between endpoints would be to compute the probability mass contained in balls of radius ϵ (measured in , not ) about a given set of points.If the mass increases rapidly as a function of ϵ, this indicates memorization.However, this is essentially noncomputable, as we'd need to integrate the density over a very high-dimensional ball, and evaluating the density requires f^-1, which is unavailable.It might be possible to use some random sampling or other methods to overcome or mitigate the objections above.However, looking at the rate of decay of p(X) along a collection of paths passing through a point seems to provide a good measure of concentration.We consider a set of lines inpassing through a given point Z and apply similar methodology to procedure <ref>.The question is which lines are informative?Note that choosing random directions or, say, every coordinate direction, will generally not work well.Section <ref> explains why: one should consider decay only in nondegenerate singular directions.Using degenerate singular directions amounts to measuring the density along paths on which the output is constant, or nearly so.This does nothing but add noise and numerical instability to the calculations, and in the worst case (which can be easily observed by choosing degenerate directions, see figure <ref>) renders memorization and generalization indistinguishable.Similar to procedure <ref>, we have procedure <ref> for computing decay: Remarks: * There are several ways to combine the decay measures from each singular direction into a single measure.In section <ref> we propose means of second differences of logp(s) (a measure of peakiness or concentration), but other variants (e.g. maximum) are possible. § COMPUTATION AND RESULTS Here we discuss some computational details and illustrate the methods of section <ref> on a pair of VAEs trained on the MNIST dataset.Identical architectures, one VAE was well-trained while the other was trained to overfit and memorize the data set.§.§ Computing the Jacobian J_f Our experiments were performed with the Keras frontend to Tensorflow.While Tensorflow supports automatic differentiation for scalar-valued functions, there is no support for automaticdifferentiation of vector-valued functions (this is awkward to implement using reverse-mode automatic differentiation).Hence, we are unable to use autodiff to compute Jacobian matrices.We instead use a simple central-difference approximation for each entry {J_f}_k,m∂ f_k/∂ z_m≈1/2ϵ(f_k(z_m + ϵ) - f_k(z_m - ϵ) )for some small ϵ.This requires 𝒪(MN) function evaluations for a latent space of size M and output space of size N.However, each evaluation is a neural network forward pass and easily parallelizable. §.§ The Intrinsic Dimension of the Manifolds The SVD of the Jacobian tells us about the intrinsic dimension of the generator function near a point.In figure <ref> we plot the 20 largest singular values in decreasing order, averaged over 1000 randomly-chosen points in the training set.For both the well-trained and overfit networks, there are no significant singular values beyond the 14th, and particularly for the overfit network the decay begins earlier.The well-trained network has larger singular values overall, reflecting the fact that its generator covers greater volumes of the output space.It seems reasonable to declare the intrinsic dimension of either map to be no greater than 14 dimensions.(Applying SVD to the point cloud of latent representations, as opposed to tangent spaces, also suggests the dimension of the latent representation is no greater than 14.)In the second plot, we show decay of singular values about 4 training examples from different classes in the well-trained network.These decay at various rates, suggesting that the effective dimension of the map varies across classes, so locally the map may be lower-dimensional in some areas. §.§ Results We use the methods of section <ref> to explore the differences between two VAEs trained on MNIST.Each used an identical architecture consisting of several convolutional layers with ReLU and max-pooling, followed by fully-connected layers to compute means and log-variances for the latent distributions, followed by fully-connected layers and transposed convolutions with strides (each with ReLU) for the generator.Data was normalized to lie in the range [-1, 1] and a final tanh output layer was applied.Both models used a 100-dimensional latent representation with a latent prior of independent and ∼𝒩(0, 1).However, the first was trained on only 100examples, while the second trained on the entire 60,000 example training set.The difference was stark.The overfit model had huge dips in p(s) between training samples, and the decay measure also showed huge concentration of mass on training examples. Figures <ref>, <ref> show the two-point interpolation method of section <ref>.The endpoints of each path correspond to two training examples in the same class; the well-trained network does a much better job of interpolating the endpoints, and this is reflected in the log-density plots.The well-trained network places significant mass in the regions between the endpoints, whereas the overtrained network places essentially all its mass at the endpoints.Figure <ref> shows the local decay method of section <ref>.These plots are obtained using a line in the largest singular direction, although similar plots are obtained using other non-degenerate singular directions.The conclusion is similar: the overtrained network concentrates the density much more closely on each example.Finally, in figure <ref> we show what happens when we look in degenerate directions: the plots become meaningless, as we're looking in directions in which f is nearly constant.Memorization and generalization become extremely difficult to distinguish based on the plots alone. We obtain a single score from both methods as follows: * We consider p(s) along lines joining each training sample to its nearest neighbor.We compute second differences δ_i = logp_i(s_0) + logp_i(s_1) - 2 * logp_i(s_1/2) p_i is the density along the i-th path, s_0, s_1 correspond to the endpoints of the path and s_1/2 is the midpoint.We then average these second differences over the training set: mean dip = 1/N∑_i=1^N δ_i* For the local decay measure, we also consider second differences, but now at several radii about the central point along the largest singular direction, normalized by radius:η_i(r) = 1/r( logp_i(r) + logp_i(-r) - 2 * logp_i(0))For a given set of radii {r_j}_1^K we compute the mean decay of the peak:mean decay = 1/NK∑_i=1^N ∑_k=1^K η_i(r_k)The motivation for the multiscale method is to obtain robustness to various shapes of decay curves by averaging over several scales.In experiments we use radii of .5 and 1. Dip and peak results are summarized in the following table (note that positive dip and negative decay indicate memorization, since the curvatures are opposite):Model Mean Dip Mean Decay Memorized 1.07 -3.40 Well-trained -0.242 .0369 The results differ by an order of magnitude in each case.The "Memorized" network exhibits a huge drop in log-probability in between samples and a very peaky density.In contrast, the dip and peak scores for the well-trained network show that, if anything, the density increases slightly away from training samples.§ CONCLUSION "Memorization" in generative models means learning an output distribution which is concentrated on a finite number of output examples.We have introduced methods for studying the output distribution and its concentration in the case where the latent density is easily to evaluate and the generator is a fixed function which is difficult to invert.The main difficulty (the apparent degeneracy of the generator function f: ^m →^n) is overcome by noting that it is in fact a smooth map between submanifolds ⊂^m and its image ⊂^n and we introduce machinery for computing the induced density on . § REFERENCES [1] Radford, A& Metz, L& Chintala, S(2015) Unsupervised Representation Learning with Deep Convolutional Generative Adversarial Networks arXiv preprint abs/1511.06434[2] Gregor, K & Danihelka, I& Graves, A& Wierstra, D(2015) DRAW: A Recurrent Neural Network For Image Generation arXiv preprint abs/1502.04623[3] Shifrin, T.(2005) Multivariable mathematics : linear algebra, multivariable, calculus, and manifolds Hoboken, NJ: Wiley[4] do Carmo, M(2013) Riemannian Geometry Boston, MA: Birkhauser Boston | http://arxiv.org/abs/1705.09303v1 | {
"authors": [
"Matt Feiszli"
],
"categories": [
"cs.LG",
"stat.ML"
],
"primary_category": "cs.LG",
"published": "20170525180019",
"title": "Latent Geometry and Memorization in Generative Models"
} |
We formulate a higher-rank version of the boundary measurement map for weighted planar bipartite networksin the disk. It sends a network to a linear combination of _r-webs, and is built upon the r-fold dimer model on the network. When r equals 1, our map is a reformulation of Postnikov's boundary measurement used to coordinatize positroid strata.When r equals 2 or 3, it is a reformulation of the _2- and_3-web immanants defined by the second author. The basic result is that the higher rank map factors through Postnikov's map. As an application, we deduce generators and relations for the space of _r-webs, reproving a result of Cautis-Kamnitzer-Morrison. We establish compatibility between our map and restriction to positroid strata, and thus between webs and total positivity. Effective Sampling: Fast Segmentation Using Robust Geometric Model Fitting Ruwan Tennakoon, Alireza Sadri, Reza Hoseinnezhad, and Alireza Bab-Hadiashar, Senior Member, IEEE R.B. Tennakoon, A. Sadri, R. Hoseinnezhad and A. Bab-Hadiashar are with the School of Engineering, RMIT University, Melbourne, Australia.E-mail: [email protected] 30, 2023 =================================================================================================================================================================================================================================================================================================================§ INTRODUCTION The Grassmannian (k,n) of k-planes in ^n is an algebraic variety which is well-loved in combinatorics. This paper links two combinatorial tools – briefly, dimer configurations and webs – which have been used to study (k,n)and its (homogeneous) coordinate ring.Both approaches have a similar flavor – each involves certain planar diagrams in a disk, and each relies heavily on local diagrammatic moves/relations amongst such diagrams. We show that this resemblance is not coincidental, and that these approaches are dual in a sense that we make precise. Moreover, statements on each side can be translated to the other to give meaningful consequences.The first approach starts with a choice of network N, meaning a planar bipartite graph in the disk whose edges are weighted by nonzero complex numbers. Such a network comes with two parameters: n (the number of boundary vertices) and k (the excedance, cf. (<ref>)). The key operation is Postnikov's boundary measurement map N ↦ (Δ_I(N))_I ∈[n]k∈(k,n)sending a network N to its n k boundary measurements. Each boundary measurement Δ_I(N) is a complex number obtained by summing over dimer configurations of N whose boundary data is I.The striking feature is that for any network, the Δ_I(N) satisfy the well-known Plücker relations, so the image of the boundary measurement is a point in the Grassmannian. The second approach considers a distinguished class of functions on (r,n), indexed by planar diagrams known as _r-webs. Let us consider the space (r,n) = __r((^r)^⊗ n, ), i.e. the vector space of _r-invariant multilinear function of n vectors. The homogeneous coordinate ring [(r,n)] can be viewed as an algebra of _r-invariants, and (r,n) is a certain subspace of [(r,n)]. We will call elements of (r,n) tensor invariants.An _r-web diagram is a planar bipartite graph in the disk, with its edges labeled by positive integers so that the labels around each internal vertex sum to r. (See the left hand side of (<ref>) for an example when r=4. Edges with label 1 are suppressed.) We denote by (r) the vector space of formal sums of _r-web diagrams (it is related to the free spider category <cit.>). Each web diagram determines an element[in fact, the diagrams defined in this introduction only determine a tensor invariant up to a sign. The sign is determined by a procedure called a tagging of the web.] of (r,n). Intuitively, web diagrams determine tensor invariants as follows: an edge in a web diagram with label a corresponds to a copy of the exterior power ⋀^a(^r), and each interior vertex v corresponds to an _r-invariant map between the exterior powers adjacent to v. The web diagram indicates how to compose these building blocks to create more complicated tensor invariants. The map (r) →(r,n) is surjective; i.e. _r-invariants coming from webs span the space of tensor invariants. Webs can be used to study more general tensor invariant spaces, and other homogeneous pieces of [(k,n)], but in this introduction, we will focus on a particular situation where our results are easiest to state. Thus, we restrict attention to networks N whose number of boundary vertices n is a multiple of its excedance k, i.e. n = kr. For these N, we construct an r-fold boundary measurement map N ↦_r(N) ∈(r,n), sending a network to a particular formal sum of webs. Its key feature (our Theorem <ref>) is that it factors through the boundary measurement map (<ref>). That is – if N and N' are two networks with the same boundary measurements (and hence which give the same point in the affine cone over the Grassmannian(k,n)), then _r(N)= _r(N') as tensor invariants,even though these will typically look different as elements of (r). When r is 2 or 3, this reduces to the construction of Temperley-Lieb immanants and web immanantsvia the double-dimer and triple-dimer models studied by the second author <cit.>.We also show (Theorem <ref>) that (<ref>) induces a natural isomorphism (k,n) ≅(r,n)^*. Moreover, both spaces (k,n) and (r,n) are irreducible S_n-modules, where S_n acts on the tensor invariant spaces by permuting the vectors. As S_n-modules, we have that (k,n) ≅(r,n)^* ⊗ϵ where ϵ is the sign representation. Thus, we get a canonically defined S_n-equivariant pairing between _k- and _r-invariant spaces. For small values of k and r, we draw the resulting “web duality pictures” in an Appendix. We will lay out the contents of this paper while mentioning the applications of our main construction. Section <ref> reviews the boundary measurement map for networks via dimer configurations <cit.>, as well as the local moves on networks which preserve the boundary measurements. We also review in this section how boundary measurements, networks, and local moves, can be used to study the positroid stratification of the Grassmannian. Section <ref> gives the basics of tensor invariants and web diagrams. In Section <ref> we make our main definition, i.e. the r-fold boundary measurement map (<ref>), and introduce the closely related immanant map. Section <ref> gives a self-contained proof that the r-fold boundary measurement factors through Postnikov's boundary measurement map. The key tool is a lemma about sign-coherence for webs that we conjecture is related to total positivity.Section <ref> discusses one of our main applications. One of the deepest results on the boundary measurement map is that if networks N and N' have the same boundary measurements, then N' is connected to N by a sequence of local moves <cit.>. Likewise, a guiding problem in the history of web combinatorics was to find a complete set of diagrammatic moves describing the kernel of the map (r) →(r,n). This problem was solved for r=3 by Kuperberg <cit.>, studied furtherby Kim <cit.> and Morrison <cit.>, and settled for all r by Cautis, Kamnitzer, and Morrison in <cit.>. We show that the completeness of the relations in <cit.> follows from Theorem <ref> and the connectedness result for networks. We remark that the results in <cit.> hold in greater generality than ours – those authors work in a quantum deformation of our setting. We are hopeful that an understanding of this will lead to a quantum deformation of the dimer model.In Section <ref>, we make a connection between webs and positroid varieties. Here, we extend the results ofSections <ref> and <ref> to the context of positroids. The totally nonnegative Grassmannian (k,n)_≥ 0 is the set of real points in (k,n) with nonnegative Plucker coordinates <cit.>.The matroidof a totally nonnegative point x ∈(k,n)_≥ 0 is known as a positroid. The Zariski-closures of the positroid strata are known as positroid varieties <cit.>. The homogeneous coordinate ring [Π]of a positroid variety Π = Π_ is a quotient of [(k,n)]. We show that the homogeneous piecesof [Π] are dual to certain naturally defined subspaces of tensor invariants (cf. (<ref>)), and suggest a way to compute the dimensions of these pieces using webs. We believe that our duality reflects deep relations between positroids and webs.We denote by [n] the set of positive integers {1,…,n}, and by S k the collection of k-element subsets of a set S. § ACKNOWLEDGEMENTSThis project, especially Section <ref>, grew from a project suggestion made by Thomas Lam and David Speyer for the AMS Snowbird '14 MRC on cluster algebras. Our results from that workshop are in an Appendix, which is jointly written with Darlayne Addabbo, Eric Bucher, Sam Clearman, Laura Escobar, Ningning Ma, Suho Oh, and Hannah Vogel, who were our group members during this workshop. Some of the work took place at the Perimeter Institute. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research & Innovation.T.L. acknowledges support from the Simons Foundation under award number 341949 and from the NSF under agreement No. DMS-1464693.§ NETWORKS, BOUNDARY MEASUREMENTS, AND POSITROID STRATA §.§ The dimer model in the diskWe denote by (k,n) the Grassmannian of k-dimensional subspaces in a fixed n-dimensional complex vector space. It has a projective embedding (k,n) ^ n k -1, and we denote by (k,n) ⊂^ n k the affine cone over (k,n) with respect to this embedding. Points in the affine cone are collections of n k coordinates satisfying the well-known Plücker relations. The homogeneous coordinate ring[(k,n)] (or equivalently, the coordinate ring of the affine cone (k,n)) is generated by Plücker coordinates (Δ_I)_I ∈[n] k. If we represent a point in x ∈(k,n) by a k × n matrix of maximal rank, then Δ_I(x) is the k × k minor of this matrix with columns I.By a planar bipartite graph in the disk we mean a graph G embedded in a closed disk, with its vertices colored in two colors (black and white) such that edges join vertices of opposite color. Furthermore, label the vertices on the boundary of the disk 1,…,n in counterclockwise order, and require that that ith boundary vertex is incident to at most one boundary edge b_i. Furthermore, for the sake of simplicity, throughout this paper we will require that each of the boundary vertices of G is black. By a network N we will mean a planar bipartite graph in the disk whose edges have been weighted by nonzero complex numbers. In examples, an edge drawn without an edge weight is implicitly assumed to have edge weightequal to 1.A dimer configuration on N (or almost perfect matching of N), is a collection π of edges of N such that each interior vertex is used exactly once in π (and each boundary vertex is used one or zero times). The boundary subset ∂(π) ⊂ [n] is the set of boundary vertices that are used in π. The cardinality k = |∂(π)| depends only on the underlying bipartite graph G, not on the choice of π or the edge weights on N. Explicitly: k = |∂(π)| = no. of interior white vertices in G minus no. of interior black vertices. We call the number k in (<ref>) the excedance of N. The requirement that boundary vertices of a planar bipartite graph G are always black simplifies our statements, but can be removed. Any graph G', possibly with white vertices on the boundary, can be turned into a graph G whose boundary vertices are all black, by adding one edge at each white boundary vertex (if i is a white boundary vertex, we drag it into the interior of the disk, and connect it to a newly created black boundary vertex by a newly created edge).Dimer configurations π on G with boundary ∂(π) = I are in bijection with dimer configurations π' of G' such that I is the union of the black vertices used in π' with the white vertices not used in π', cf. <cit.> for further details. For a network N and a dimer cover π on N, the weight (π) is the product of the weights of the edges used in π. The boundary measurement Δ_I(N) is a weight generating function for dimer configurations with boundary I:Δ_I(N) = ∑_π ∂(π) = Iwt(π). Let N be a network of excedance k, with n boundary vertices, and with at least one almost perfect matching. Then the boundary measurements (Δ_I(N))_I ∈[n] k∈^ n k determine a point (N) in the affine cone (k,n), and thus a point X(N) ∈(k,n).That is, the boundary measurements satisfy the Plücker relations. If N has no almost perfect matchings, then all of its Plücker coordinates are zero, so X(N) is not a well-defined point in (k,n). We make the standing assumption that all networks N considered in this paper have an almost perfect matching.§.§ Local moves It is easy to verify that the following local moves can be applied to a network N to yield a new network N' satisfying X(N') = X(N). Thus the Plücker coordinates for (N) and (N') differ by a common scalar å and we write (N) = å(N'). (G) Gauge equivalence: If edges e_1,e_2,…,e_d are the edges incident to an interior vertex v, we can rescale all of their edge weights by the same constant å∈^*. The resulting network N' satisfies (N) = å(N').(M1) Spider move, square move, or urban renewal: assuming the leaf edges of the spider have been gauge fixed to 1, the transformation isa'=a/ac+bd b'=b/ac+bd c'=c/ac+bd d'=d/ac+bd[scale=0.6](-2,0) – (0,1)–(2,0)–(0,-1)–(-2,0); (0,1) – (0,2); (0,-1) – (0,-2); at (-1.6,0.9) a; at (-1.6,-0.9) d; at (1.6,0.9) b; at (1.6,-0.9) c;[black] (0,1) circle (0.1cm); [black] (0,-1) circle (0.1cm); [white] (-2,0) circle (0.1cm); (-2,0) circle (0.1cm); [white] (0,2) circle (0.1cm); (0,2) circle (0.1cm); [white] (2,0) circle (0.1cm); (2,0) circle (0.1cm); [white] (0,-2) circle (0.1cm); (0,-2) circle (0.1cm);[scale=0.7] (0,-2) – (1,0)– (0,2)– (-1,0)– (0,-2); (1,0) – (2,0); (-1,0) – (-2,0); at (0.9,-1.4) a'; at (-0.9,-1.4) b'; at (0.9,1.4) d'; at (-0.9,1.4) c';[black] (1,0) circle (0.1cm); [black] (-1,0) circle (0.1cm); [white] (-2,0) circle (0.1cm); (-2,0) circle (0.1cm); [white] (0,2) circle (0.1cm); (0,2) circle (0.1cm); [white] (2,0) circle (0.1cm); (2,0) circle (0.1cm); [white] (0,-2) circle (0.1cm); (0,-2) circle (0.1cm);and satisfies (N) = (ac+bd) (N').(M2) two-valent vertex removal.If v has degree two, we can gauge fix both incident edges (v,u) and (v,u') to have weight 1, then contract both edges (that is, we remove both edges, and identify u with u').Note that if v is a two-valent vertex adjacent to boundary vertex i, with edges (v,i) and (v,u), then this move can only be applied when u has degree at most two.(R1) Multiple edges with same endpoints is the same as one edge with sum of weights.(R2) Leaf removal:Suppose v is a leaf, and (v,u) the unique edge incident to it.Then we can remove both v and u, and all edges incident to u. If (u,v) is a boundary edge b_i, then the leaf cannot be removed. (R3) Dipoles (two degree one vertices joined by an edge) can be removed. On the other hand, the following is one of the deepest results on the combinatorics of networks:[Postnikov <cit.>] If N and N' satisfy X(N) = X(N'), then they are connected to each other by a finite sequence of these moves. If furthermore both of these networks have the minimal number of faces in their move-equivalence class, then they are connected by a finite sequence using (M1) and (M2) only. §.§ Positroids Considering boundary measurement maps for various bipartite graphs G leads to a special stratification of the Grassmannian by positroid varieties.The positroid varieties are distinguished in many senses – they are exactly the varieties that can be obtained by projecting Richardson varieties from the flag variety <cit.>, they are exactly the compatibly split Frobenius subvarieties of (k,n) with respect to the standard splitting <cit.>, the (open versions of) positroid varieties are exactly the symplectic leaves with respect to a Poisson structure on (k,n) <cit.>. We only review what we will need here and refer the reader to <cit.>. The totally nonnegative Grassmannian (k,n)_≥ 0 consists of points in the Grassmannian (k,n) that can be given by k × n matrices with real entries, all of whose Plücker coordinates are nonnegative.For any point x ∈(k,n), the matroid of x is the realizable matroid (x) formed by the subsets I ∈[n] k such that Δ_I(x) ≠ 0. The matroid variety associated to a matroidis the closure in (k,n) of {x ∈(k,n)(x) = }.A matroid is a positroid if it is the matroid of a totally nonnegative point x ∈(k,n)_≥ 0. Its corresponding matroid variety Π = Π_ is called a positroid variety. The open positroid variety ∘Π_ is defined to be the subset of Π not belonging to a lower-dimensional positroid variety; it is a Zariski open subset of Π. The intersection ∘Π∩(k,n)_≥ 0 is called a positroid cell and is homeomorphic to _>0^Π.Unlike general matroid varieties which can have arbitrarily bad singularities, positroid varieties are normal and Cohen-Macaulay. Likewise, while the problem of indexing matroid varieties (that is, recognizing representable matroids) is essentially hopeless, positroids are indexed by a well-behaved family of combinatorial structures.Here is one way of indexing positroid strata:Let G be a planar bipartite graph in the disk.Then as N varies over all possible edge-weightings of G by nonzero complex numbers, the boundary measurements X(N) sweep out a Zariski dense subset of a single positroid variety Π = Π(G). Furthermore, every positroid variety Π arises in this way from some planar bipartite graph G. If we restrict attention to networks N with positive real edge weights, then the boundary measurements sweep out the entire positroid cell (k,n)_≥ 0∩∘Π. In particular, if a given boundary measurement Δ_I is not identically zero on G, then Δ_I(N) is positive for each choice of network N with _>0 edge weights. We denote by (G) the positroid such that Π_ = Π(G).We say that G (or N) represents the top cell if its positroid (G) is the uniform matroid, or equivalently, if its positroid variety Π(G) is equal to the Grassmannian (k,n). § TENSOR INVARIANTS AND WEBSLet U be an r-dimensional vector space. We denote by ⋀^a(U) the ath exterior power of U. Throughout this paper we assume we have chosen a basis E_1…,E_r for U, thus giving an isomorphism ⋀^r(U) ≅ under which the volume form E_1 ∧⋯∧ E_r ↦ 1.Let V_1,…,V_n be a sequence of irreducible representations of (U). A tensor invariant is an element of the space_(U)(⊗_i=1^n V_i,). In this paper, we will be interested in tensor invariants of fundamental representations. Let = (_1,…,_n) be a sequence of integers satisfying0 ≤_i ≤ rfori=1, …,n,and _1 + _2 + ⋯ + _n = krfor some k.We are interested in the space _(U) = _(U)(⊗_i=1^n ⋀^_i(U),). The dimension of the space (<ref>) is the coefficient of the Schur polynomial s_((k^r))(x_1,…,x_r) in the product of r elementary symmetric polynomials e_a_1(x_1,…,x_r) ⋯ e_a_r(x_1,…,x_r). By iterating the dual Pieri Rule, this is the number of r × k tableaux, of content , whose entries are strictly increasing along rows and weakly increasing down columns. The simplest example, which the reader is encouraged to have in mind throughout, is when n= kr and = (1,…,1). In this case we prefer to denote _(1,…,1)(U) by (r,n). The space (r,n) is the vector space of _r-invariant multilinear functions on n vectors.Its dimension is the number of standard Young tableaux with r rows and k columns. Webs are a particular way of encoding tensor invariants by using planar diagrams. We will define these tensor invariants in two steps, by introducing untagged webs followed by tagged webs.An untagged web W is a planar graph in the disk, with each directed edge e = (u,v) labeled by a multiplicity m(e) = m(u,v) ∈ [r-1], so that reversing directions replaces a by r-a, that is m(u,v)+m(v,u) = r for all directed edges (u,v). Furthermore, at each interior vertex, there exists a way todirect the edges so that the sum of the incoming multiplicities equals the sum of the outgoing multiplicities. The degree (W) is the sequence of multiplicities (m(b_1),…,m(b_n)) obtained by directing all boundary edges away from the boundary.We will see that an untagged web W determines a tensor invariant in _(U), up to a sign. The sign is determined by choosing a tagging of the web, which we now define. A tagged _r-web Ŵis a planar graph in the disk subject to the following. First, each edge or half-edge of Ŵ is directed and labeled by a multiplicity in one of the three following ways[scale = 1][thick,decoration=markings,mark=at position 1 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (-1,-1)–(-.4,0);[thick] (-.4,0)–(.2,1);at (.2,0) a;[xshift = -5cm]at (8.7,-.5) a;at (9.7,.5) r-a;[thick, decoration=markings,mark=at position 1 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (8,-1)–(8.3,-.5); [thick] (8.3,-.5)–(8.9,.5); [thick,decoration=markings,mark=at position 1 with [scale=2]>, postaction=decorate, shorten >=0.4pt] (9.2,1)–(8.9,.5);[thick] (8.6,0)–(8.4,.17);at (8,-2) pair tag; [xshift = -2cm]at (8.7,-.5) a;at (9.7,.5) r-a;[thick](8,-1)–(8.3,-.5); [thick, decoration=markings,mark=at position 1 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (8.6,0)–(8.3,-.5); [thick, decoration=markings,mark=at position 1 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (8.6,0)–(8.9,.5); [thick] (9.2,1)–(8.9,.5);[thick] (8.6,0)–(8.4,.17);at (8,-2) source tag; . The “tiny edges” decorating the second and third edge types are called tags and they provide us with a preferred choice of side for the given edge. Second, we require thateach interior vertex of Ŵ is modeled on one of the following two pictures: [scale = 1][thick,decoration=markings,mark=at position 1 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,0)–(0,.6);[thick] (0,.6)–(0,1.2);[thick,decoration=markings,mark=at position 1 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (-1.2,-1.2)–(-.6,-.6);[thick] (-.6,-.6)–(0,0);[thick,decoration=markings,mark=at position 1 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (-.6,-1.2)–(-.3,-.6);[thick] (-.3,-.6)–(0,0);[thick,decoration=markings,mark=at position 1 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (.6,-1.2)–(.3,-.6);[thick] (.3,-.6)–(0,0);[thick,decoration=markings,mark=at position 1 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (1.2,-1.2)–(.6,-.6);[thick] (.6,-.6)–(0,0); at (-1.25,-1.4) a_1;at (-.65,-1.4) a_2;at (0,-1.4) …;at (0,-.85) ...;at (.65,-1.4) a_s-1;at (1.25,-1.4) a_s;at (1.2,.6) a_1+⋯ a_s;[fill= white] (0,0) circle [radius = .1];at (0,-2) wedge;[xshift = 4.5cm][thick,decoration=markings,mark=at position 1 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,-1.2)–(0,-.6);[thick] (0,-.6)–(0,0);[thick] (-1.2,1.2)–(-.6,.6);[thick,decoration=markings,mark=at position 1 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,0)–(-.6,.6);[thick] (-.6,1.2)–(-.3,.6);[thick,decoration=markings,mark=at position 1 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,0)–(-.3,.6);[thick] (.6,1.2)–(.3,.6);[thick,decoration=markings,mark=at position 1 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,0)–(.3,.6);[thick] (1.2,1.2)–(.6,.6);[thick,decoration=markings,mark=at position 1 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,0)–(.6,.6); at (-1.25,1.4) a_1;at (-.65,1.4) a_2;at (0,1.4) …;at (0,.85) ...;at (.65,1.4) a_s-1;at (1.25,1.4) a_s;at (1.2,-.5) a_1+⋯ a_s;[fill= black] (0,0) circle [radius = .1];at (0,-2) shuffle; . Thirdly and finally, we require that each boundary edge in Ŵ is a source. The degree (Ŵ) = (_1,…,_n) of Ŵ is the sequence of edge multiplicities of the boundary edges.We adopt the convention throughout that Ŵ stands for a tagged web as in Definition <ref>, and will reserve the symbol W for untagged webs, or for the weblike subgraphs that we discuss later. We will refer to the data of tagging by the letter , writing Ŵ = Ŵ(W,).The definition of a tagged web could in principle allow for “wedge and shuffle” vertices, in which there are multiple edges flowing in both the inward and outward direction in (<ref>). However, these will not come up in practice in our paper, so we do not work with this definition.It will be important for us to note that by tagging sufficiently many edges in Ŵ, one can always ensure that there are no oriented cycles in Ŵ. We will always assume that Ŵ has been tagged in this way, because this simplifies the association of a tensor invariant to each web. Denote by _(r) the space of finite formal -linear combinations of tagged _r-web diagrams of degree .Now we explain how an _r-web Ŵ of degreedetermines a tensor invariant Ŵ∈_(U).This induces a linear map _(r) →_(U) that is known to be surjective. An edge with multiplicity m(e) in Ŵ encodes a copy of ⋀^m(e)(U). Each of the interior vertices (<ref>) encodes an (U)-invariant map between the indicated tensor powers of fundamental representations. The first vertex encodes the exterior product map ⋀^a_1(U) ⊗⋯⊗⋀^a_s(U) →⋀^a_1+⋯ +a_s(U)given by x_1 ⊗ x_2 ⊗⋯⊗ x_s ↦ x_1 ∧ x_2 ∧⋯∧ x_s.The second vertex encodes the map ⋀^a_1+⋯ +a_s(U) →⋀^a_s(U) ⊗⋯⊗⋀^a_1(U)given by sending the wedge product x_1 ∧⋯∧ x_b ∈⋀^b(U) to the signed sum of shuffles∑± (x_i_1∧ x_i_2∧⋯∧ x_i_a_s) ⊗⋯⊗(x_i_b-a_1+1∧ x_i_b-a_1+2∧⋯∧ x_i_b) ∈⋀^a_s(U) ⊗⋯⊗⋀^a_1(U)where the summation is over permutations (i_1,i_2,…,i_b) of (1,2,…,b) such that indices are increasing in each block: i_1 < i_2 < ⋯ < i_a_s, …,i_b-a_1+1< ⋯ < i_b.The sign ± is the sign of the permutation (i_1,i_2,…,i_b), multiplied by the “global sign” of the permutation (b-a_s+1,…,b,…,1,…,a_1)Note that in both cases, the cyclic order of the edges in (<ref>) is crucial for specifying signs. The tagged edges in (<ref>) should be thought of as degenerate cases of the maps (<ref>) and (<ref>), where the tag stands encodes a copy of ⋀^r(U), which we canonically identify withusing the volume form. Thus the pair tag ⋀^a ⊗⋀^r-a→⋀^r≅ produces a number obtained by pairing the two incoming tensors, and source tag “creates” two tensors using the shuffle ⋀^r→⋀^a⊗⋀^r-a. Note that the side of the edge that the tag occurs on matters, because, for example, the maps ⋀^a ⊗⋀^r-a→⋀^r≅ and ⋀^r-a⊗⋀^a→⋀^r≅ differ by a sign.To evaluate Ŵ on a simple tensor x_1 ⊗⋯⊗ x_n ∈⊗_i=1^n ⋀^_i(U), we imagine placing each tensor x_i at boundary vertex i. We repeatedly compose the four basic morphisms – wedge, shuffle, pair, and source – as indicated by the arrows in Ŵ to obtain the value Ŵ(x_1 ⊗⋯⊗ x_n). [Taggings and perfect orientations] In practice, the following recipe will work to specify a tagging for the webs we consider in this paper. Suppose that we have an untagged web W whose underlying graph G is a bipartite graph.Orient all the edges in W from white to black, and further suppose that the weights of the directed edges around each vertex sum to r. These conditions characterize the r-weblike subgraphs that appear in this paper. In this situation, we can tag the web by choosing a perfect orientation of G. A perfect orientation of a planar bipartite graph G in the disk is a choice of direction on each edge of G such that every white vertex has outdegree 1 and every interior black vertex has indegree 1. To get a perfect orientation of G, just choose an almost perfect matching of G: given an almost perfect matching π, we obtain a perfect orientation by directing each edge e from white to black if e ∈π (resp. black to whiteif e ∉π).The r-weblike subgraphs defined below always have almost perfect matchings (in fact, they are a union of r almost perfect matchings), so they can be perfectly oriented. Furthermore, if G has excedance k, then any perfect orientation for G will have exactly k boundary sinks (and exactly n-k boundary sources). By adding a pair tag at each of the k boundary sinks, we get a tagged web. The only reason we don't use this construction in general is that there is no guarantee that the resulting web does not contain oriented cycles. To obtain such a web, we may need to add more tags along interior edges, introducing sources and pairings. Consider the _4-web Ŵ in (<ref>). The edge mutiplicities are red, and multiplicities equal to oneare omitted. The underlying bipartite graph for Ŵ has excedance 2. In the second figure, we show how to evaluate Ŵ on the tensor product of basis vectorsE = E_1 ⊗ E_2 ⊗ E_3 ⊗ E_4 ⊗ E_3 ⊗ E_2 ⊗ E_1 ⊗ E_4 ∈ U^⊗ 8. We abbreviate E_123 = E_1 ∧ E_2 ∧ E_3 etc. [scale=.85]at (190:2.4cm) 3;at (145:2.4cm) 2;at (100:2.4cm) 1; at (50:2.4cm) 8;at (5:2.4cm) 7;at (-40:2.4cm) 6;at (-85:2.4cm) 5;at (-130:2.4cm) 4;[dashed] (0,0) circle (2cm); [thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,0)–(15:.9cm);[red] at (55:.65cm) 2;[thick,decoration=markings,mark=at position .7 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (15:1cm)–(40:1.35cm);[red] at (50:1.1cm) 3;[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (50:1.9cm)–(40:1.35cm) ;[thick] (40:1.35cm)–(50:1.35cm); [thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (5:2cm)–(15:1.15cm);[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (135:1cm)–(135:.15cm);[red] at (165:.7cm) 3;[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (95:2cm)–(130:1.1cm);[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (145:2cm)–(136:1.15cm);[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (190:2cm)–(140:1cm);[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt](0,0)–(250:.9cm);[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (235:2cm)– (245:1.13cm);[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (280:2cm)–(255:1.13cm);[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (250:1cm)–(310:1.2cm) node[pos = .5, red, above]3;[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (325:1.9cm)–(310:1.2cm);[thick] (310:1.2cm)–(323:1cm); [black] (0,0) circle (0.1cm); [white] (15:1cm) circle (.1cm);(15:1cm) circle (.1cm); [white] (135:1cm) circle (.1cm);(135:1cm) circle (.1cm); [white] (250:1cm) circle (.1cm);(250:1cm) circle (.1cm);[xshift = 7cm]at (190:2.4cm) E_3;at (145:2.4cm) E_2;at (100:2.4cm) E_1;at (50:2.4cm) E_4;at (5:2.4cm) E_1;at (-40:2.4cm) E_2;at (-85:2.4cm) E_3;at (-130:2.4cm) E_4; [dashed] (0,0) circle (2cm);[dashed] (0,0) circle (2cm); [thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,0)–(15:.9cm);at (55:.65cm) E_23;[thick,decoration=markings,mark=at position .7 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (15:1cm)–(40:1.45cm);at (58:1.2cm) E_231;at (210:.6cm) E_1;[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (50:1.9cm)–(40:1.45cm) ;[thick] (40:1.45cm)–(50:1.45cm); [thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (5:2cm)–(15:1.15cm);[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (135:1cm)–(135:.15cm);at (165:.7cm) E_123;[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (95:2cm)–(130:1.1cm);[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (145:2cm)–(136:1.15cm);[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (190:2cm)–(140:1cm);[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt](0,0)–(250:.9cm);[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (235:2cm)– (245:1.13cm);[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (280:2cm)–(255:1.13cm);[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (250:1cm)–(310:1.3cm) node[pos = .7, above] E_143;[thick,decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (325:1.9cm)–(310:1.3cm);[thick] (310:1.3cm)–(323:1.2cm); [black] (0,0) circle (0.1cm); [white] (15:1cm) circle (.1cm);(15:1cm) circle (.1cm); [white] (135:1cm) circle (.1cm);(135:1cm) circle (.1cm); [white] (250:1cm) circle (.1cm);(250:1cm) circle (.1cm); [black] (0,0) circle (0.1cm); [white] (15:1cm) circle (.1cm);(15:1cm) circle (.1cm); [white] (135:1cm) circle (.1cm);(135:1cm) circle (.1cm); [white] (250:1cm) circle (.1cm);(250:1cm) circle (.1cm); .To begin, place the ith vector in E at boundary vertex i. Vectors flow along directed arrows until they reach a black vertex, which in this case happens when E_1 ∧ E_2 ∧ E_3 arrives at the interior black vertex. The map (<ref>) splits this tensor up as a signed sum E_1 ⊗ (E_2 ∧ E_3)- E_2 ⊗ (E_1 ∧ E_3)+ E_3 ⊗ (E_1 ∧ E_2). The signs come from (<ref>).The evaluation is a sum of contributions from three terms, but only the first term pairs nontrivially with the vectors at 6 and 8. The pairings for this term are (E_1 ∧ E_4 ∧ E_3) ∧ E_2 = -1 and (E_2 ∧ E_3 ∧ E_1) ∧ E_4 = 1. The evaluation is Ŵ|_E = 1 · -1 · 1 = -1, obtained as the product of the sign at the shuffle, at vertex 6, and at vertex 8. Definition <ref> is not the usual definition of a web <cit.>. Typically, the definition comes with an added requirement that every interior vertex be trivalent (as a degenerate case of this, bivalent interior vertices are also allowed). Requiring trivalence is reasonable, because it can be shown that all possible (U)-equivariant maps amongst tensor products of various ⋀^a(V) come from compositions of maps involving three tensor factors <cit.>. Likewise, using (M2), any network N can be transformed to a network N' with interior vertices of valence at most three, without changing the boundary measurements. However, the trivalent restriction is not necessary for the current work.[Webs in small rank] Let us describe _r-webs Ŵ in the multilinear case = (1,…,1), for r = 1, 2,3. In each case, we have vectors v_1,…,v_n sitting at the boundary vertices 1,…,n. When r=1, then U ≅, and Ŵ is a union of isolated interior edges (these do not affect the tensor invariant and can be removed) and edges based at boundary vertices. Thus Ŵ encodes a monomial in the vectors v_1,…,v_n.When r =2, an _2-web Ŵ consists of a disjoint union of a) tagged cycles, and b) directed paths v_i→ v_j between boundary vertices. An oriented cycle contributes a multiplicative factor of ± 2 (depending on the tagging) when Ŵ is evaluated on v_1,…,v_n. Changingthe orientation on a path changes the web by a minus sign. Thus, ignoring these signs, and removing all cycles, _2-webs are spanned by crossingless matchings on the boundary vertices. In fact, these crossingless matchings are a basis for (2,2r). A result in similar spirit is true when r=3. In this case, the sign of a web does not depend on tagging. _3-webs are typically drawn as bipartite graphs without directed edges, with the convention that the edge multiplicities are given by m(b,w)=1, where b is a black vertex and w is a white vertex. The _3 skein relations provide diagrammatic rules for expressing any _3-web in terms of a basis of non-elliptic webs <cit.>, i.e. webs that are without 2-valent vertices or interior faces bounded by four or fewer sides (cf. <cit.> for a summary of these results). For r ≥ 4, the set of _r-webs is a distinguished spanning set, but the existence of a web basis satisfying enough “desirable” properties is unknown.Finally, let us recall the relationsip between webs and the Grassmannian. Each Plücker coordinate Δ_I ∈[(k,n)] is an _k-invariant function of the column vectors v_1,…,v_n ∈^k representing a point in (k,n). In general, there is a ^n-grading of [(k,n)] given by the degree in each column. For example, Δ_123Δ_256∈[(3,6)] has degree (1,2,1,0,1,1). The coordinate ring decomposes as [(k,n)] = ⊕_r=1^∞⊕__1+⋯_n = rk[(k,n)]_,where the inner direct sum is overas in (<ref>). Note that [(k,n)]_ is given by the invariant space__k(⊗_j=1^n ^_j(^k),). Our main construction will give a duality between the graded piece [(k,n)]_ of the homogeneous coordinate ring of the Grassmannian and the _r-invariant space _(U), wheresatisfies (<ref>). We will discuss in Section <ref> that these spaces are naturally dual using purely representation-theoretic considerations. However, our work manifests this duality explicitly in terms of webs and dimers.If n = rk, the invariant space (k,n) sits inside [(k,n)] as the multilinear piece = (1,…,1). It is spanned by the r-fold products of Plücker coordinates Δ_I_1⋯Δ_I_r satisfying I_1 ∪⋯∪ I_r = [n]. Thus our results will provide a duality between(k,n) and (r,n). We examine this duality pictorially in the Appendix.§ THE MAIN CONSTRUCTIONIn this section we make the connection between dimer configurations and webs. Suppose we are given r dimer configurations π_1,…,π_r on N. By superimposing them we naturally obtain what we call an r-weblike subgraph:An r-weblike subgraph W ⊂ G (alternatively,W ⊂ N) is a subgraph of G, using all the vertices of G, with each edge e of W labeled by a multiplicity m(e) ∈ [r], in such a way that the sum of the multiplicities around each interior vertex is r. The weight of W is the product (W) = ∏_e (e)^m(e) of the edge weights raised to the indicated multiplicity. The degree of W is its sequence (W) = (m(b_1),…,m(b_n)) of boundary multiplicities,where b_1,…,b_n are the boundary edges.Each r-weblike subgraph W determines an untagged _r-web by giving each black-white edge (b,w) the multiplicity of this edge in W. Thus, one can get a tensor invariant Ŵ(W,) from W by choosing a tagging .For any two such choices of taggingand ', the resulting tensor invariants are equal up to a sign: Ŵ(W,) = ±Ŵ(W,').In Section <ref>, we define a sign (W,) ∈{± 1} so that the tensor invariantW := (W,)Ŵ(W,).does not depend on .In other words, there is a “correct” choice of sign for the tensor invariant represented by W.With this in mind, we can formulate the main definition of this paper.Let N be a network of excedance k andsatisfy (<ref>). We denote by_r(N;) := ∑_W ⊂ G, (W) = (W)W∈_(U),the weighted sum over the r-weblike subgraphs of G with degree . It is a -linear combination of the various boldface W∈_(U), with the (W)'s serving as coefficients.We think of _r(N;) as an r-fold boundary measurement.When r =1, the choice ofis equivalent to the choice of I ∈[n]k, and _1(N;I) is a variant of the boundary measurement Δ_I(N) (<ref>).Whereas Δ_I(N) is a number, our _1(N;I) lies in _(U), which in this case is isomorphic to .We denote by _r(N) := ∑__r(N;) ∈⊕__(U). Consider the pair of networks N and N' given by[scale=.65] [dashed] (1,1) circle (3cm);[thick] (0,0) – (0,2)– (2,2)– (2,0)– (0,0);[thick] (2.5,-.5) – (2,0);[thick] (-.5,2.5) – (0,2);[thick] (-.5,2.5)–(-1.6,2.5);[thick] (-.5,2.5)–(-.8,3.4);[thick] (2.5,-.5)–(3.6,-.5);[thick] (2.5,-.5)–(2.8,-1.4);[thick] (2,2)–(3.1,3.1);[thick] (-1.1,-1.1)–(0,0);at (-.4,1) a;at (2.4,1) c;at (1,2.4) b;at (1,-0.4) d;at (-1.2,2.2) e;at (3.1,-.1) f; [black] (2,0) circle (0.1cm); [black] (0,2) circle (0.1cm); [white] (0,0) circle (0.1cm);(0,0) circle (0.1cm); [white] (2,2) circle (0.1cm);(2,2) circle (0.1cm); [white] (-.5,2.5) circle (0.1cm);(-.5,2.5) circle (0.1cm); [white] (2.5,-.5) circle (0.1cm);(2.5,-.5) circle (0.1cm); [scale=0.7] [thick] (0,0) – (2,0)–(2,2)–(0,2)–(0,0);[thick] (2.5,2.5) – (2,2);[thick] (2.5,2.5)–(3.1,3.1);[thick] (-.5,-.5)–(-1.1,-1.1);[thick] (0,2)–(-1.6,2.5);[thick] (0,2)–(-.8,3.4);[thick] (-.5,-.5)–(-1.1,-1.1);[thick] (2,0)–(3.6,-.5);[thick] (2,0)–(2.8,-1.4);[thick] (-.5,-.5) – (0,0);at (-.4,1) c';at (2.4,1) a';at (1,2.4) b';at (1,-0.4) d';at (-1.2,2.1) e;at (3.2,0) f;[black] (0,0) circle (0.1cm); [black] (2,2) circle (0.1cm); [white] (2,0) circle (0.1cm);(2,0) circle (0.1cm); [white] (0,2) circle (0.1cm);(0,2) circle (0.1cm); [white] (2.6,2.6) circle (0.1cm);(2.6,2.6) circle (0.1cm); [white] (-.6,-.6) circle (0.1cm);(-.6,-.6) circle (0.1cm); [dashed] (1,1) circle (3cm); ,where a',b',c',d' are related to a,b,c,d according to the spider move (<ref>)a'=a/ac+bd b'=b/ac+bd c'=c/ac+bd d'=d/ac+bd. Their parameters are k=2 and n=6, and the letters a,a',b,b',…,e,f ∈^× denote edge weights. Let us consider r =3 and = (1,1,1,1,1,1). The network N has three r-weblike subgraphs with degree . The 3-fold boundary measurement _3(N;) is a linear combination [scale=.45] [dashed] (1,1) circle (3cm);[thick] (0,0) – (0,2)– (2,2)– (2,0)– (0,0);[thick] (2.5,-.5) – (2,0);[thick] (-.5,2.5) – (0,2);[thick] (-.5,2.5)–(-1.6,2.5);[thick] (-.5,2.5)–(-.8,3.4);[thick] (2.5,-.5)–(3.6,-.5);[thick] (2.5,-.5)–(2.8,-1.4);[thick] (2,2)–(3.1,3.1);[thick] (-1.1,-1.1)–(0,0);at (-4,1) abcdef;[black] (2,0) circle (0.1cm); [black] (0,2) circle (0.1cm); [white] (0,0) circle (0.1cm);(0,0) circle (0.1cm); [white] (2,2) circle (0.1cm);(2,2) circle (0.1cm); [white] (-.5,2.5) circle (0.1cm);(-.5,2.5) circle (0.1cm); [white] (2.5,-.5) circle (0.1cm);(2.5,-.5) circle (0.1cm);[scale=0.45] [dashed] (1,1) circle (3cm);[thick] (0,0) – (0,2);[thick] (2,0) – (2,2);[thick] (2.5,-.5) – (2,0);[thick] (-.5,2.5) – (0,2);[thick] (-.5,2.5)–(-1.6,2.5);[thick] (-.5,2.5)–(-.8,3.4);[thick] (2.5,-.5)–(3.6,-.5);[thick] (2.5,-.5)–(2.8,-1.4);[thick] (2,2)–(3.1,3.1);[thick] (-1.1,-1.1)–(0,0);at (-4.2,1) +a^2c^2ef;[red] at (-.5,1)2;[red] at (2.5,1)2; [black] (2,0) circle (0.1cm); [black] (0,2) circle (0.1cm); [white] (0,0) circle (0.1cm);(0,0) circle (0.1cm); [white] (2,2) circle (0.1cm);(2,2) circle (0.1cm); [white] (-.5,2.5) circle (0.1cm);(-.5,2.5) circle (0.1cm); [white] (2.5,-.5) circle (0.1cm);(2.5,-.5) circle (0.1cm);[scale=0.45] [dashed] (1,1) circle (3cm);[thick] (0,0) – (2,0);[thick](0,2) – (2,2);[thick](2.5,-.5) – (2,0);[thick](-.5,2.5) – (0,2);[thick](-.5,2.5)–(-1.6,2.5);[thick](-.5,2.5)–(-.8,3.4);[thick](2.5,-.5)–(3.6,-.5);[thick](2.5,-.5)–(2.8,-1.4);[thick](2,2)–(3.1,3.1);[thick](-1.1,-1.1)–(0,0);at (-4.2,1) +b^2d^2ef;[red] at (1,-.5)2;[red] at (1,2.5)2; [black] (2,0) circle (0.1cm); [black] (0,2) circle (0.1cm); [white] (0,0) circle (0.1cm);(0,0) circle (0.1cm); [white] (2,2) circle (0.1cm);(2,2) circle (0.1cm); [white] (-.5,2.5) circle (0.1cm);(-.5,2.5) circle (0.1cm); [white] (2.5,-.5) circle (0.1cm);(2.5,-.5) circle (0.1cm); with coefficients depending on a,…,f. On the other hand, _3(N';) is a linear combination of two webs[scale=0.45] [thick](0,0) – (0,2);[thick](2,2)–(2,0);[thick](2.5,2.5) – (2,2);[thick](2.5,2.5)–(3.1,3.1);[thick](-.5,-.5)–(-1.1,-1.1);[thick](0,2)–(-1.6,2.5);[thick](0,2)–(-.8,3.4);[thick](-.5,-.5)–(-1.1,-1.1);[thick](2,0)–(3.6,-.5);[thick](2,0)–(2.8,-1.4);at (-4,1) a' c'ef;[red] at (2.05,2.7)2;[red] at (-.7,-.05)2;(-.5,-.5) – (0,0);[black] (0,0) circle (0.1cm); [black] (2,2) circle (0.1cm); [white] (2,0) circle (0.1cm);(2,0) circle (0.1cm); [white] (0,2) circle (0.1cm);(0,2) circle (0.1cm); [white] (2.6,2.6) circle (0.1cm);(2.6,2.6) circle (0.1cm); [white] (-.6,-.6) circle (0.1cm);(-.6,-.6) circle (0.1cm); [dashed] (1,1) circle (3cm); [scale=0.45] [thick](0,0) – (2,0);[thick](2,2)–(0,2);[thick](2.5,2.5) – (2,2);[thick](2.5,2.5)–(3.1,3.1);[thick](-.5,-.5)–(-1.1,-1.1);[thick](0,2)–(-1.6,2.5);[thick](0,2)–(-.8,3.4);[thick](-.5,-.5)–(-1.1,-1.1);[thick](2,0)–(3.6,-.5);[thick](2,0)–(2.8,-1.4);[thick](-.5,-.5) – (0,0);at (-4,1) +b' d' ef;[red] at (2.05,2.7)2;[red] at (-.7,-.05)2; [black] (0,0) circle (0.1cm); [black] (2,2) circle (0.1cm); [white] (2,0) circle (0.1cm);(2,0) circle (0.1cm); [white] (0,2) circle (0.1cm);(0,2) circle (0.1cm); [white] (2.6,2.6) circle (0.1cm);(2.6,2.6) circle (0.1cm); [white] (-.6,-.6) circle (0.1cm);(-.6,-.6) circle (0.1cm); [dashed] (1,1) circle (3cm); .The following theorem says that the r-fold boundary measurements factors through Postnikov's boundary measurement map.If N and N' are two networks satisfying (N) = (N'), then _r(N;)= _r(N';) ∈_λ(U). That is, even though the right hand sides of (<ref>) are different elements of _(r), they are equal as tensor invariants in _λ(U). We prove Theorem <ref> in Section <ref>. Continuing with Example <ref>, recall that a',b',c',d' are related to a,b,c,d by (<ref>). The networks N and N' are related by (M1), and (N) = (ac+bd)(N').It follows that _3(N;) = (ac+bd)^3_3(N';). By equating the coefficient of abcdef in (<ref>) with the coefficient of abcdef/(ac+bd)^3 in (<ref>), we deduce the square move for _3-webs: [scale=.45] [dashed] (1,1) circle (3cm);[thick](0,0) – (0,2)– (2,2)– (2,0)– (0,0);[thick](2.5,-.5) – (2,0);[thick](-.5,2.5) – (0,2);[thick](-.5,2.5)–(-1.6,2.5);[thick](-.5,2.5)–(-.8,3.4);[thick](2.5,-.5)–(3.6,-.5);[thick](2.5,-.5)–(2.8,-1.4);[thick](2,2)–(3.1,3.1);[thick](-1.1,-1.1)–(0,0);at (6,1) =; [black] (2,0) circle (0.1cm); [black] (0,2) circle (0.1cm); [white] (0,0) circle (0.1cm);(0,0) circle (0.1cm); [white] (2,2) circle (0.1cm);(2,2) circle (0.1cm); [white] (-.5,2.5) circle (0.1cm);(-.5,2.5) circle (0.1cm); [white] (2.5,-.5) circle (0.1cm);(2.5,-.5) circle (0.1cm); [scale=0.45] [thick](0,0) – (0,2);[thick] (2,2)–(2,0);[thick](2.5,2.5) – (2,2);[thick](2.5,2.5)–(3.1,3.1);[thick](-.5,-.5)–(-1.1,-1.1);[thick](0,2)–(-1.6,2.5);[thick](0,2)–(-.8,3.4);[thick](-.5,-.5)–(-1.1,-1.1);[thick](2,0)–(3.6,-.5);[thick](2,0)–(2.8,-1.4);[red] at (2.05,2.7)2;[red] at (-.7,-.05)2;[thick](-.5,-.5) – (0,0);[black] (0,0) circle (0.1cm); [black] (2,2) circle (0.1cm); [white] (2,0) circle (0.1cm);(2,0) circle (0.1cm); [white] (0,2) circle (0.1cm);(0,2) circle (0.1cm); [white] (2.6,2.6) circle (0.1cm);(2.6,2.6) circle (0.1cm); [white] (-.6,-.6) circle (0.1cm);(-.6,-.6) circle (0.1cm); [dashed] (1,1) circle (3cm); [scale=0.45] [thick](0,0) – (2,0);[thick](2,2)–(0,2);[thick](2.5,2.5) – (2,2);[thick](2.5,2.5)–(3.1,3.1);[thick](-.5,-.5)–(-1.1,-1.1);[thick](0,2)–(-1.6,2.5);[thick](0,2)–(-.8,3.4);[thick](-.5,-.5)–(-1.1,-1.1);[thick](2,0)–(3.6,-.5);[thick](2,0)–(2.8,-1.4);[thick](-.5,-.5) – (0,0);at (-3,1) +;[red] at (2.05,2.7)2;[red] at (-.7,-.05)2; [black] (0,0) circle (0.1cm); [black] (2,2) circle (0.1cm); [white] (2,0) circle (0.1cm);(2,0) circle (0.1cm); [white] (0,2) circle (0.1cm);(0,2) circle (0.1cm); [white] (2.6,2.6) circle (0.1cm);(2.6,2.6) circle (0.1cm); [white] (-.6,-.6) circle (0.1cm);(-.6,-.6) circle (0.1cm); [dashed] (1,1) circle (3cm); .§.§ ImmanantsNow consider any functional φ∈_(U)^* on the _r tensor invariant space, wheresatisfies (<ref>). From Theorem <ref>, the number φ(_r(N;))is independent of the choice of network N representing the point (N) ∈(k,n). Now let G be a bipartite graph that represents the top cell.As we vary over networks N with underlying graph G, the map (N) ↦φ(N) defines a function on the subset of (k,n) swept out by (N)'s.The function (N) ↦φ(_r(N;)) extends to a (uniquely defined) element of [(k,n)]_; this function does not depend on the choice of G. We will prove Proposition <ref> in Section <ref>. It leads naturally to the following definition: The linear map : _(U)^* →[(k,n)]_ defined by (φ)((N)) = φ(_r(N;)) is called the immanant map. Another of our main theorems is the following.The immanant map : _(U)^* →[(k,n)]_ is an isomorphism.Thus, we obtain a canonical pairing of _(U) with [(k,n)]_, described more explicitly in (<ref>) and (<ref>). In the multilinear case = (1,…,1), this pairing is the unique (up to scalars) S_n-equivariant pairing of the web spaces (r,n) and (k,n), cf. Section <ref>.When r is 2 or 3, the second author previously defined _2- and _3-web immanants <cit.>. These are obtained via (<ref>) when the functional φ∈(r,n)^* is the dual functional to an element of the web basis. In the case that r ≥ 4, since we are without a notion of a web basis, we believe the r-fold boundary measurement _r(N) is the more fundamental object.§ PROOF OF THE MAIN THEOREMSLet W be an r-weblike subgraph of a planar bipartite graph G with n boundary vertices, and with boundary edges b_1,…,b_n. A consistent labeling of W is a labeling of each edge e in W by a subset S(e) ⊂ [r] so that |S(e)| = m(e), and such that the union of the sets around each interior vertex is [r]. Equivalently, one can require that the sets S(e) are disjoint around each vertex. Let ℓ be a consistent labeling. From the following pair of equalities of multisets⋃_all edges S(e) = ⋃_white vertices [r]⋃_all non-boundary edges e S(e) = ⋃_interior black vertices [r],it follows that the multiset S(b_1) ∪⋯∪ S(b_n) satisfies S(b_1) ∪⋯∪ S(b_n) = {1^k,2^k,…,r^k}.We refer to = (S(b_1),…,S(b_n)) as a list of boundary label subsets. We can also associate to any labeling ℓ a list of boundary location subsets = (I_1,…,I_r) ⊂ [n], defined by I_i = {j ∈ [n] i ∈ S(b_j)}. From the preceding argument, we know that each |I_i| has size k. Furthermore,I_1 ∪⋯∪ I_r = {1^_1,2^_2,…,n^_n}as multisets, whereis the degree of W. The data of boundary labels = (S_1,…,S_n) ⊂ [r] satisfying (<ref>), and the data of boundary locations = (I_1,…,I_r) ⊂ [n], satisfying (<ref>), are dual to each other (in the sense of combinatorial design theory), i.e. one can be recovered from the other.Recall we have a vector space U with basis E_1,…,E_r satisfying E_1 ∧ E_2 ∧⋯∧ E_r = 1. To any subset S ⊂ [r] with |S| = a we can associate the tensor E_S ∈⋀^a(U) by taking the wedge of the basis vectors labeled by S, in ascending order. If = (S_1,…,S_n) is a list of boundary label subsets,we obtain in this way a tensor E_∈⊗_j=1^n ⋀^_j(U) by E_ = E_S_1⊗ E_S_2⊗⋯⊗⋯⊗E_S_n. It will be convenient to replace the multiset {1^k,2^k,…,r^k} by an honest set = {1_1,1_2,…,1_k,2_1,2_2,…,2_k,…,r_1,…,r_k} which we refer to as the alphabet. We introduce the obvious total order 1_1 < 1_2 < ⋯ < 1_k < 2_1 < ⋯ < r_k on . We letbe the free vector space with basis , and with volume form 1_1 ∧ 1_2 ∧⋯∧ r_k = 1 ∈⋀^rk(). We will think of the numbers 1,…,r as colors: for i_s,j_t ∈ we say that i_s and j_t have the same color if i=j. If we read the indices of the basis vectors in (<ref>) from left to right, we get a word w'() in which each of the numbers 1,…,r appears exactly k times.We let ()= 1 or -1 depending on whether the number of inversions of w'() is even or odd. Equivalently, we can think of the entries of w'() as elements ofby making the subscripts increase by one from left to right.We let w() ∈ denote the word obtained from w'() in this way. Then () is the value of the wedge product of w() ∈⋀^rk().Let G be the bipartite graph underlying Example <ref>. Here is a particular 4-weblike subgraph W⊂ G, and a consistent labeling ℓ of W: [scale=.85]at (190:2.4cm) 3;at (145:2.4cm) 2;at (100:2.4cm) 1; at (50:2.4cm) 8;at (5:2.4cm) 7;at (-40:2.4cm) 6;at (-85:2.4cm) 5;at (-130:2.4cm) 4;at (-4,0) W =;[dashed] (0,0) circle (2cm); [thick] (0,0)–(15:.9cm) node[pos = .5, red, above]2;[thick] (15:1cm)–(50:1.9cm);[thick] (5:2cm)–(15:1.15cm);[thick] (135:1cm)–(135:.15cm); [thick] (95:2cm)–(130:1.1cm);[thick] (145:2cm)–(136:1.15cm);[thick] (190:2cm)–(140:1cm);[thick](0,0)–(250:.9cm);[thick] (235:2cm)– (245:1.13cm);[thick] (280:2cm)–(255:1.13cm);[thick] (250:1cm)–(325:1.9cm);[black] (0,0) circle (0.1cm); [white] (15:1cm) circle (.1cm);(15:1cm) circle (.1cm); [white] (135:1cm) circle (.1cm);(135:1cm) circle (.1cm); [white] (250:1cm) circle (.1cm);(250:1cm) circle (.1cm);[xshift = 7cm]at (190:1.5cm) 3;at (150:1.5cm) 2;at (115:1.5cm) 1; at (0:1.5cm) 1; at (-80:1.5cm) 3;at (-130:1.5cm) 4;at (-3,0) ℓ =;[dashed] (0,0) circle (2cm); [thick] (0,0)–(15:.9cm) node[pos = .5, above] 23;[thick] (15:1cm)–(50:1.9cm) node[pos = .7, left] 4;[thick] (5:2cm)–(15:1.1cm);[thick] (135:1cm)–(135:.15cm) node[pos = .8, left] 4;[thick] (95:2cm)–(130:1cm);[thick] (145:2cm)–(136:1.15cm);[thick] (190:2cm)–(140:1cm);[thick](0,0)–(250:.9cm) node[pos = .5, left] 1;[thick] (235:2cm)– (245:1.13cm);[thick] (280:2cm)–(255:1.13cm);[thick] (250:1cm)–(325:1.9cm) node[pos = .5, above] 2; [black] (0,0) circle (0.1cm); [white] (15:1cm) circle (.1cm);(15:1cm) circle (.1cm); [white] (135:1cm) circle (.1cm);(135:1cm) circle (.1cm); [white] (250:1cm) circle (.1cm);(250:1cm) circle (.1cm); .The labeling ℓ of W has boundary location subsets = ({1,7},{2,6},{3,5},{4,8}) and boundary label subsets = ({1},{2},{3},{4},{3},{2},{1},{4}). The tensor E_ (<ref>) is the 8-fold tensor E from Example <ref>. The sign () is given by1_1 ∧ 2_1∧ 3_1∧ 4_1 ∧ 3_2 ∧ 2_2 ∧ 1_2 ∧ 4_2 = -1. The _4-web Ŵ from Example <ref> is a tagging of W, obtained by directing the edges of G according to (<ref>) (cf. Remark <ref>). The labeling ℓ of W determines the flow of tensors along the edges of Ŵ pictured in (<ref>), specifically we replace S in ℓ by E_S for edges directed black to white and by E_[4] \ S for edges directed white to black.As Example <ref> suggests, consistent labelings of W with boundaryare closely related to evaluating Ŵ on the tensor E_. Denote by a(;W) the number of consistent labelings of W with fixed boundary label subsets .The following seemingly innocuous lemma is the key technical result underpinning our main theorems. The proof of the lemma is somewhat intricate, and has minimal bearing on the rest of the paper, so it may be skipped on a first reading. Let W be an r-weblike subgraph of a planar bipartite graph G, and Ŵ = Ŵ(W,) a choice of tagging for W.Then there is a sign (W,) ∈{± 1}, such that for any list of boundary label subsets ,we have Ŵ|_E_ =()(W,) a(; W).It is fairly easy to see (and we elaborate on this below) that consistent labelings with boundarygive rise to terms in the evaluation of Ŵ on E_. Therefore, the key assertion in this lemma is that each of these terms contributes to the evaluation with the same sign. Let us also emphasize that the sign (W,) is independent of the boundary input E_. Thus for any r-weblike subgraph W, the tensor invariant W = (W,)Ŵ(W,) ∈_(U) of (<ref>) is characterized by the equationW|_E_ =() a(; W).for every list of boundary label subsets . Let us consider a consistent labeling ℓ with boundary label subsets . The choice of tagging on Ŵ prescribes how to evaluate Ŵ on E_. Our first goal is to explain why the consistent labeling ℓ gives a term in the evaluation of Ŵ on E_. This evaluation involves many terms because of the vertices where we perform the shuffle operation. For a given labeling ℓ, we can place tensors on the edges of Ŵ as governed by ℓ. If an edge (or half-edge if the edge is tagged) is labeled by the set S in ℓ, let us place the tensor E_S along that edge if it is directed black to white and place the complementary tensor E_[r] \ S along the edge if it is directed white to black. A tensor along a given edge in a given term represents some partial stage of the evaluation Ŵ on E_. Because ℓ is consistent, the incoming flow of basis vectors equals the outgoing flow at each interior vertex, and no two basis vectors flow through the same interior vertex. Thus, once we have placed the appropriate tensor on each edge of Ŵ as indicated by ℓ, we get a sign for this term which contributes to the final evaluation. Thus, each consistent labeling corresponds to a term in Ŵ|_E_, and the nonzero terms in the evaluation of Ŵ on E_ are counted by a(; W). The nontrivial assertion is that onceis fixed, every term contributes with the same sign. The content of the proof lies in understanding these signs. The signs come from two places: the shuffles at interior vertices and the pairings at each tag.We begin our sign analysis by associating to a consistent labeling ℓ a flow of vectors along the directed edges of Ŵ. Specifically, if i ∈ S ⊂ [r] and E_S is at a given edge (or half-edge) of this term, then we say that the basis vector E_i flows along that edge. The evaluation begins with tensors E_S_1,…,E_S_n at the boundary vertices. Each of these E_S_i is itself a wedge product of basis vectors. By keeping track of how such a boundary basis vector flows along the web, we get a path from the boundary to a tag. In this way, the labeling ℓ provides us with rk paths starting at boundary sources and ending at tags. These paths are naturally indexed by the multiset {1^k,…,r^k}, and we index them by elements of the alphabetby adding subscripts in counterclockwise order along the boundary. For the sake of simplicity, we will assume that there are no interior sources in Ŵ, so that the term is completely described by the union of these rk paths (if there are interior sources, the argument below can be repeated with slight modifications). Let us take these rk paths from the boundary to the tags and isotope them slightly so that they are disjoint at the boundary, and are noncrossing except in transverse crossings in small neighborhoods of interior shuffle vertices. Thus when a of these paths flow along an edge of multiplicity a, the a paths have a natural order from left to right. For each tag there will be exactly r paths ending at the tag.A flow on a tagged web Ŵ is a collection of rk paths starting at the boundary and ending at tags, where the number of paths flowing along each edge is the multiplicity of the edge, and where the paths are labeled by the alphabet . We will usually isotope these paths so that we get a parallel paths along each edge of multiplicity a and so that the crossings of these paths are concentrated in a neighborhood of shuffle vertices.To summarize, a consistent labeling ℓ provides us with a flow, which we will also call ℓ. The consistent labeling from Example <ref>, which corresponds to the term in the evaluation computed in Example <ref>, has the following flow:[scale = .8]at (190:2.4cm) 3_1;at (145:2.4cm) 2_1;at (100:2.4cm) 1_1;at (50:2.4cm) 4_2;at (5:2.4cm) 1_2;at (-40:2.4cm) 2_2;at (-85:2.4cm) 3_2;at (-130:2.4cm) 4_1; [dashed] (0,0) circle (2cm); [rounded corners] (100:2cm)–(130:1cm)–(0,.1)–(250:1cm)–(330:1.3cm);[rounded corners,red] (145:2cm)–(135:1cm)–(0,0)–(17:.85cm)–(60:1.5cm);[rounded corners,blue] (190:2cm)–(140:1cm)–(0,-.1)–(10:1cm)–(60:1.5cm);[rounded corners,green] (-130:2cm)–(250:1.1cm)–(330:1.3cm);[rounded corners,blue] (-85:2cm)–(250:1.2cm)–(330:1.3cm);[rounded corners] (5:2cm)–(15:1.1cm)–(60:1.5cm);[red] (-40:2cm)–(330:1.3cm);[green] (50:2cm)–(60:1.5cm); We will now consider flows which do not necessarily come from consistent labelings. In a general flow, it is not required that the paths along each edge are of different colors (likewise for the paths flowing into a tag). This requirement holds precisely for flows coming from consistent labelings. As an intermediate step, we will associate a sign (ℓ) = (ℓ,) to any flow ℓ on Ŵ, not necessarily coming from a consistent labeling. The sign (ℓ) is a topological invariant of the rk paths from the boundary to the tags. It knows nothing about structure of the setother than that it consists of rk elements (and thus it does not know about the colors of these rk elements).Let t_1,…,t_k be a list of the tags in Ŵ. For each tag t_i, we let C_i ∈ r be the set of labels of the r paths ending at t_i. We refer to C_i as a tag subset. Within each tag subset, the terms are ordered (from left to right) according to how they flow into the tag, where we recall that if x and y are complementary tensors flowing into a tag and x is left of y (when the tag points north) then the tensors are ordered x ∧ y. The ordered set C_i naturally gives rise to an element of ⋀^r, which by abuse of notation we also call C_i. Then the sign (ℓ) is defined by(ℓ) := C_1 ∧ C_2 ∧⋯∧ C_k ∈⋀^rk. For any list of boundary subsets , there is a canonical way of flowing according to , namely the one in which strands cross maximally at each interior shuffle vertex.According to (<ref>), there is no sign at interior shuffle vertices associated with such a flow. We let ℓ_canon() denote the canonical flow with boundary . This canonical flow is a device that gives us a reference point for analyzing signs. Let _0 be list of boundary label subsets whose word is lexicographically smallest, i.e. w(_0) = 1_1 1_2 1_3…r_k. For any other choice of boundary label subsets , the canonical flows have signs related by (ℓ_canon(_0)) =(ℓ_canon()) ().Now suppose we have a consistent labeling ℓ with boundary subset . We can get from ℓ_canon() to ℓ by performing a sequence of swaps at shuffle vertices. Clearly, each such swap changes the signby a factor of -1. We have that (ℓ) = (ℓ_canon()) (-1)^# of shuffle swaps ℓ_canon() →ℓ = (ℓ_canon()) (sign of shuffle swaps). Now we compute (ℓ) by computing the number of transpositions required to transform (ℓ) into1_1 ∧ 1_2 ∧⋯∧ r_k. First, within each tag subset, we can rearrange the terms so that they are increasing in . We will refer to this as the inversions within tags. In addition to alphabetizing within each tag, we must perform a transposition for each inversion involving two elements of different tag subsets. We will refer to this latter number of inversions as the inversions between tags. Thus, we have that (ℓ) = (-1)^inversions within tags(-1)^inversions between tags = (sign within tags) ·(sign between tags). We need one final observation: the contribution of ℓ to Ŵ|_E_S has sign equal to(sign of shuffle swaps) times (sign within tags). Putting this together with (<ref>) through (<ref>), the sign of the contribution of ℓ is thus(sign within tags) (sign of shuffle swaps) = (ℓ_canon(_0)) (sign between tags) ().Thus the proof is finished once we show the following: the sign between tags is the same for any consistent labeling ℓ of Ŵ, regardless of the choice of boundary label subsets . Recall that for a consistent labeling ℓ the rk paths are labeled by the alphabetby adding subscripts in counterclockwise order. This last claim follows from two subclaims: first, since ℓ is a consistent labeling, all of the tag subsets are copies of [r]. Thus, the number of inversions between tags involving elements of a different color does not depend on ℓ. For example, if 4 is in the tag subset C_3, then it will be in inversion with the copies of 1,2,3 in the tag subsets C_4,C_5,…. Second, we need to check that the total number of inversions between elements of the same color (and in different tags) does not depend on the consistent labeling ℓ. Thus we have reduced the proof of the lemma to the second subclaim, which we now state formally: Fix a tagged web Ŵ. Consider any consistent labeling ℓ of Ŵ. Suppose that ℓ has boundary label subset . Adding subscripts in counterclockwise order, we get a flow on Ŵ with paths labeled by . Then the signs between tags for elements ofof the same color is independent of ℓ (and hence ). Up to this point, the sign analysis has been mostly “soft” reasoning. Proving this second subclaim requires using the planarity of Ŵ in an essential way. Let us cut the boundary of the disk between boundary vertices n and 1, flattening the web Ŵ so that the boundary vertices are drawn on the x-axis in ^2 in the order 1,…,n, and the web Ŵ is drawn in the upper half plane.The flow ℓ gives rk directed paths from the x-axis to the tags, colored by the elements of [r].Let us make the assumption that the x-coordinates of the boundary vertices, the tags t_i, and the intersection points between the rk paths are pairwise distinct.Let us assume furthermore that the tags are ordered t_1,…,t_k by the x-coordinates, from left to right. Fixing the boundary data , the flow ℓ_canon() is the one in which these paths cross each other maximally, and any consistent labeling ℓ with boundaryis obtained by resolving certain of the crossings in ℓ_canon(). Furthermore, a key observation that we use later is that if the labeling is consistent, then two paths of the same color never intersect. For each pair (p,t_j) of a path p ending at t_i, and a tag t_j (where i ≠ j), we define the nesting number n(p,t_j) of the pair (p,t_j) to be the total number of intersection points of p with the vertical ray R_j going upwards from t_j to ∞. We define n(p,t_i) = 0 if p ends at t_i.For any labeling ℓ, we can define the total nesting number of ℓ as the sum of the nesting numbers for all pairs (p,t_j).We now argue that when we resolve a crossing in a flow, the total nesting number does not change.Let p, q be two paths ending at t_i and t_j respectively.Since we have assumed that Ŵ has no oriented cycles, there are no oriented cycles in the flow.It follows thatWhen we resolve a crossing of p and q,the quantity n(p,t_k) + n(q,t_k) for k ≠ i,j clearly remains unchanged. Let us decompose the path p (resp. q) into two segments p_bp_a (resp. q_bq_a) so that the crossing happens when p_a ends and p_b begins (likewise for q_a and q_b). We will be done if we show that n(p,t_j)+n(q,t_i) ≡ n(q_bp_a,t_i)+n(p_bq_a,t_j)2. We can decompose n(p,t_j) as # p_a ∩ R_j + # p_b ∩ R_j and so on for the other paths. Certain of these contributions cancel modulo 2, and our claim reduces to checking that # (p_a∪ q_a) ∩ (R_i ∪ R_j) ≡ 02.Indeed, each of # (p_a∪ q_a) ∩ (R_i) and# (p_a∪ q_a) ∩ (R_i) is even, which follows from the fact that t_j (resp. t_i) cannot lie inside a bounded region enclosed by p_a, q_a and the x-axis by (<ref>) and (<ref>).Thus the total nesting number of any consistent labeling ℓ with boundaryis the same as the total nesting number of ℓ_canon(). The total nesting number of ℓ_canon() is independent ofbecause the flow of the paths for ℓ_canon is defined topologically (without regard to color).Moreover, because ℓ is a consistent labeling, and therefore two paths of the same color never cross, the total nesting number coincides with the number of inversions between elements of the same color modulo 2. This completes the proof of the claim and the lemma. The following result relates tensor evaluation of webs to boundary measurements of networks; it serves as the fundamental link between webs and dimers.Let N be a network with excedance k. Let = (I_1,…,I_r) be a list of subsets satisfying (<ref>), dual to = (S_1,…,S_n) satisfying (<ref>). Then Δ_I_1(N) ⋯Δ_I_r(N) = () _r(N;) |_E_.We still find the statement of Proposition <ref> surprising, yet the importance of this statement for us is belied by the simplicity of the proof.We have thatΔ_I_1(N) ⋯Δ_I_r(N)= ∑_r-weblike subgraphs W, degree= a(; W)(W) = () ∑_WW(W) = () _r(N;) |_E_.The first equality follows by considering superpositions of dimer configurations (π_1,…,π_r) with boundary subsets (I_1,…,I_r) and grouping them according to the r-weblike subgraph they determine. The second equality follows from Lemma <ref>, and the third equality from the definition (<ref>). From (<ref>), the evaluation of _r(N;) on a basis tensor E_ can be expressed in terms of boundary measurements, which only depend on (N). Evaluation at E_ for varyingdetermines an element of _(U). This proves Theorem <ref>. First note that for each list of boundary label subsets , there is a functional eval(E_)∈_(U)^* given by evaluation at E_. Furthermore, by (<ref>), the function (N) →eval(E_) (_r(N; )) agrees with the function (N) ↦() Δ_I_1(N) ⋯Δ_I_r(N). This latter function is a regular function on (k,n). Since the set of points of the form (N) is Zariski dense in (k,n), the function (N) →eval(E_) (_r(N; )) extends uniquely to a regular function on all of (k,n). The resulting extension is the image of the immanant map. Now we note the functionals eval(E_) span _(U)^*, so we conclude that every element of φ∈_(U)^* gets sent to a a regular function on (k,n) by the immanant map.This proves Proposition <ref>.Since r-fold products Δ_I_1⋯Δ_I_r of Plücker coordinates span the graded piece of [(k,n)]_, the immanant map is surjective. A dimension count verifies that the map is an isomorphism. We will see another argument for the equality of dimensions in Section <ref>.This proves Theorem <ref>.Finally, let us extract some consequences.First, the pairing _(U) ⊗[(k,n)]_→ can be given in simple terms as follows:W, Δ_I_1⋯Δ_I_r = () W|_E_ = a(;W)_r(N;),f= f((N)) .In (<ref>) we let (I_1,…,I_r) be a list of subsets, dual to , and let W be an r-weblike subgraph with tensor invariant W∈_(U).In (<ref>) we let f ∈[(k,n)]_ and N be any network of excedance k.Note in particular that (<ref>) is always nonnegative. The equation (<ref>) follows from checking the special case where f=Δ_I_1⋯Δ_I_r, which was the content of Proposition <ref>.Second, we can give another interpretation of the duality between [(k,n)]_ and _(U). Let G be a bipartite graph representing the top cell. Let us pick a collection of edges of G so that varying the weights w_e of those edges (keeping the rest of the weights as 1) gives a a rational parametrization (^×)^k(n-k)→(k,n).We now think of the weights w_e as variables, and form the sum_r(G;) := ∑_W ⊂ G(W) = (W)W.Here, (W) are monomials in the edge weight variables w_e. Specializing the edge-weights to complex numbers gives an element of _(U). Thus we get elements of _(U) depending algebraically on points of (k,n). Our above analysis shows that we can then view _r(G;) as giving rise to an element_r(G;) ∈[(k,n)]_⊗_(U)which is the pairing between these two spaces. We have that_r(G;) |_E_ = () Δ_I_1⋯Δ_I_rfor any boundary label subsets , and in fact, this characterizes _r(G;) ∈[(k,n)]_⊗_(U). In particular, we emphasize that _r(G;) is independent of G (as long as G represents the top cell).Since _r(G;) realizes the duality between [(k,n)]_ and _(U), it must be a full-rank tensor in [(k,n)]_⊗_(U). From this, it can be seen that as N varies, the various _r(N;) span _(U). This will also follow from Theorem <ref>.Let us also formulate the following positivity conjecture. Under the isomorphism _(U) ≅[(r,n)]_, the functions W take nonnegative values on the cone (r,n)_≥ 0 over the totally nonnegative Grassmannian. In Section <ref>, we show that webs in (r,n) are naturally dual to certain elements of [(k,n)], and that this duality behaves well under restriction to positroid subvarieties Π⊂(k,n). Our positivity conjecture <ref> says that _r-webs are positive when thought of as functions on the Grassmannian (r,n). To briefly explain the reasoning, we first note that the positivity property of Conjecture <ref> is known to hold for the canonical basis by the work of Lusztig <cit.>.In the r = 2 or r = 3 cases, W expands positively as a sum of non-elliptic web basis elements.These web basis elements are known to coincide with the canonical basis for r = 2, and share many similar properties with the canonical basis in the case r = 3. Finally, we view the sign coherence in Lemma <ref> as a manifestation of positivity. As a simple example of Conjecture <ref>, consider the single cycle web W ∈(3,9) (the second web in the first row of Figure <ref>), with its boundary vertices numbered so that vertices 1,4,7 connect to the interior hexagon of W. Multiplying by the Plücker coordinate Δ_479 and using the skein relations, one sees that Δ_479· W is a sum of two cluster monomials. It follows that W can be expressed as a Laurent monomial with positive coefficients in Plücker coordinates for [(3,9)], and therefore takes positive values on (3,9)_≥ 0. See <cit.>),for a more general version of this argument (that does not explicitly mention positivity) in the case r=3.§ DEDUCINGSKEIN RELATIONS FROM MOVES ON NETWORKSAccording to Theorem <ref>, the element _r(N) only depends on (N). Thus it is unchanged when we perform the local moves from Theorem <ref> to N. However, the expression for _r(N) as a weighted sum of r-weblike subgraphs will change after each local move, since the weblike subgraphs change. In this way, we obtain relations between certain linear combinations of _r-webs. As it turns out, the diagrammatic relations on webs we obtain in this way are exactly the Cautis-Kamnitzer-Morrison relations <cit.>. This will be the content of Theorem <ref>] and Corollary <ref>. We can interpret this fact in two ways: On the one hand, it is possible (though perhaps tedious) to use the Cautis-Kamnitzer-Morrison relations to give an alternative proof Theorem <ref> – one checks that if N and N' are related by a local move on networks, then _r(N) and _r(N') differ by a skein relation. We believe that our approach to Theorem <ref>, based on Proposition <ref> and Lemma <ref>, is somewhat more illuminating. On the other hand, in this paper we will use Theorem <ref> to derive diagrammatic relations amongst webs, and use our duality setup to prove that the relations we obtain in this way are a complete set of diagrammatic relations.Our r-weblike graphs are closely related to, but slightly different from the webs considered in <cit.>. These differences require some care when comparing moves on bipartite graphs and diagrammatic relations on webs. Thus we will begin with diagrammatic relations on r-weblike subgraphs before proceeding to the case of webs. §.§ r-weblike graph relationsWe begin by illustrating how the various skein relations on r-weblike graphs can be derived from relations on networks. Consider the space _(r) of finite formal -linear combinations of r-weblike graphs (i.e, labeled bipartite graphs such that the sum of labels around every interior vertex equals r). We will later compare this with the space _(r) of formal -linear combinations of _r-web diagrams.There is a surjection _(r)_(U) from the evaluation equation (<ref>), that is, by sending a r-weblike graph W to the tensor invariant given by its boldface version W.We begin by listing all the diagrammatic moves that can be performed to an r-weblike graph W without changing the value of the tensor invariant W. Of these moves, the last move is the only truly interesting one. * Two-valent vertex removal (<ref>). There is also an analogous move to (<ref>) which has the colors reversed. For simplicity, we drew all neighboring edges in (<ref>) with multiplicity 1, though in general they may have any multiplicity. * Bigon removal (<ref>). Parallel edges, with multiplicities a and b, can be replaced by one edge with multiplicity a+b, at the cost of a multplicative factor a+b b. As before, colors can be reversed. * Leaf and dipole removal. Edges with multiplicity r or 0 can be removed from r-weblike graphs. * The square move for r-weblike graphs (<ref>). Notice that the diagrams in (<ref>) connect to the “outside” with multiplicity j from the southwest, ℓ from the southeast, r+v-s-ℓ from the northeast,and r+s-v-j from the northwest. If j,ℓ,v are fixed, there is a square move for each s satisfying max(0,v-ℓ,v+j-r) ≤ s ≤min(j,r-ℓ).[scale =1](-1,0)–(1,0);(-1.5,1)–(-1,0)–(-1.7,0);(-1.5,-1)–(-1,0);(1.5,1)–(1,0)–(1.5,.25);(1.5,-.25)–(1,0)–(1.5,-1);at (-.5,.3) a;at (.5,.3) r-a; [black] (-1,0) circle (0.1cm);(-1,0) circle (0.1cm); [black] (1,0) circle (0.1cm);(1,0) circle (0.1cm); [white] (0,0) circle (0.1cm);(0,0) circle (0.1cm); at (2.25,0) =;(3,1)–(3.5,0)–(2.8,0);(3,-1)–(3.5,0);(4,1)–(3.5,0)–(4,.25);(4,-.25)–(3.5,0)–(4,-1); [black] (3.5,0) circle (0.1cm);(3.5,0) circle (0.1cm); [xshift = 7.5cm](-1.7,0)–(-1,0);(-1.7,.5)–(-1,0);(-1.7,-.5)–(-1,0);(0,0)–(.7,0);(.7,.5)–(0,0);(.7,-.5)–(0,0);at (-.5,.7) a;at (-.5,-.7) b; [black] (-1,0) circle (0.1cm);(-1,0) circle (0.1cm);[rounded corners] (-1,0)–(-.75,.2)–(-.5,.3)–(-.25,.2)–(0,0); [rounded corners] (-1,0)–(-.75,-.2)–(-.5,-.3)–(-.25,-.2)–(0,0);[white] (0,0) circle (0.1cm);(0,0) circle (0.1cm);at (2.24,0) = a+b b;(3.55,0)–(4.25,0);(5.25,0)–(5.95,0);(5.25,0)–(4.25,0);(3.55,.5)–(4.25,0);(3.55,-.5)–(4.25,0);(5.95,.5)–(5.25,0);(5.95,-.5)–(5.25,0); at (4.75,.5) a+b; [black] (4.25,0) circle (0.1cm);(4.25,0) circle (0.1cm); [white] (5.25,0) circle (0.1cm);(5.25,0) circle (0.1cm);[scale=0.7] (0,0) – (2,0)–(2,2)–(0,2)–(0,0);(3,3) – (2,2);(-1,-1) – (0,0);at (-1,1) j-s;at (3.1,1) r-ℓ-s;at (1,2.4) v;at (1,-0.4) s;at (-1.2,-.3) r-j;at (2.5,3.45) ℓ-v+s;[black] (0,0) circle (0.1cm); [black] (2,2) circle (0.1cm); [white] (2,0) circle (0.1cm);(2,0) circle (0.1cm); [white] (0,2) circle (0.1cm);(0,2) circle (0.1cm); [white] (3,3) circle (0.1cm);(3,3) circle (0.1cm); [white] (-1,-1) circle (0.1cm);(-1,-1) circle (0.1cm); at (7.3,1) =∑_t j-ℓ+v-st;[xshift =13.5cm](0,0) – (0,2)– (2,2)– (2,0)– (0,0);(3,-1) – (2,0);(-1,3) – (0,2);at (-1.6,1) r-j-v+t;at (-2.2,2.4) v+j-s;at (3.1,1) ℓ-v+t;at (3.3,0) r-ℓ;at (1,2.4) s-t;at (1,-0.4) v-t; [black] (2,0) circle (0.1cm); [black] (0,2) circle (0.1cm); [white] (0,0) circle (0.1cm);(0,0) circle (0.1cm); [white] (2,2) circle (0.1cm);(2,2) circle (0.1cm); [white] (-1,3) circle (0.1cm);(-1,3) circle (0.1cm); [white] (3,-1) circle (0.1cm);(3,-1) circle (0.1cm);The relations amongst networks imposed by Theorem <ref> generate the kernel _(r) _(U).The four diagrammatic relations listed above generate the kernel _(r) _(U).Before proving Theorem <ref>, we will show how Corollary <ref> follows from it.Let us briefly explain how the relations (<ref>) and (<ref>) can be deduced from network moves. The two-valent vertex removal (M2) for networks implies two-valent vertex removal (<ref>) for r-weblike graphs. The parallel edge removal (R1) for networks implies the bigon removal move (<ref>) for r-weblike graphs. Leaf and dipole removal (R2), (R3) for networks give leaf and dipole removal for r-weblike graphs.Thus we now focus on the most interesting move on networks (M1), and show that it implies the square move for r-weblike graphs (<ref>). Suppose N and N' are are related by a square move involving edge weights a,b,c,d ∈^* [scale=0.7] (0,0) – (2,0)–(2,2)–(0,2)–(0,0);(3,3) – (2,2);(-1,-1) – (0,0);at (-.4,1) a;at (2.4,1) c;at (1,2.4) b;at (1,-0.4) d;[black] (0,0) circle (0.1cm); [black] (2,2) circle (0.1cm); [white] (2,0) circle (0.1cm);(2,0) circle (0.1cm); [white] (0,2) circle (0.1cm);(0,2) circle (0.1cm); [white] (3,3) circle (0.1cm);(3,3) circle (0.1cm); [white] (-1,-1) circle (0.1cm);(-1,-1) circle (0.1cm);[scale=0.7](0,0) – (0,2)– (2,2)– (2,0)– (0,0);(3,-1) – (2,0);(-1,3) – (0,2);at (-.4,1) c';at (2.4,1) a';at (1,2.4) d';at (1,-0.4) b';[black] (2,0) circle (0.1cm); [black] (0,2) circle (0.1cm); [white] (0,0) circle (0.1cm);(0,0) circle (0.1cm); [white] (2,2) circle (0.1cm);(2,2) circle (0.1cm); [white] (-1,3) circle (0.1cm);(-1,3) circle (0.1cm); [white] (3,-1) circle (0.1cm);(3,-1) circle (0.1cm); . We need to compare the r-weblike subgraphs coming from the networks N and N'. Each such weblike subgraph can be built by fixing the edge multiplicites m(e) for edges e outside the local fragment (j from the southwest, ℓ from the southeast, and so on, as described above), then choosing the edge multiplicitiesinside the local fragment in a way that is compatible with the outside. Thus the numbers j,ℓ and s-v are fixed throughout this computation. We will let s be variable (so that s determines v). [scale=0.7] (0,0) – (2,0)–(2,2)–(0,2)–(0,0);(3,3) – (2,2);(-1,-1) – (0,0);at (-1,1) j-s;at (3.1,1) r-ℓ-s;at (1,2.4) v;at (1,-0.4) s;at (-1.2,-.3) r-j;at (2.5,3.4) ℓ-v+s;[black] (0,0) circle (0.1cm); [black] (2,2) circle (0.1cm); [white] (2,0) circle (0.1cm);(2,0) circle (0.1cm); [white] (0,2) circle (0.1cm);(0,2) circle (0.1cm); [white] (3,3) circle (0.1cm);(3,3) circle (0.1cm); [white] (-1,-1) circle (0.1cm);(-1,-1) circle (0.1cm); .The contribution to (W) from the edges in the local fragment is a^j-sb^v c^r-ℓ-sd^s.On the other hand, in N', for the same value of outside multiplicities, the filling inside will look like [scale=0.7](0,0) – (0,2)– (2,2)– (2,0)– (0,0);(3,-1) – (2,0);(-1,3) – (0,2);at (-1.2,1) r-j-u;at (-2.4,2.4) v+j-s;at (2.8,1) ℓ-u;at (3.3,0) r-ℓ;at (1,2.4) u-v+s;at (1,-0.4) u; at (-.5,-.3) ;[black] (2,0) circle (0.1cm); [black] (0,2) circle (0.1cm); [white] (0,0) circle (0.1cm);(0,0) circle (0.1cm); [white] (2,2) circle (0.1cm);(2,2) circle (0.1cm); [white] (-1,3) circle (0.1cm);(-1,3) circle (0.1cm); [white] (3,-1) circle (0.1cm);(3,-1) circle (0.1cm);where the values of j,ℓ,v-s agree with the fixed values, but u can vary. Notice that s and v only occur in (<ref>) together via their difference s-v.Now we denote by W_s the weblike subgraph indexed by s in (<ref>) and denote by W'_u the one indexed by u in (<ref>).We sum up the boldface W_s with their weights and the W_u with their weights. Since the points in the affine cone are related by(N) = (ac+bd)(N'), it follows that _r(N) = (ac+bd)^r_r(N'). Writing out what this means: ∑_s a^j-sb^v c^r-ℓ-sd^sW_s= (a c +b d)^r ∑_u (a')^ℓ-u(b')^u(c')^r-j-u(d')^u-v+s W'_u= (a c +b d)^j-ℓ+v-s∑_u a^ℓ-ub^u c^r-j-ud^u-v+s W'_u. Now we fix a value of s, and we also fix the exponent t of b d in the binomial formula expansion of (a c +b d)^v+j-ℓ-s. In order to match the weights on the left and right hand side, the value of u must be v-t. The corresponding equality is W_s = ∑_t j-ℓ+v-stW'_v-t∈_(U),which is the square move (<ref>). Thus we see that the local moves on networks imply the four diagrammatic moves for r-weblike graphs that preserve the corresponding tensor invariants. Now, we will give an “abstract proof” that diagrammatic moves amongst r-weblike subgraphs we obtain in this way are a complete set of relations. Let us denote by K the kernel _(r) →_(U) and abbreviate = _(r). Let (k) denote the set of networks of excedance k whose underlying graph G represents the top cell. We will denote by [(k)] the linear space of functions f (k) → with the property that for any fixed graph G, the map f is a polynomial in the edge weights on G. The space [(k)] is a very large space of functions.We say that f ∈[(k)] is consistent of order r if f(N) = å^r f(N') whenever (N) = å(N').The space of consistent functions of order r is canonically identified with the space[(k,n)]_(r) := ⊕_[(k,n)]_()where the sum is oversuch that _1+⋯+_n=kr.There is a map ^† : ^* →[] sending a functional φ∈^* to the function on networks N ↦φ(_r(N;)), where we now think of _r(N;) as an element of . It is defined similarly to the immanant mapbut on the space of formal sums of weblike graphs. This map obviously fits into a commutative diagram ( / K)^*@^(->[d] [drr]^≅ ^*[r]_^† [(k)]@_(->[l][(k,n)]_(). The diagonal arrow in this diagram is the immanant map : _(U)^* →[(k,n)]_. Going diagonally and then left, we arrive at the space of consistent functions of order r on networks inside [(k)] which have degree . The image of the downward arrow is K^⊥.The main claim which needs to be checked is that ^† is an injection. Let us first see that the fact that ^† is an injection allows us to finish the proof.Suppose that ^† is an injection.Then it follows that φ∈^* defines a consistent function on networks if and only ifφ∈ K^⊥. To put it another way, whether or not φ lies in K^⊥ is detected by whether ^†(φ) is a consistent function on networks. A function in [(k)] is consistent if an only if it transforms appropriately under local moves. Therefore, the relations defining K^⊥ are exactly those forced on φ by the local moves on networks (cf. Proposition <ref>). In other words, we need that^†(φ)(N)=α^r^†(φ)(N') whenever(N)=α(N').We now prove the injectivity of ^†.Namely, if a functional φ∈^* satisfies φ(_r(N;)) = 0 for all networks N of excedance k, then φ must be 0.Equivalently, we must prove that the various {_r(N;)} spanas we vary N over all networks representing the top cell.Now let G be a fixed bipartite graph that represents the top cell.The graph G has a finite number of r-weblike subgraphs. LetW_1,…,W_ℓ be the list of all of the r-weblike subgraphs of G having degree . We claim that there exists a list N_1,…,N_ℓ of networks with underlying graph G such that the matrix(_N_i(W_j))_i,j=1,…,ℓ is invertible. First suppose that such N_1,…,N_ℓ exist. The rows of the matrix give the terms of _r(N_i;), so for each j there exists a linear combination of the {_r(N_i;)} giving the r-weblike subgraph W_j. By varying G, we then conclude that any r-weblike subgraph W is in the span of {_r(N;)}, and {_r(N;)} spans .The existence of N_1,…,N_ℓ follows from a standard argument. The expression for _r(N; ) consists of a monomial in the edge weights of G multiplied by each weblike subgraph W_i. Because the W_i all differ, the monomial attached to each web has a different multidegree. Then a standard Vandermonde-type argument shows that one can specialize the edge-weights appropriately so that the matrix (_N_i(W_j)) is invertible. §.§ Cautis-Kamnitzer-Morrison relations Recall we denote by _(r) the space of formal sums of tagged _r-web diagrams of degree . Any tagged web gives a tensor invariant, so that we get a surjection (r) _(r). Cautis, Kamnitzer, and Morrison <cit.> gave a set of diagrammatic relations describing the kernel of this surjection. Let us briefly write down these relations (see <cit.> for pictures and discussion): * Switching a tag <cit.>. If e is a tagged edge with multiplicities a and r-a, then changing which side the tag is on contributes a factor (-1)^a(r-a). * Tag migration <cit.>. Suppose an interior white vertex v has an incident edge e with a tag pointing in the clockwise direction around v (the tag can be either a pair or a source tag, and the vertex v can be either a shuffle or wedge vertex). Let e' be the next edge incident to v in the clockwise direction from v. Then the tag on e can be migrated to a tag on e' pointing in the counterclockwise direction, without any change of sign. * Wedge product is associative <cit.> and shuffle is coassociative.* Bigon removal <cit.>. A pair of directed edges v → u, of multiplicities a and b, can be replaced by a single edge v → u with multiplicity a+b, at the cost of a factor (-1)^aba+bb. * The square move for tagged webs: [scale=0.5] at (8.5,1) =∑_t j-ℓ+v-st;[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,0) – (2,0);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (2,0)–(2,2);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,0) – (0,2);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (2,2) – (3,3);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,2) – (-1,3);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (-1,-1) – (0,0);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (3,-1)–(2,0);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (2,2)–(0,2);at (-1,1) j-s;at (3.3,1) ℓ+s;at (1,2.4) v;at (1,-0.4) s;at (-1.6,-1) j;at (3,3.6) ℓ-v+s;at (-1,3.6) j-s+v;at (3.5,-1) ℓ; [black] (0,0) circle (0.1cm); [black] (2,2) circle (0.1cm); [white] (2,0) circle (0.1cm);(2,0) circle (0.1cm); [white] (0,2) circle (0.1cm);(0,2) circle (0.1cm);[xshift = 15cm][decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (2,0)–(.2,0);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,0)–(0,2);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,2)–(2,2);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (2,0)–(2,2);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt](3,-1) – (2,0);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt](-1,-1) – (0,0);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (2,2) – (3,3);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,2) – (-1,3);at (-1.5,1) j+v-t;at (3,3.6) ℓ-v+s;at (-1,3.6) j-s+v;at (3.5,1) ℓ-v+t;at (3.5,-1) ℓ;at (1,2.4) s-t;at (1,-0.4) v-t;at (-.5,-.3) ;at (-1.6,-1) j; [black] (2,0) circle (0.1cm); [black] (0,2) circle (0.1cm); [white] (0,0) circle (0.1cm);(0,0) circle (0.1cm); [white] (2,2) circle (0.1cm);(2,2) circle (0.1cm); , [Cautis-Kamnitzer-Morrison] The five relations listed above generate the kernel (_r) _(U). We note that the results in <cit.> are more general than ours – they describe all diagrammatic relations amongst fundamental representations of the quantum group U_q(𝔰𝔩_n), whereas our current proof only makes sense in the “classical” (q=1) setting. Additionally, they provide certain redundant relations that are needed to describe the kernel if one wishes to work over [q,q^-1] instead of (q). In our version, we work over .Now we explain how Theorem <ref> can be deduced from our Corollary <ref>.There are three differences between r-weblike graphs and webs. First, and most important, r-weblike graphs do not come with a tagging. Recall that tagging an r-weblike graph W gives a web Ŵ = Ŵ(W,), whererefers to the data of the tagging. From any r-weblike graph W, we associate a canonical invariant W := (W,)Ŵ. One of the advantages of working with our tensor invariants W is that the signs come naturally built-in.The second difference between r-weblike graphs and webs is purely a matter of convention and is less serious: Cautis-Kamnitzer-Morrison require that vertices in their webs are of degree three, whereas ours can be arbitrary. For this reason, they need additional associativity relations that we do not. These relations follow from two-valent vertex removal on networks. The third difference is also less serious. We require our graphs to be bipartite, so that in order to go from a web to an r-weblike graph, we will sometimes need to contract edges. The fact that edge contractions give r-weblike graphs that are related by allowable moves again follows from two-valent vertex removal.We now proceed with the proof. The first two relations in Theorem <ref> have to do with tagging. These tagging relations generate a subspace of relations ⊂_(r). There is a linear map _(r) _(r) defined by replacing a tagged web Ŵ = Ŵ(W,) by its “underlying r-weblike subgraph” with a sign: Ŵ↦(W,)W ∈_(r). Let us be more specific about how to obtain an r-weblike graph from Ŵ. Any edges in Ŵ joining vertices of the same color should be contracted, so that we end up with a bipartite graph. If necessary, we may need to use tag switches and migrations before contracting. Once this is done, if e is an edge (or half-edge) with multiplicity a, and if e points from a white vertex to a black vertex, then e has multiplicity r-a in W, while if it points away from a black vertex towards a white vertex, it has multiplicity a. Finally, forget about all the directions on the edges. The resulting labeled diagram is an r-weblike graph.The quotient _(r) / is exactly the image of _(r) in _(r). We have that the map _(r) _(r) factors as _(r) _(r) _(r) There are relations on _(r) that come from the kernel of the first map. These are precisely the relations ⊂_(r) coming from tagging. These relations follow from our analysis of tags in Lemma <ref>.The remaining relations on _(r) come from “tagging” (i.e., adding tags to) relations in _(r). Thus, to find a complete set of diagrammatic relations amongst tagged webs, we need to lift the relations defining (_(r) _(r)) to _(r). Let us do this in the most interesting case of the square move. The square move for r-weblike graphs (<ref>) can be lifted to a relation in _(r) by tagging as follows [scale=0.5] at (-4.3,1) (W,);at (10.5,1) =∑_t (W'_t,'_t)j-ℓ+v-st;[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,0) – (2,0);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (2,0)–(2,2);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,0) – (0,2);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (2,2) – (3,3);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,2) – (-1,3);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (-1,-1) – (0,0);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (3,-1)–(2,0);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (2,2)–(0,2);at (-1,1) j-s;at (3.3,1) ℓ+s;at (1,2.4) v;at (1,-0.4) s;at (-1.6,-1) j;at (3,3.6) ℓ-v+s;at (-1,3.6) j-s+v;at (3.5,-1) ℓ; [black] (0,0) circle (0.1cm); [black] (2,2) circle (0.1cm); [white] (2,0) circle (0.1cm);(2,0) circle (0.1cm); [white] (0,2) circle (0.1cm);(0,2) circle (0.1cm);[xshift = 19cm][decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (2,0)–(.2,0);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,0)–(0,2);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,2)–(2,2);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (2,0)–(2,2);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (3,-1) – (2,0);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (-1,-1) – (0,0);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (2,2) – (3,3);[decoration=markings,mark=at position .6 with [scale=1.7]>, postaction=decorate, shorten >=0.4pt] (0,2) – (-1,3);at (-1.5,1) j+v-t;at (3,3.6) ℓ-v+s;at (-1,3.6) j-s+v;at (3.5,1) ℓ-v+t;at (3.5,-1) ℓ;at (1,2.4) s-t;at (1,-0.4) v-t;at (-.5,-.3) ;at (-1.6,-1) j; [black] (2,0) circle (0.1cm); [black] (0,2) circle (0.1cm); [white] (0,0) circle (0.1cm);(0,0) circle (0.1cm); [white] (2,2) circle (0.1cm);(2,2) circle (0.1cm); ,where we have chosen a tagging outside the local fragment that is compatible with the tagging inside the fragment shown above. Then (<ref>) follows once one checks that all of the signs, (W,) and (W'_t,'_t), are equal and do not depend on t. This can be proved by analyzing the canonical flows through the local fragments, as described in the proof of Lemma <ref>. We briefly mention how to lift the remaining relations. Two-valent vertex removal for r-weblike graphs allows us to contract and expand edges to derive the associativity of wedge for tagged webs. In a similar fashion, bigon removal for r-weblike graphs lifts to the bigon removal for tagged webs. § POSITROIDS AND WEBSIn this section we fix a positroidand consider a graph G such that (G) = (see Section <ref>).Let Π_ = Π_G be the corresponding positroid variety. §.§ The immanant map for a positroidLet () ⊂[(k,n)] be the (homogeneous) ideal generated by{Δ_I: I ∉ M}.By <cit.>, () is a prime ideal, and its vanishing set in (k,n) is the positroid variety Π_.The homogeneous coordinate ring Π_ is thus given as the quotient[Π_] =[(k,n)]/(),noting that by <cit.>, Π_⊆(k,n) is projectively normal. The grading (<ref>) descends to a grading on [Π_]. One of our goals will be to give a description of the graded pieces [Π_]_ via tensor invariants.Recall that the immanant map (for the top cell) is an isomorphism: _(U)^* →[(k,n)]_. We now define an immanant map _: _(U)^* →[Π_]_ for the positroidby the equation_(φ)((N))=φ(_r(N;))for N any assignment of edge weights to G. The function (N) ↦φ(_r(N;)) extends to a (uniquely defined) element of [Π_]_.The proof is the same as the proof of Proposition <ref> after noting that the set of points of the form (N) is Zariski dense in _. We get that _(φ) is a regular function on _, and that thus the immanant map _ is well-defined. Let us note that the immanant map is characterized as the map that (up to sign) sends (E_) to the function Δ_I_1⋯Δ_I_r whereand =(I_1,… I_r) are dual. Therefore _ =π_∘, where π_: [(k,n)]_[Π_]_ denotes the quotient map. §.§ Partial evaluation of websLet ()_ denote the degreepart of the positroid ideal ().We now give a description of the subspace (()_)^⊥:= { x ∈ W_(U) : x, f= 0for allf ∈()_}⊆ W_(U)where ·, · denotes the pairing between W_(U) and [(k,n)]_.Thus the inclusion (()_)^⊥↪ W_(U) is dual to the surjection π_: [(k,n)]_[Π_]_.We now define the notion of the partial evaluation of a web. Let Ŵ∈ W_(U) be an _r-web, where _1+ … +_n = kr. The copies of ⋀^_1(U),…,⋀^_n(U) gives us kr locations to plug in vectors v_1,…,v_kr. For example, the first _1 vectors are plugged into boundary vertex 1 as the wedge v_1 ∧⋯∧ v__1, and so on.Let I = {i_1,…,i_k}∈[n] k. Let U' ⊂ U be the subspace spanned by E_1,…,E_r-1. Then the partial evaluation map along I _(U)→_μ(U') x↦ x |_I ↦ E_ris the linear map obtained by specializing the input vectors v_i_1,v_i_2,…,v_i_k to the last basis vector E_r, and then restricting the remaining vectors to lie in the subspace U'. The resulting function is _r-1-invariant. We have μ_i = λ_i-1 or μ_i = λ_i depending on whether i ∈ I or i ∉ I.It satisfies μ_1+…+μ_n = (r-1)k.In the multilinear case = (1,…,1), partial evaluation along I is a linear map (r,n) ↦(r-1,n-k).We can now make the key definition of this section. For a positroid , we denote by _(U)() = {x ∈_(U) x |_I ↦ E_r = 0for allI ∉}.We get a collection of subspaces _(U)() of the _r-tensor invariant space indexed by positroidsfor the Grassmannian (k,n). If W is an r-weblike subgraph, its partial evaluation along I is zero if and only if W has no consistent labelings in which there are r's at the boundary edges indicated by I, cf. Example <ref>. We note that the partial evaluation of a web W will often decompose as a sum of several _r-1-web invariants.Letbe a positroid with positroid variety Π_. Then we have _(U)() = (()_)^⊥.In particular, these two spaces have the same dimension.()_ is spanned by the r-fold products of Plücker coordinates Δ_I_1⋯Δ_I_r satisfying (<ref>) such that at least one of the I_i satisfies I_i ∉. By reordering indices, let us assume that I_r ∉.Thus, the dual space (()_)^⊥ is the subspace of tensor invariants x ∈_(U) that pair to zero with each of these r-fold products. From (<ref>), pairing to 0 with (I_1,…,I_r) means that x |_E_ = 0 whereis dual to (I_1,…,I_r). If we fix I_r, this holds for all choices of I_1,…,I_r-1 if and only if the partial evaluation x |_I_r ↦ E_r = 0 ∈_μ(U'). By varying I_r over all subsets not in , we see that the dual space is exactly (<ref>) as claimed.Let G be a planar bipartite graph with positroid .Then the subspace _(U)() is spanned by either of the following sets:* the elements _r(N;), as N varies over the (infinitely many) networks whose underlying graph is G;* the elements W, as W varies over the (finitely many) r-weblike subgraphs of G with degree .This characterization of [Π_]_ may be easier to work with than the characterization using promotion and cyclic Demazure crystals given in <cit.>. First we prove that the various _r(N;) span the subspace _(U)(). From (<ref>), we see immediately that _r(N;) pairs to zero with any r-fold product as in the proof of Theorem <ref> (since Δ_I_r(N) = 0 on Π_).Thus _r(N;) ∈_(U)(). On the other hand, if f ∈[(k,n)]_ pairs to 0 with every _r(N;), then by (<ref>) we have f((N)) = 0 for every network N representing G. Since the set of points of the form (N) is Zariski dense in the affine cone over Π_, we deduce that f is the zero polynomial, i.e. f ∈()_. This shows the various _r(N;) span the dual space (()_)^⊥. Finally, the span of the various _r(N;) coincides with the span of the weblike subgraphs of G, as follows from the from the argument inverting the transition matrix (<ref>) in the proof of Theorem <ref>.Thus for a fixed G associated to a positroid , the r-weblike subgraphs contained in G are forced to evaluate to 0 on certain basis tensors. This seems to be a completely new phenomenon. Also surprising is the fact that the elements W, for r-weblike subgraphs of a bipartite graph G representing the top cell, must span the invariant space (k,n).The statement that the weblike subgraphs of G span the subspace (<ref>) also implies some non-obvious compatibilities between webs, the partial evaluation map (<ref>) and positroids. We indicate one such compatibility in the following examples.Let I_last = {n-k+1,…,n} so that Δ_I_last is the Plücker coordinate occupying the last k columns. We consider the positroid = {[n]k}\ I_last. It is a positroid, because it can be written as an appropriate intersection of cyclically rotated Schubert matroids <cit.>. The ideal (Π)_ is the set of multiples of Δ_I_last inside (Π)_, i.e. (Π)_ = Δ_I_last[(k,n)]_μ, where μ is obtained fromby decrementing along I.If G is a bipartite graph with positroid , let W_1,…,W_s be a set of r-weblike subgraphs of G of degreesuch that W_1,…,W_s is a basis for _(U)(). We can extend W_1,W_2,…,W_s to a basis W_1,…,W_s,W_s+1,…,W_s+t of _(U) consisting entirely of weblike graphs. Then the number t is given by t = ((Π)_) = ([(k,n)]_μ) = (W_μ(U')),where the last equality follows from duality. On the other hand, since (W_s+1,…,W_s+t) ∩_(U)() = {0}, we deduce that the partial evaluations of W_s+1,…,W_s+t along I_last are linearly independent in W_μ(U'). Thus, these partial evaluations are a basis for W_μ(U').As we have already said, these partial evaluations are not guaranteed to be _r-1-webs.Let us the illustrate the previous example in a particular case. We let k=r=3 and = (1,…,1), so we are considering the space (3,9) of _3-invariant multilinear functions of 9 vectors v_1,…,v_9. Its dimension is the number of 3 × 3 standard Young tableaux, which is 42.The non-elliptic basis webs (up to rotation) are listed in the top row of Figure <ref>. We will focus on the positroid = [9]3\{Δ_789} as in the preceding example. The subspace (3,9)() is the kernel of the linear map (3,9) →(2,6) given by specializing v_7=v_8=v_9=E_3.Concretely, this means placing 3's at boundary vertices 7,8, and 9, which will force certain interior edges to also be labeled by 3's, and then studying consistent labelings of the leftover edges once these 3's have been placed.A web will be in the subspace (3,9)() if and only if it either has a fork between vertices 7 and 8 or has a fork between vertices 8 and 9. By examining the forks in Figure <ref>, we see that there are exactly 37 webs that are in (3,9)(), and five webs that are not. We have placed 3's at the appropriate locations in four of the nonvanishing webs. The fifth nonvanishing web (not pictured) is obtained by reflecting the third web in Figure <ref> along the vertical axis (with 3's at the same boundary vertices). From Theorem <ref>, it follows that ([Π()]_(1,…,1)) = 37. To see that that ([Π()]_(1,…,1)) = 37 directly, let W ∉(3,9)() be one of the five non-vanishing webs. As we said above, the placement of 3's at the boundary forces certain interior edges to be 3's. The remaining edges in W must be labeled by 1's and 2's. These remaining edges naturally decompose into a union of three noncrossing paths joining boundary vertices in pairs, as we demonstrate schematically via the downward arrows in Figure <ref>. The five _2-webs we get in this way are exactly the 5 crossingless matchings on 6 vertices, i.e. a set of basis webs for (2,6). In confirmation of Theorem <ref>, we can see concretely that no nontrivial linear combination of the five _3-webs combines to give an element of (3,9)() (and thus, (3,9)() is spanned by the 37 webs that vanish under partial evaluation). § DUALITY OF SYMMETRIC GROUP REPRESENTATIONSIn this section, we deal only with the multilinear web spaces (r,n) and (k,n) where n=kr. These spaces are [(r,n)]_ and for [(k,n)]_ for = (1,…,1), so we no longer use the symbolto denote the degree of a web.Insteadwill be used to denote a partition of n. Both tensor invariant spaces (k,n) and (r,n) carry an action of the symmetric group by permutations of the vectors. Furthermore, these S_n-modules are irreducible, and are related to each other by tensoring with the sign representation ϵ. The immanant map (r,n)^* →(k,n) ⊗ϵ is an isomorphism of S_n-modules.Since the S_n-modules in Theorem <ref> are irreducible, the map in Theorem <ref>is unique up to a scalar factor. The S_n-equivariance of the immanant map does not seem obvious, since there is not a natural action of S_n on the space of networks. Before proving Theorem <ref> we recall the relevant notions from S_n representation theory.Let = (_1,…,_r) ⊢ n be a Young diagram, with _1 the number of boxes in the first row, _2 the number of boxes in the second row, and so on. Let S_ = S__1×⋯× S__r⊂ S_n be the corresponding Young subgroup. A tableau for a shape ⊢ n is a filling ofby the numbers 1,…,n. A tabloid is an equivalence class of tableaux, where we identify two tableaux if their entries differ by a permutation in each row. The free vector space M_ on the set of tabloids of shapeis a right S_n-module. It is the induced representation1_S_^S_n where 1 is the trivial representation. We use the notation [T] for the tabloid determined bya tableau T. We denote by N_ = M_⊗ = _S_^S_n. It has a basis consisting of anti-tabloids, which are defined similarly to tabloids but with the added requirement that swapping two entries in a given row contributes a multiplicative factor of -1.We use the notation {T} for the antitabloid determined by T. Both of the S_n-modules M_ and N_ are reducible. To obtain S_n irreducibles, we recall that the polytabloid determined by T is the signed sum of tabloidspoly(T) = ∑_σ∈col(T)(σ)[T ·σ] ∈ M_,where col(T) is the subgroup of S_n consisting of permutations that fix the entries in each column of T.Then the Specht Module S_ is the subspace of M_ spannedby the polytabloids T, as T varies over all fillings of . The various Specht modules S_, for ⊢ n, are exactly the irreducibles for S_n. For a Young diagram , we let ^t denote the conjugate or transpose partition (likewise T^t denotes the conjugate tableau to T). Then the Specht modules S_ and S_^t are related by S_ = S_^t⊗.For the space (k,n), a list of boundary location subsets (I_1,…,I_r) is the same as an r× k tableau T(I_1,…,I_r) whose rows are I_1,…,I_r. The boundary label subsets = (S_1,…,S_n) are singletons such that S_j records the row of T(I_1,…,I_r) in which j appears. The sign of , which we can denote by (T), is determined by the well-known descent number of T modulo 2, where we recall that the number of descents of a tableaux is the number of pairs i>j but i is in a lower row of T than j.Lettingdenote the r × k rectangle, there is an isomorphism S_n-modules(k,n)↔ S_^t Δ_I_1Δ_I_2⋯Δ_I_r ↦poly(T^t)where T = T(I_1,…,I_r). Indeed, consider a k × n matrix of indeterminates x_ij.The homogeneous coordinate ring [(k,n)] can be identified with the invariant subring [x_ij]^_k⊂[x_ij].For a set J={j_1, …, j_r}∈[n]r, denote by x_iJ the monomial x_ij_1x_ij_2… x_ij_r. For a tableau T = T(I_1,…,I_r) whose columns are J_1,…,J_k, we can consider the monomial∏_i=1^k x_iJ_i.Clearly this monomial only depends on [T^t]. The expansion of Δ_I_1⋯Δ_I_r as a signed sum of such monomials agrees with the expression of (<ref>) of poly(T^t) as a signed sum of tabloids. Thus the subspace spanned by poly(T^t), which is the Specht module S_^t, is isomorphic to (k,n) as a representation of S_n.The idea is to fit the immanant map into a commuting diagram of maps of S_n-modules, from which we deduce that the immanant map is S_n-equivariant. We remark that, all the maps in this diagram do not requireto be of rectangular shape, and are unique up to scalars. The only assertion that requiresto be of rectangular shape is the interpretation of one of these maps as the immanant map.First, it easy to check that the pairing M_⊗ N_ →[S] ⊗{T} ↦δ_S,T (S).is S_n-equivariant, where δ_S,T is the Kronecker delta. In this way, we get an S_n-equivariant map N_→ M_^* ⊗. Since S_⊂ M_, we can further composeN_→ M_^* ⊗→ S_^* ⊗.On the other hand, there is always an S_n-equivariant map N_→ S_^t induced by {T}↦poly(T^t). The map (<ref>) can be thought of as a special case of this. (The space of formal sums of Δ_I_1⋯Δ_I_r – before imposing Plücker relations – is naturally identified with N_).Then the immanant map (S_)^* ⊗ϵ→ S_^t is the unique (up to scalars) map making the diagram N_[d] [r] (M_)^* ⊗[r] (S_)^* ⊗[dll]^S_^tLet us explain why this diagonal map is the immanant map. Going down in (<ref>) is the map that replaces a formal sum of products Δ_I_1⋯Δ_I_r with their images in [(k,n)] (i.e., going down is imposing the Plücker relations).Let us now examine the top row of the commutative diagram. The space M_ can be identified with monomials in y_ij which are entries in an r × n matrix, and which are multilinear in each column, by the analysis above. The map N_→ M_^* ⊗ takes an antitabloid {T} to (T) times the functional that picks out the coefficient of the tabloid [T]. Putting this together, the antitabloid {T} gets sent to (T) times the coefficient of ∏_i y_iI_i. When we restrict this to the Specht module, we get exactly the functional ()eval(E_), whereeval(E_) is as in the proof of Proposition <ref> . The commutativity now follows since the immanant map is characterized by (<ref>). § APPENDIX: WEB DUALITY PICTURESThis appendix is jointly written with Darlayne Addabbo, Eric Bucher, Sam Clearman, Laura Escobar, Ningning Ma, Suho Oh, and Hannah Vogel. For simplicity, we restrict attention to the multilinear case = (1,…,1) and n = kr. The immanant map provides us with an isomorphism (r,n)^* →(k,n). When r is equal to 2 or 3, the space (r,n) has a distinguished choice of web basis(given by crossingless matchings, and non-elliptic _3 webs, respetively). For a given basis web W, we denote by φ_W ∈(r,n)^* the dual basis element with respect to . Its image (φ_W) ∈(k,n) is called a web immanant in <cit.>. Our main observation in this appendix is the following: Let r=2 or 3, and W ∈(r,n) be a basis web. Then the web immanants (φ_W) ∈(k,n) are also web invariants (up to a sign).That is, the unique S_n-equivariant pairing between the tensor invariant spaces (k,n) and (r,n) induces a duality between basis webs in these spaces, in small cases. We verified this observation in small cases by straightforward calculations which we omit here (see <cit.> for some examples). The resulting pairing is rotation-equivariant, and we find the resulting pictures appealing, cf. Figure <ref>. Our second observation is as follows. When r=2 or 3, there is a well-known bijection between standard young tableaux and non-elliptic webs, due to Khovanov and Kuperberg <cit.> (see also <cit.>). It has the property that rotation of webs is given by promotion of standard young tableaux, and this can be used to give an elementary proof <cit.> of the cyclic sieving phenomenon for rectangular tableaux with 2 or 3 or rows. Our next observation is that under this bijection, dual basis webs correspond to transposed tableaux:In small cases, duality between basis webs is given by transposing standard young tableaux.Specifically, we checked that this is true for the dualities (3,6) ↔(2,6) and (3,9) ↔(3,9). For r ≥ 4, Westbury has given a basis of _r-webs indexed by standard young tableaux <cit.>. In the special case of (4,8), the basis of web immanants listed in the second column ofFigure <ref> is different than Westbury's basis for (4,8). The web immanant basis consists of the 4 rotation classes of the first type of web, the 8 rotation classes of the second type of web, and the 2 rotation classes of the third type of web. We remark that this third type of _4-web is fixed by rotating two units, which is not obvious, but is an instance of the _4-square move. The web immanant basis disagrees with Westbury's basis in these last two elements – the element Westbury assigns to these two tableaux has order 4 under rotation. Once the elements of Westbury's basis are replaced by the corresponding web immanants, we again have that rotation of webs is given by promotion. This suggests that there might be a slightly different choice of web basis, indexed by tableaux, that is better behaved with respect to promotion. Let us also note that the last _5-web in Figure <ref> must be fixed by rotating two units, because this is true of its dual _2-web. And indeed, this can be checked using the _5 diagrammatic relations – applying a square move to the top square of this web produces the same web, but rotated 4 units clockwise (so applying the square move three times, we get rotation by 2 units).We mention that Observation 8.3 is probably implied by similar statements guaranteeing duality between the canonical bases for (r,n) and (k,n) <cit.>, and from an agreement between the web basis and canonical basis in these instances. Nonetheless, the pictures in Figure <ref> are new. When r and k are both >3, the immanant map gives us a bilinear pairing of (k,n) with (r,n). It would be interesting to understand the resulting pairing between _k-webs and _r-webs combinatorially.abbrvnat | http://arxiv.org/abs/1705.09424v2 | {
"authors": [
"Chris Fraser",
"Thomas Lam",
"Ian Le"
],
"categories": [
"math.CO",
"math.RT"
],
"primary_category": "math.CO",
"published": "20170526033632",
"title": "From dimers to webs"
} |
A Sampling Theory Perspective of Graph-based Semi-supervised Learning Aamir Anis, Student Member, IEEE, Aly El Gamal, Member, IEEE, Salman Avestimehr, Senior Member, IEEE, and Antonio Ortega, Fellow, IEEEThis work is supported in part by NSF under grants CCF-1410009, CCF-1527874, CCF-1408639, NETS-1419632 and by AFRL and DARPA under grant 108818. S. Avestimehr and A. Ortega are with the Ming Hsieh Department of Electrical Engineering, University of Southern California. A. Anis is currently with Google Inc., he was affiliated with the University of Southern California at the time this work was completed. A. El Gamal is with the Department of Electrical and Computer Engineering, Purdue University. E-mail: [email protected], [email protected], [email protected], [email protected]. Copyright (c) 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]. ============================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================Graph-based methods have been quite successful in solving unsupervised and semi-supervised learning problems, as they provide a means to capture the underlying geometry of the dataset. It is often desirable for the constructed graph to satisfy two properties: first, data points that are similar in the feature space should be strongly connected on the graph, and second, the class label information should vary smoothly with respect to the graph, where smoothness is measured using the spectral properties of the graph Laplacian matrix. Recent works have justified some of these smoothness conditions by showing that they are strongly linked to the semi-supervised smoothness assumption and its variants. In this work, we reinforce this connection by viewing the problem from a graph sampling theoretic perspective, where class indicator functions are treated as bandlimited graph signals (in the eigenvector basis of the graph Laplacian) and label prediction as a bandlimited reconstruction problem. Our approach involves analyzing the bandwidth of class indicator signals generated from statistical data models with separable and nonseparable classes. These models are quite general and mimic the nature of most real-world datasets. Our results show that in the asymptotic limit, the bandwidth of any class indicator is also closely related to the geometry of the dataset. This allows one to theoretically justify the assumption of bandlimitedness of class indicator signals, thereby providing a sampling theoretic interpretation of graph-based semi-supervised classification.§ INTRODUCTION The abundance of unlabeled data in various machine learning applications, along with the prohibitive cost of labeling, has led to growing interest in semi-supervised learning. This paradigm deals with the task of classifying data points in the presence of very little labeling information by relying on the geometry of the dataset. Assuming that the features are well-chosen, a natural assumption in this setting is to consider the marginal density p() of the feature vectors to be informative about the labeling function f() defined on the points. This assumption is fundamental to the semi-supervised learning problem both in the classification and the regression settings, and is also known as the semi-supervised smoothness assumption <cit.>, which states that the label function is smoother in regions of high data density. There also exist other similar variants of this assumption specialized for the classification setting, namely, the cluster assumption <cit.> (points in a cluster are likely to have the same class label) or the low density separation assumption <cit.> (decision boundaries pass through regions of low data density). Most present day algorithms for semi-supervised learning rely on one or more of these assumptions to predict the unknown labels.In practice, graph-based methods have been found to be quite suitable for geometry-based learning tasks, primarily because they provide an easy way of exploiting information from the geometry of the dataset. These methods involve constructing a distance-based similarity graph whose vertices (nodes) represent the data points and whose edge weights are in general a decreasing function of the distances between them. The learning task then involves predicting the labels of the unknown nodes, given the known labels, often called the transductive learning paradigm. The key assumption here is that the label function is “smooth” over the graph, in the sense that labels of vertices do not vary much over edges with high weights (i.e., edges that connect close or similar points). There are numerous ways of quantitatively imposing smoothness constraints over label functions defined on the vertices of a similarity graph. Most graph-based semi-supervised classification algorithms incorporate one of these criteria as a penalty against the fitting error in a regularization problem, or as a constraint term while minimizing the fitting error in an optimization problem. For example, a commonly used measure of smoothness for a label functionis the graph Laplacian regularizer ^T ( being the graph Laplacian), and many algorithms involve minimizing this quadratic energy function while ensuring thatsatisfies the known set of labels <cit.>. Another example is the graph total variation <cit.>. There also exist higher-order variants of the smoothness measure such as iterated graph Laplacian regularizers ^T ^m <cit.> and the p-Laplacian regularizer <cit.>, that have been shown to make the problem more well-behaved. On the other hand, a spectral theory based classification algorithm restrictsto be spanned by the first few eigenvectors of the graph Laplacian <cit.>, that are known to form a representation basis for smooth functions on the graph. In each of the examples, the criterion enforces smoothness of the labels over the graph – a lower value of the regularizer ^T, and a smaller number of leading eigenvectors to modelimply that vertices that are close neighbors on the graph are more likely to have the same label. A more recent approach, derived from Graph Signal Processing (GSP) <cit.>, considers the semi-supervised learning problem from the perspective of sampling theory for graph signals <cit.>. It involves treating the class label functionas a bandlimited graph signal, and label prediction as a bandlimited reconstruction problem.The advantage of this approach is that one can also analyze, using sampling theory, the label complexity of graph-based semi-supervised classification, that is, the fraction of labeled vertices on the graph required for predicting the labels of the unlabeled vertices. A key ingredient in this formulation is the bandwidth ω() of signals on the graph, which is defined as the largest Laplacian eigenvalue for which the projection of the signal over the corresponding eigenvector is non-zero. Signals with lower bandwidth tend to be smoother on the graph and have a lower label complexity. Label prediction using bandlimited reconstruction then involves estimating a graph signal that minimizes prediction error on the labeled set under a bandwidth constraint. This can also be carried out without explicitly computing the eigenvectors of the Laplacian, and has been shown to be quite competitive in comparison to state-of-the-art graph-based semi-supervised learning methods <cit.>.Although graph-based semi-supervised learning methods are well-motivated, their connection to the underlying geometry of the dataset had not been clearly understood so far in a theoretical sense. Recent works focused on justifying these approaches by exploring their geometrical interpretation in the limit of infinitely available unlabeled data. This is typically done by assuming a probabilistic generative model for the dataset and analyzing the graph smoothness criteria in the asymptotic setting for certain commonly-used graph construction schemes. For example, it has been shown that for data points drawn from a smooth distribution with an associated smooth label function (i.e., the regression setting), the graph Laplacian-based regularizers converge in the limit of infinite data points to some density-weighted variational energy functional that penalizes large variations of the labels in high density regions <cit.>. A similar connection ensues for semi-supervised learning problems in the classification setting (i.e., when labels are discrete in the feature space). If points drawn from a smooth distribution are separated by a smooth boundary into two classes, then the graph cut for the partition converges to a weighted volume of the boundary <cit.>. This is consistent with the low density separation assumption – a low value of the graph cut implies that the boundary passes through regions of low data density. To our knowledge, no such connections have been drawn for the sampling theoretic approach to learning. A geometrical interpretation of this approach would help complete our theoretical understanding of graph-based semi-supervised learning approaches and strengthen their link with the semi-supervised smoothness assumption and its variants. Therefore, in this work, we seek answers for the following questions: * What is the connection between the bandwidth of class indicator signals over the similarity graph and the underlying geometry of the data set? * What is the interpretation of the bandlimited reconstruction approach for label prediction?* How many labeled examples does one require for predicting the unknown labels?To answer these questions, our work analyzes the asymptotic behavior of an iterated Laplacian-based bandwidth estimator for class indicator signals on similarity graphs constructed from a statistical model for the feature vectors. To make our analysis as general as possible, we consider two data models: separable and nonseparable. These generative models are quite practical and can be used to mimic most datasets in the real world. The separable model assumes that data points are independently drawn from an underlying probability distribution in the feature space and each class is separated from the others by a smooth boundary. On the other hand, the nonseparable model assumes a mixture distribution for the data where the data points are drawn independently with certain probability from separate class conditional distributions. We also introduce a notion of “boundaries” for classes in the nonseparable model in the form of overlap regions (i.e., the region of ambiguity), defined as the set of points where the probability of belonging and not belonging to a class are both non-zero. This definition is quite practical and useful for characterizing the geometry of such datasets.Using the data points, we consider a specific graph construction scheme that applies the Gaussian kernel over Euclidean distances between feature vectors for computing their similarities (our analysis can be generalized easily to arbitrary kernels under simple assumptions). In order to compute the bandwidth of any signal on the graph, we define an estimator based on the iterated Laplacian regularizer. A significant portion of this paper focuses on analyzing the stochastic convergence of this bandwidth estimate (using variance-bias decomposition) in the limit of infinite data points for any class indicator signal on the graph. The analysis in our work suggests a novel sampling theoretic interpretation of graph-based semi-supervised learning and the main contributions can be summarized as follows:* Relationship between bandwidth and data geometry.For the separable model, we show that under certain rate conditions, the bandwidth estimate for any class indicator signal over the graph converges to the supremum of the data density over the class boundary. Similarly, for the nonseparable model, we show that the bandwidth estimate converges to the supremum of the density over the overlap region. Based on these results, we conjecture, with supporting experiments, that the bandwidths also converge to the same values.* Interpretation of bandlimited reconstruction.Using the geometrical interpretation of the bandwidth, we conclude that bandlimited reconstruction allows one to choose the complexity of the hypothesis space while predicting unknown labels (i.e., a larger bandwidth allows more complex class boundaries).* Quantification of label complexity for sampling theory-based learning. For both the separable and nonseparable models, we conjecture, with supporting arguments and experiments, that the fraction of labeled nodes on the graph for reconstructing class indicator signals converges, in the asymptotic limit, to the probability mass of the sublevel set that entirely encompasses the boundary. Our analysis has significant implications: Firstly, class indicator signals have a low bandwidth if class boundaries lie in regions of low data densities, that is, the semi-supervised assumption holds for graph-based methods. And secondly, our analysis also helps quantify the impact of bandwidth and data geometry in semi-supervised learning problems. Specifically, it enables us to theoretically assert that for the sampling theoretic approach to graph-based semi-supervised learning, the label complexity of class indicator signals over the graph is indeed lower if the boundary lies in regions of low data density, as demonstrated empirically in earlier works <cit.>.The rest of this paper is organized as follows: In Section <ref>, we formally introduce the statistical data models and the graph construction scheme for analysis, along with a precursor of concepts from graph sampling theory. In Section <ref>, we review prior work and underline their connections with our work. In Section <ref>, we state our main results and outline their implications. In Section <ref>, we prove the major building blocks for our results. We finally conclude with numerical validation in Section <ref>, followed by discussion and an outline of future work in Section <ref>. It is worth noting that the bandwidth convergence result for the separable model and an interpretation of bandlimited reconstruction were given in our preliminary work <cit.>. This paper presents complete formal proofs for those results, extends them to the nonseparable model, and also analyzes label complexity.§ PRELIMINARIES §.§ Data models §.§.§ The separable modelIn this model, we assume that the dataset consists of a pool of n random, d-dimensional feature vectors = {_1, _2, …, _n} drawn independently from some probability density function p() supported on ℝ^d (this is assumed for simplicity, the analysis can be extended to subsets D ⊂ℝ^d and low-dimensional manifolds ℳ in ℝ^d, but would more technically involved). To simplify our analysis, we also assume that p() is bounded from above, Lipschitz continuous and twice differentiable. We assume that a smooth hypersurface ∂ S, with radius of curvature lower bounded by a constant τ, splits ℝ^d into two disjoint classes S and S^c, with indicator functions 1_S():ℝ^d →{0,1} and 1_S^c():ℝ^d →{0,1}. This is illustrated in Figure <ref>.Thus, the n-dimensional class indicator signal for class S is denoted by the bold-faced vector notation _S ∈{0,1}^n, and defined as (_S)_i := 1_S(_i), i.e., the i^th entry of _S is 1 if _i ∈ S and 0 otherwise.§.§.§ The nonseparable modelIn this model, we assume that each class has its own conditional distribution supported on ℝ^d (that may or may not overlap with other distributions of other classes). The data set consists of a pool of n random and independent d-dimensional feature vectors = {_1, _2, …, _n} drawn independently from any of the distributions p_i() with probabilities α_i, such that ∑_i α_i = 1.For our analysis, we consider a class denoted by an index A with selection probability α_A, class conditional distribution p_A() and an n-dimensional indicator vector _A whose i^th component takes value 1 if _i is drawn from class A. Note that _A does not have a continuous domain counterpart, unlike _S which is sampled from the indicator function 1_S() on points in . We illustrate the nonseparable model in Figure <ref>. Further, we denote by α_A^c = 1- α_A the probability that a point does not belong to A and by p_A^c() = ∑_i ≠ Aα_i p_i() / α_A^c the density of all such points. The marginal distribution of data points is then given by the mixture densityp() = α_A p_A() + α_A^c p_A^c(). Once again, to simplify our analysis, we assume that all distributions are Lipschitz continuous, bounded from above and twice differentiable in ℝ^d. Next, we introduce the notion of a “boundary” for classes in the nonseparable model as follows: for class A, we define its overlap region ∂ A as∂ A := {∈ℝ^d|p_A() p_A^c() > 0 }.Intuitively, ∂ A can be considered as the region of ambiguity, where both points belonging and not belonging to A co-exist. In other words, ∂ A can be thought of as a “boundary” that separates the region where points can only belong to A from the region where points can never belong to A. Since class indicator signals on graphs will change values only within the overlap region, one would expect that the indicators will be smoother if there are fewer data points within this region. We shall show later that this is indeed the case, both theoretically and experimentally. Note that the definition of the boundary is not very meaningful for class conditional distributions with decaying tails, such as the Gaussian, since the boundary in this case technically encompasses the entire feature space. However, in such cases, one can approximate the boundary with appropriate thresholds in the definition and this approximation can also be formalized for distributions with exponentially decaying tails. §.§ Graph construction Using the n feature vectors, we construct an undirected distance-based similarity graph where nodes represent the data points and edge weights are proportional to their similarity, given by the Gaussian kernel:w_ij := K_σ^2(_i,_j) = 1/(2πσ^2)^d/2 e^-_i - _j^2/2σ^2 ,where σ is the variance (bandwidth) of the Gaussian kernel. Further, we assume w_ii = 0, i.e., the graph does not have self-loops.The adjacency matrix of the graphis an n× n symmetric matrix with elements w_ij, while the degree matrix is a diagonal matrix with elements _ii = ∑_j w_ij.We define the graph Laplacian as = 1/n(-). Normalization by n ensures that the norm ofis stochastically bounded as n grows. Since the graph is undirected,is a symmetric matrix with non-negative eigenvalues 0 ≤λ_1 ≤…≤λ_n and an orthogonal set of corresponding eigenvectors {_1, …, _n}. It is known that for a larger eigenvalue λ, the corresponding eigenvectorexhibits greater variation when plotted over the nodes of the graph <cit.>. Thus, one of the fundamental postulates of Graph Signal Processing consists of using the eigen-decomposition ofto provide a notion of frequency for graph signals, with the eigenvalues acting as graph frequencies and the eigenvectors forming the graph Fourier basis <cit.>. §.§ Graph sampling theory: bandwidth, bandlimited reconstruction and label complexityIn traditional sampling theory, bandwidth plays an important role in specifying the inherent dimensionality of a signal and therefore determines the sampling rate required for perfect reconstruction.A similar notion exists for signals defined over graphs – the bandwidth ω() of any signalon the graph is defined as the largest eigenvalue for which the projection of the signal on the corresponding eigenvector is non-zero <cit.>, i.e.,ω() := max_i {λ_i||_i^T | > 0 }.Signals with lower bandwidth have low frequency content, and tend to be smoother on the graph. Bandwidth plays a central role in the sampling theoretic approach to semi-supervised learning, where the class indicator signals are assumed to be bandlimited over the similarity graph and interpolated through bandlimited reconstruction. For a ground-truth signalthat we are trying to reconstruct, and whose values are known only on a subset L ⊂{1,2,…,n}, this approach involves solving the following least-squares problem <cit.>: min_ _L - _L ^2 subject toω() ≤θ, where _L and _L denote the values ofand , respectively, on the set L. The constraint restricts the hypothesis space to a set of bandlimited signals with bandwidth less than θ, which is equivalent to enforcing smoothness of the labels over the graph. This method essentially improves upon the Fourier eigenvector approach suggested in <cit.> in two ways: first, label prediction can be carried out without explicitly computing the eigenvectors ofusing efficient iterative approaches implemented via graph filtering operations <cit.>. And second, one can also use the sampling theorem for graph signals to set θ as the cutoff frequency ω_c(L) associated with the labeled set <cit.>, which, for a given L, is defined as the bandwidth below which any bandlimited signal is uniquely represented by its values on L. This approach is taken in <cit.>, and is particularly useful when ω() < ω_c(L), in which case the minimizer ^* of (<ref>) exactly equals , i.e., ^* -= 0. Alternatively, one can also reconstructusing the variational problem: min_ ω() subject to_L = _L; the minimizer in this case is also exactly equal toif ω() < ω_c(L) <cit.>. Further, it also possible to provide error bounds for both methods when ω() < ω_c(L) is not satisfied <cit.>. The bandwidth of any indicator signal is also useful in specifying the amount of labeling required for its recovery in the context of sampling theory, as demonstrated by the following key result <cit.>: Let 𝒩_(t) denote the number of eigenvalues ofless than or equal to t. Then, for any signalwith bandwidth ω(), there exists a subset of nodes T ⊆ V of size |T| = 𝒩_(ω()) such thatcan be perfectly recovered from its values _T on T.Sincehas bandwidth ω(), it is spanned by the first 𝒩_(ω()) eigenvectors of , i.e., let R := {1,…,r}, then we have= ∑_i=1^𝒩_(ω()) c_i _i = _:,R,where c_i ≠ 0 for i = 𝒩_(ω()) and _:,R denotes the rectangular matrix formed using the first r eigenvectors of . Since the eigenvectors {_i} are orthogonal, _:,R has rank r = 𝒩_(ω()). Therefore, there exists a subset of rows, indexed by a set T, with cardinality |T| = r = 𝒩_(ω()), such that the r × r matrix _T,R is full-rank, and thus invertible. Using this in (<ref>), we get = _T,R^-1_T and thuscan be perfectly recovered from _T as = _V,R_T,R^-1_T, thereby proving our claim. Note that this is exactly the closed-form solution of (<ref>), for L = T and θ = ω(), when the eigenvectors ofare known. We shall use this result later, to compute the label complexity of any signalon the graph as 1/n_(ω()). Note, however, that this quantity only specifies the fraction of nodes to label on the graph – selecting which nodes to label is another question altogether. This problem has been well-studied as part of graph sampling theory <cit.>, with consideration of other important issues such as stability of reconstruction and computational complexity. §.§ Estimating bandwidth for graph signals Ideally, computing the bandwidth ω() of a graph signalrequires obtaining the eigenvectors {_i} ofand the corresponding projections _i = _i^T. However, analyzing the convergence of these coefficients is technically challenging. Therefore, we resort to the following estimate of the bandwidth <cit.>:ω_m() := ( ^T ^m /^T )^1/m, where we call ω_m() the m^th-order bandwidth estimate.It can be shown that the bandwidth estimates satisfy the property: for all 0 < m_1 < m_2, ω_m_1() ≤ω_m_2() ≤ω(). In other words, {ω_m()} forms a monotonically improving sequence of estimates of the true bandwidth ω(). Further, we can also show <cit.>:∀,ω() = lim_m →∞ω_m().§.§ Focus of this paper The discussion in Section <ref> indicates that in the discrete setting, with finite number of data points, the notions of bandwidth, bandlimited reconstruction and label complexity are well-motivated and quite useful in highlighting a sampling theory perspective of graph-based semi-supervised learning. However, there is a lack of understanding of these concepts in terms of their geometrical interpretation, i.e., their connection with the underlying geometry of the dataset. Thus, inspired by existing analysis in the literature for popular graph-based smoothness measures, we seek to bridge this gap by analyzing these concepts in the asymptotic regime of infinite data points for the data models and graph construction scheme described earlier.Analyzing the convergence of the bandwidth estimates of class indicator signals for the separable and the nonseparable models constitutes the main subject for the rest of this paper. Our approach, similar to existing results in the literature, starts in the discrete domain by drawing n samples from the data models, constructs a sequence of graphs G_n,σ from the data points, and considers the behavior ofω_m(_S) = ( _S^T ^m _S/_S^T _S)^1/mandω_m(_A) = ( _A^T ^m _A/_A^T _A)^1/mover the graphs as n →∞, σ→ 0 and m →∞. Intuitively, the condition n →∞ implies an abundance of unlabeled data, σ→ 0 dictates that the connectivity around each node is meaningful and does not blow up, and m →∞ translates to improving estimates of the bandwidth. Our analysis relates ω_m(_S) and ω_m(_A) to the underlying data distribution p() and class boundaries – the hypersurface ∂ S in the separable case and the overlap region ∂ A in the nonseparable case. Using these results, we also comment on the label complexities of reconstructing _S and _A over the graph in the asymptotic limit.§ RELATED WORK AND CONNECTIONSExisting convergence analyses of the graph-based smoothness measures for various graph construction schemes appear in two different settings – classification and regression. The classification setting assumes that labels indicate class memberships and are discrete, typically with 1/0 values. Note that both the separable and nonseparable data models considered in our paper are in the classification setting. On the other hand, in the regression setting, one allows the class label signalto be sampled from a smooth function on ℝ^d with soft values, such that ∈ℝ^n, and later applies some thresholding mechanism to infer class memberships. For example, in the two class problem, one can assign +1 and -1 to the two classes and thresholdat 0. Convergence analysis of smoothness measures in this setting requires different scaling conditions than the classification setting, and leads to fundamentally different limit values that require differentiability of the label functions in the continuum. Applying these to class indicator functions may lead to ill-defined results. A summary of convergence results in the literature for both settings is presented in Table <ref>. Although these results do not focus on analyzing the bandwidth of class indicator signals, the proof techniques used in this paper are inspired by some of these works. We review them in this section and discuss their connections to our work. §.§ Classification setting Prior work under this setting assumes the separable data model where the feature space is partitioned by smooth decision boundaries into different classes. When m=1, the bandwidth estimate ω_m(_S) for the separable model in our work reduces (within a scaling factor) to the empirical graph cut for the partitions S and S^c of the feature space, i.e.,Cut(S,S^c) := ∑__i ∈ S, _j ∈ S^c w_ij = n _S^T _S.Convergence of this quantity has been studied before in the context of spectral clustering, where one tries to minimize it across the two partitions of the nodes. It has been shown in <cit.> that the cut formed by a hyperplane ∂ S in ℝ^d converges with some scaling under the rate conditions σ→ 0 and nσ^d+1→∞ as1/nσ_S^T _S 1/√(2π)∫_∂ S p^2()d,where d ranges over all (d-1)-dimensional volume elements tangent to the hyperplane ∂ S, and p. denotes convergence in probability.The analysis has also been extended to other graph construction schemes such as the k-nearest neighbor graph and the r-neighborhood graph, both weighted and unweighted. The condition σ→ 0 in (<ref>) is required to have a clear and well-defined limit on the right hand side. We borrow this convergence regime in our work, since it allows a succinct interpretation of the bandwidth of class indicator signals. Intuitively, it enforces sparsity in the similarity matrixby shrinking the neighborhood volume as the number of data points increases. As a result, one can ensure that the graph remains sparse even as the number of points goes to infinity. A similar result for a similarity graph constructed with normalized weights w'_ij = w_ij/√(d_i d_j) was shown earlier for an arbitrary hypersurface ∂ S in <cit.>, where d_i denotes the degree of node i. In this case, normalization of the graph weights results in convergence to 1/√(2π)∫_∂ S p()d. Similarly, in <cit.>, the convergence of normalized cuts is analyzed for points drawn from a uniform density.All of these results aim to provide an interpretation for spectral clustering – up to some scaling, the empirical cut value converges to a weighted volume of the boundary. Thus, spectral clustering is a means of performing low density separation on a finite sample drawn from a distribution in feature space. Note that these works provide little insight for the convergence analysis of higher-order regularizers, i.e., ω_m(_S) for m > 1 in our case, since these require different scaling factors and rate conditions. Further, we get no clue about the continuum limit values of ω_m(_S) and ω_m(_A) from any of these results. However, the definition and some of the proof techniques we use for the separable models in this paper have been inspired by <cit.>.§.§ Regression setting To predict the labels of unknown samples in the regression setting, one generally minimizes the graph Laplacian regularizer ^T subject to the known label constraints <cit.>:min_ ^Tsuch that (L) = (L),One particular convergence result in this setting assumes that n data points are drawn i.i.d. from p() and are labeled by sampling a smooth function f() on ℝ^d. Here, the graph Laplacian regularizer ^T can be shown to converge in the asymptotic limit under the conditions σ→ 0 and nσ^d→∞ as in <cit.>: 1/n σ^2^TC ∫_ℝ^d∇ f() ^2 p^2() d, where for each n,is the n-dimensional label vector representing the values of f() at the n sample points, ∇ is the gradient operator and C is a constant factor independent of n and σ. The right hand side of the result above is a weighted Dirichlet energy functional that penalizes variation in the label function weighted by the data distribution. Similar to the justification of spectral clustering, this result justifies using the formulation in (<ref>) for semi-supervised classification: given label constraints, the predicted label function must vary little in regions of high density. The work of <cit.> generalizes this result by using arbitrary kernel functions for defining graph weights, and defining data distributions over manifolds in ℝ^d. Convergence results for another regularizer called Graph Total Variation, defined as GTV() = ∑_i,jw_ij|f_i - f_j|, are presented in <cit.>. For data points drawn from p() defined over a domain D ⊂ℝ^d, graph weights given by w_ij = 1/ε^dη( _i-_j/ε), one has as n→∞ and ε→ 0:1/n^2ε GTV()C∫_D ∇ f()p^2()d, where the limit is analyzed in the setting of Γ-convergence <cit.>. These results extend to the classification setting when f() is an indicator function, for example, the limit for f() = 1_S() reduces to that of (<ref>). This approach is used in <cit.> to analyze convergence of Cheeger and ratio cuts. Similar convergence results have also been derived for the higher-order Laplacian regularizer ^T ^m obtained from uniformly distributed data <cit.>. In this case, it was shown that for data points obtained from a uniform distribution on a d-dimensional submanifold ⊂ℝ^N such that Vol () = 1 and 2m-differentiable functions f(), one has as n →∞: 1/nσ_n^m^T ^mC ∫_ f() Δ^m f() d, where Δ is the Laplace operator and σ_n = n^-1/(2d + 4 + α) is a vanishing sequence with α > 0. Extensions for non-uniform probability distributions p() over the manifold can be obtained using the weighted Laplace-Beltrami operator <cit.>. More recently, an ℓ_p-based Laplacian regularization has been proposed for imposing smoothness constraints in semi-supervised learning problems <cit.>. This is similar to a higher-order regularizer but is defined as J_p() := ∑_i,j ∈ E w_ij^p |f_i - f_j|^p, where w_ij = ϕ(_i - _j / h) and ϕ(.) is a smoothly decaying Kernel function. It has been shown for a bounded density p() defined on [0,1]^d that for every p ≥ 2, as n →∞, followed by h → 0, 1/n^2 h^p + d J_p()C ∫_[0,1]^d∇ f()^p p^2() d. The work of <cit.> generalizes this result over an open, bounded and connected set Ω⊂ℝ^d and analyzes rate conditions such that the scalings n→∞, h→ 0 occur jointly.Note that although our work also uses higher powers ofin the expressions for ω_m(_S) and ω_m(_A), we cannot use theconvergence results in (<ref>) and the proof techniques of (<ref>), since they are only applicable for smooth functions (i.e., differentiable up to a certain order) on ℝ^d. Specifically, in our case, _S in the separable model is sampled from a discontinuous indicator function 1_S(), hence plugging it into existing results does not give a meaningful result for higher values of m. Further, the nonseparable model can only be defined in the classification setting, i.e., _A in the nonseparable model does not have a continuum counterpart. Therefore, our analysis has to take a different route that has more similarities with the proof techniques used for the classification setting. We shall later see that a bulk of the effort in proving our results goes into expanding ω_m(_S) and ω_m(_A) for any m by keeping track of every term in the expansion. This is followed by a careful evaluation of the integrals in their expected values by reducing them term-by-term.§ MAIN RESULTS AND DISCUSSION§.§ Interpretation of bandwidth and bandlimited reconstruction We first show that under certain conditions, the bandwidth estimates of class indicator signals for both the data models, i.e., ω_m(_S) and ω_m(_A), over Gaussian kernel-based similarity graphs G_n,σ constructed from data points in , converge to quantities that are functions of the underlying distribution and the class boundary for both data models. This convergence is achieved under the following asymptotic regime:* Increasing size of dataset: n →∞.* Shrinking neighborhood volume: σ→ 0.* Improving bandwidth estimates: m →∞. Note that an increasing size of the dataset n →∞ is required for the stochastic convergence of the bandwidth estimate. σ→ 0 ensures that the limiting values are concise and have a simple interpretation in terms of the data geometry. Intuitively, as the number of data points increases, the neighborhood around each data point shrinks – as a result, the degree of each node in the graph does not blow up. Finally, m →∞ leads to improving values of the bandwidth estimate.The convergence results are precisely stated in the following theorems: If n →∞, σ→ 0 and m →∞ while satisfying the following rate conditions * (n σ^md+1)/(m^2C^m) →∞, where C = 2/(2π)^d/2,* m 2^m σ→ 0,then for the separable model, one has1/σ^1/mω_m(_S) sup_∈∂ S p(),where “p." denotes convergence in probability. If n →∞, σ→ 0 and m →∞ while satisfying the following rate conditions * (n σ^md)/(m^2C^m) →∞, where C = 2/(2π)^d/2,* m2^mσ^2 → 0, then for the non-separable model, one hasω_m(_A) sup_∈∂ A p(). The dependence of the results on the rate conditions will be explained later in the proofs section. An example of parameter choices for scaling laws to hold simultaneously is illustrated in the following remark: Equations (<ref>) and (<ref>) hold if for each value of n, we choose m and σ as follows:m= [m_0 (logn)^y],σ = σ_0 n^-x/md ,for constants m_0, σ_0 > 0, 0 < y < 1/2 and 0 < x < 1. [ . ] indicates taking the nearest integer value.Theorems <ref> and <ref> give an explicit connection between bandwidth estimates of class indicator signals and class boundaries in the dataset. This interpretation forms the basis of justifying the choice of bandwidth as a smoothness constraint in graph-based learning algorithms. Theorem <ref> suggests that for the separable model, if the boundary ∂ S passes through regions of low probability density, then the bandwidth of the corresponding class indicator vector ω(_S) is low. A similar conclusion is suggested for the nonseparable model from Theorem <ref>, i.e., if the density of data points in the overlap region ∂ A is low, then the bandwidth ω(_A) is low. In other words, low density of data in the boundary regions leads to smooth indicator functions.From our results, we also get an intuition behind the smoothness constraint imposed in the bandlimited reconstruction approach (<ref>) for semi-supervised learning. Basically, enforcing smoothness on classes in terms of indicator bandwidth ensures that the algorithm chooses a boundary passing through regions of low data density in the separable case. Similarly, in the nonseparable case, it ensures that variations in labels occur in regions of low density. Further, the bandwidth constraint θ in (<ref>) effectively imposes a constraint on the complexity of the hypothesis space – a larger value increases the size of the hypothesis space and opens up choices consisting of more complex boundaries.Note that Theorems <ref> and <ref> can be improved and their assumptions generalized in several ways:* The convergence results can be generalized to graphs with edge weights computed using any non-increasing kernel η_σ() = 1/σ^dη(), where σ is a scaling parameter that controls the kernel width and goes to zero as n→∞. The limits of ω_m(_S) and ω_m(_A) stay the same as in (<ref>) and (<ref>), up to a constant factor. * The domain of the data density p() can be generalized to open, bounded and connected sets D ⊂ℝ^d with Lipschitz boundary similar to the work of <cit.>, or a low dimensional compact manifold embedded in ℝ^d as in <cit.>. * Convergence of the bandwidth estimates ω_m(_S) and ω_m(_A) does not imply convergence of the actual bandwidths ω(_S) and ω(_A), respectively, to the same continuum limiting values. This is because the scaling of m is tied to n and σ in our rate conditions, whereas ideally, one should take the limit m →∞ first, and independently of n and σ while analyzing the estimates. In this case, the scaling factor 1/σ^1/m in the left hand side of (<ref>) also disappears. The analysis for this interchange of limits is challenging and we do not know how to approach this problem at the moment, so we leave it for future work. However, based on experiments in Section <ref>, where we use actual bandwidths instead of their estimates to validate convergence, we conjecture that the same results hold for both, i.e., As n →∞ and σ→ 0 at appropriate rates, ω(_S) →sup_∈∂ S p() and ω(_A) →sup_∈∂ A.* Note that Theorems <ref> and <ref> show pointwise convergence for fixed underlying data models, i.e., convergence is proven for a given indicator signal _S specified by { p(), ∂ S }, and _A specified by {p_A(), p_A^c()}. This is not sufficient when we want to interpret the behavior of a bandwidth-based learning algorithm, since we cannot guarantee that the solution returned by the algorithm matches the solution of its continuum limit version. We need stronger convergence results for this case, such as those recently covered in <cit.>.Finally, as a special case of our analysis, we also get a convergence result for the graph cut in the nonseparable model analogous to the results of <cit.> for the separable model. Note that the cut in this case equals the sum of weights of edges connecting points that belong to class A to points that do not belong to class A, i.e.,Cut(A,A^c):= ∑__i ∈ A, _j ∈ A^c w_ij = n_A^t _A. With this definition, we have the following result:If n →∞, σ→ 0 such that nσ^d→∞, then1/nCut(A,A^c) ∫α_A α_A^c p_A() p_A^c() d.The result above indicates that if the overlap between the conditional distributions of a particular class and its compliment is low, then the value of the graph cut is lower. This justifies the use of spectral clustering in the context of nonseparable models.§.§ Label complexity In the context of our work, we define the label complexity of learning class indicators over the graph using a sampling theoretic approach, as the fraction of labeled nodes required for perfectly predicting the labels of the unlabeled nodes. Formally, for a given class indicator _C ∈{ 0,1,}^n over the graph G_n, we define it as the fraction of points that need to be labeled so that a sampling theory-based reconstruction algorithm (such as bandlimited reconstruction of (<ref>)) outputs a solution ^* with zero reconstruction error: ^* - _C= 0. Note that perfect reconstruction is a strong requirement that can be relaxed by allowing an error tolerance ϵ, in which case the amount of labeling required is lower. However, this requirement simplifies our analysis since we can directly use results from sampling theory to evaluate this quantity. Specifically, we can simply use Lemma <ref> to calculate the label complexity for _C over the graph as 1/n_(ω(_C)). In our context, label complexity is essentially an indicator of how “good" the semi-supervised problem is, i.e., how much help we get from geometry while predicting the unknown labels. A low label complexity is indicative of a favorable situation, where one is able to learn from only a few known labels by exploiting data geometry. Note that our definition of label complexity is concerned with reconstructing class indicators only on the nodes of the graph. This pertains to the transductive learning philosophy, a common setting considered in most graph-based semi-supervised learning literature, where the goal is to simply predict the labels of the unlabeled points and not learn a general labeling rule/classifier. Further, our definition is different and simpler than the more general (ϵ,δ) definition of sample/label complexity in Probably Approximately Correct (PAC) learning <cit.>, i.e., it is concerned with reconstructing only a given class indicator, with zero error, using a sampling theory-based learning approach, over a graph constructed from a given data model.§.§.§ Ideal label complexitiesA simple way to compute the label complexity, for the data models we consider, is to find the fraction of points belonging to a region that fully encompasses the boundary. To formalize this, let us define the following two sublevel sets in ℝ^d:_S:= { : p() ≤sup_∈∂ S p()}, _A:= { : p() ≤sup_∈∂ A p()}.Note that by definition, ∂ S is fully contained in _S and ∂ A is fully contained in _A (see Figure <ref> for an example in ℝ^1). Therefore, to perfectly reconstruct the indicator signals _S and _A for any n, it is sufficient to know the labels of all points in _S and _A, respectively, as this strategy removes all ambiguity in labeling the two classes; a good learning algorithm can simply propagate the known labels on to the unlabeled points. Based on this and using the law of large numbers, we arrive at the following conclusion:The ideal label complexities of learning _S and _A in the asymptotic limit are given by P(_S) and P(_A), respectively, where P(Ω) = ∫_Ω p()d. §.§.§ Label complexity of _S and _A using a sampling theory-based approach Note that from Lemma <ref>, we know that the label complexitiesfor _S and _A are given as 1/n_(ω(_S)) and 1/n_(ω(_A)), respectively. Since our bandwidth convergence results relate the bandwidth of indicators for the two data models with data geometry, we only need to asymptotically relate the fraction of eigenvalues ofbelow any constant. This is achieved by first proving the following:Let _(t) be the number of eigenvalues ofbelow a constant t. Then, as n →∞ and σ→ 0, we have1/n𝒩_(t)⟶ P( { : p() ≤ t}).See Section <ref>. Note that Theorem <ref> can be strengthened by proving convergence of 1/n_(t) rather than its expected value. This requires further analysis, which we leave for future work. Plugging in ω(_S) and ω(_A) in place of t in Theorem <ref>, and using the convergence results from Theorems <ref> and <ref>, and Conjecture <ref>, we speculate the following convergence for the label complexities of _S and _A: As n →∞, σ→ 0, we have1/n_(ω(_S)) → P(_S),1/n_(ω(_A)) → P(_A).The limiting values in (<ref>) and (<ref>) are the same as those predicted by Remark <ref>; this is encouraging as far as the validity of Conjecture <ref> is concerned. Additionally, we see strong evidence in our experiments to support our claims; specifically, the average error of predicting the labels of the unlabeled nodes goes to zero as the fraction of labeled examples crosses the limit values of (<ref>) and (<ref>) (see Figure <ref>). The limiting values in (<ref>) and (<ref>) essentially indicate how the low density separation assumption can benefit semi-supervised learning, since in this case, one can forgo the task of labeling a significant fraction of the points and still reconstruct the indicator by exploiting data geometry. A classic example of where this can be useful is the two-step learning process, where the first step uses semi-supervised learning in a transductive setting to create a large training set using a combination of unlabeled and labeled data, and the second step involves learning a classifier using supervised learning. If the low density separation is satisfied by the data, then semi-supervised learning using a sampling theory-based approach effectively reduces the sample complexity of the supervised learning step by a constant fraction, equal to the limiting values in (<ref>) and (<ref>). § PROOFSWe now present the proofs[A partial sketch of the proof for the separable model is also provided in our parallel work <cit.>; here we provide the complete proof.] of Theorems <ref> and <ref>. The main idea is to perform a variance-bias decomposition of the bandwidth estimate and then prove the convergence of each term independently. Specifically, for any indicator vector _R ∈{0,1 }^n, we consider the random variable:(ω_m(_R))^m = _R^T ^m _R/_R^T _R = 1/n_R^T ^m _R /1/n_R^T _R . We study the convergence of this quantity by considering the numerator and denominator separately (it is easy to show that the fraction converges if both the numerator and denominator converge). By the strong law of large numbers, the following can be concluded for the denominator as n →∞: 1/n_R^T _R ∫_∈ R p()d, where a.s. denotes almost sure convergence. For the numerator, we decompose it into two parts – a variance term for which we show stochastic convergence using a concentration inequality, and a bias term for which we prove deterministic convergence.§.§ Expansion of 1/n_R^T ^m _RLet V := 1/n_R^T ^m _R. We begin by expanding V asV= 1/n^m+1_R^T (- )^m _R = 1/n^m+1_R^T ( ∑_k = 0^2^m - 1_k ) _R,where _k denotes the k^th term out of the 2^m terms in the expansion of ( - )^m. _k is composed of a product of m matrices, each of which can be eitheror -. In order to write it down explicitly, one can use the m-bit binary representation of the index k and replace 0s withand 1s with -, i.e., if b_v(k) denotes the v^th most-significant bit in the m-bit binary representation of k for v ∈{1,…,m} and s(k) denotes the number of ones in it (i.e., s(k) := ∑_v=1^m b_v(k)), then _k= ∏_v=1^m(^1- b_v(k).(-)^b_v(k)) = (-1)^s(k)∏_v=1^m(^1- b_v(k).^b_v(k)), where the product notation assumes that the ordering of the matrices is kept fixed, i.e., ∏_p=1^m_p = _1 _2 …_m.Noting thatandare composed of the edge weights w_ij = 1/(2πσ^2)^d/2K(_i,_j), we now describe how to expand the quadratic form V by considering each term _R^T _k _R individually: * The sign of the term _R^T _k _R is determined by the number of (-) matrices in the product _k.* By using the definitions ofandin the product expansion of _k, the absolute value of _R^T _k _R can be expressed through the following template:∑_i_1,…,i_m+1 (_R)_i_1 w_i_1 i_2 w_* i_3… w_* i_m w_* i_m+1 (_R)_*,where (_R)_i denotes the i^th element of of the indicator vector, and the locations with a “*” need to be filled with appropriate indices in {i_1,…,i_m+1}. Note that the template consists of a product of m edge weights w_ij, each contributed by either aordepending on its location in the expression. * By performing an explicit matrix multiplication, we fill the locations from left to right one-by-one using the following rule: let a term containing a * be preceded by an edge-weight w_ab, then, * If w_ab is contributed by , then * = a.* If w_ab is contributed by , then * = b. Since the binary representation of k is closely tied to the ordering ofandin the product term _k, we can once again use it to explicitly express _R^T _k _R. In order to populate any “*” location according to the rules above, we require a quantity that depends on the position of the last occurringwith respect to any location in the product expression of _k. Therefore, using the m-bit binary representation of k, we define for location u ∈{1,…,m}: c_u(k) := 1 + max( {0}∪{v | 1 ≤ v ≤ u, b_v(k) = 1 }), where max({.}) returns the maximum element in a set of numbers. The template described in (<ref>) can then be completed using the rules to obtain _R^T _k _R = (-1)^s(k)∑_i_1,…,i_m+1[ (_R)_i_1 w_i_1 i_2 w_i_c_1(k) i_3 w_i_c_2(k) i_4…… w_i_c_m-2(k) i_m w_i_c_m-1(k) i_m+1 (_R)_i_c_m(k)]. Finally, the expansion of V can be obtained by summing the 2^m quadratic forms in (<ref>): V = 1/n^m+1∑_k=0^2^m-1_R^T _k _R = 1/n^m+1∑_k=0^2^m-1 (-1)^s(k)∑_i_1,…,i_m+1 (_R)_i_1 w_i_1 i_2 w_i_c_1(k) i_3…… w_i_c_m-2(k) i_m w_i_c_m-1(k) i_m+1 (_R)_i_c_m(k)= 1/n^m+1∑_i_1,i_2,…,i_m+1 g( _i_1,_i_2,…,_i_m+1), where we defined g( _i_1,_i_2,…,_i_m+1) :=∑_k=0^2^m-1 (-1)^s(k)[ (_R)_i_1 w_i_1 i_2 w_i_c_1(k) i_3 w_i_c_2(k) i_4…… w_i_c_m-2(k) i_m w_i_c_m-1(k) i_m+1 (_R)_i_c_m(k)].§.§ Convergence of variance terms For V = 1/n_R^T ^m _R, we have the following concentration result: For every ϵ > 0, we have:( | V - V | > ϵ)≤ 2 exp( -[n/(m+1)]σ^mdϵ^2/2C^m V + 2/3| C^m - σ^mdV|ϵ) ,where C = 2/(2π)^d/2.Note that the expansion of V in (<ref>) has the form of a V-statistic. Further, as defined in (<ref>), g is composed of a sum of 2^m terms, each a product of m kernel functions K that are non-negative. Therefore, we have the following upper bound:g ≤ 2^mK_∞^m = (2/(2πσ^2)^d/2)^m = C^m/σ^md. In order to apply a concentration inequality for V, we first re-write it in the form of a U-statistic by regrouping terms in the summation in order to remove repeated indices, as given in <cit.>:V= 1/n^(m+1)∑_(n,m+1) g^*( _i_1,_i_2,…,_i_m+1),where ∑_(n,m+1) denotes summation over all ordered (m+1)-tuples (i_1,…,i_m+1) of distinct indices taken from the set {1,…,n}, n^(m+1) = n.(n-1)…(n-m) is the falling factorial (or number of (m+1)-permutations of n) and g^* is a weighted arithmetic mean of specific instances of g that avoids repeating indices:g^*( _1,_2,…,_m+1) = ∑_j=0^m+1n^(j)/n^m+1∑_(j)^* g( _l_1,_l_2,…,_l_m+1),where ∑_(j)^* denotes summation over all (m+1)-tuples (l_1,l_2,…,l_m+1) formed from {1,…,j} with exactly j distinct indices. Note that the number of such (m+1)-tuples is given by {}0ptm+1j, which is a Stirling number of the second kind. Hence, we have g_*_∞≤∑_j=0^m+1n^(j)/n^m+1{}0ptm+1jg_∞ = g_∞, where we used the property ∑_j=0^m+1 n^(j){}0ptm+1j = n^m+1. Therefore, g^* has the same upper bound as that of g derived in (<ref>). Moreover, using the fact that V = g^*(_i_1,_i_2,…,_i_m+1), we can bound the variance of g^* asg^*≤g^*_∞g^* = C^m/σ^mdV. Finally, plugging in the bound and variance of g^* in Bernstein's inequality for U-statistics as stated in <cit.>, we arrive at the desired result of (<ref>).Note that as n →∞ and σ→ 0 with rates satisfying (n σ^md) / (mC^m) →∞, we have P(|V - V| > ϵ) → 0 for all ϵ > 0. The continuous mapping theorem then allows us to conclude that V^1/m (V)^1/m.§.§ Expansion of 1/n_R^T ^m _R The V-statistic expansion of V = 1/n_R^T ^m _R in (<ref>) has summands with repeating indices, hence we first define a U-statistic counterpart that avoids these repetitions: U := 1/n^(m+1)∑_(n,m+1) g(_i_1, _i_2, …, _i_m+1), where g(_i_1, _i_2, …, _i_m+1) are the kernels defined in (<ref>), and the definitions of ∑_(n,m+1) and n^(m+1) are the same as those for (<ref>). The U-statistic definition is convenient sinceU = g(_i_1, _i_2, …, _i_m+1),as opposed to V, where one would have to deal with terms with repeated indices separately. Further, note that n^m+1V = n^(m+1) U + ∑_(n,m+1)^* g(_i_1, _i_2, …, _i_m+1), where ∑_(n,m+1)^* denotes summation over all ordered (m+1)-tuples (i_1,…,i_m+1) of indices obtained from {1,2,…,n} such that at least two of them are equal. Note that there are n^m+1 - n^(m+1) terms in the summation ∑_(n,m+1)^*. Therefore, we have V = n^(m+1)/n^m+1U + 1/n^m+1∑_(n,m+1)^* g(_i_1, …, _i_m+1) = g + n^m+1 - n^(m+1)/n^m+1.g+ 1/n^m+1∑_(n,m+1)^* g(_i_1, …, _i_m+1) = g + O( m^2C^m/nσ^md), where we used n^m+1 - n^(m+1) = O(m^2 n^m), g≤g_∞ and g_∞ = C^m/σ^md from (<ref>). We now focus on computing g(_i_1, _i_2, …, _i_m+1). Based on (<ref>), we can express it as follows: g(_i_1, _i_2, …, _i_m+1) = ∑_k=0^2^m-1 h_k, where we define: h_k:= (-1)^s(k)∫__1∫__2…∫__m+1[ 1_R(_1) K(_1, _2) K(_c_1(k), _3) K(_c_2(k), _4) …… K(_c_m-2(k), _m) K(_c_m-1(k), _m+1) 1_R(_c_m(k)) ]p(_1)d_1 p(_2) d_2 … p(_m+1) d_m+1,with c_u(k) defined as in (<ref>). §.§ Convergence of bias term for the separable model To evaluate the convergence of bias terms, we shall require the following properties of the d-dimensional Gaussian kernel:If p() is twice differentiable, then∫ K_σ^2(,)p()d = p() + O( σ^2 ). Using the substitution =+ followed by a Taylor series expansion about , we have∫ K_σ^2(,)p()d= ∫1/(2πσ^2)^d/2 e^-^2/2σ^2 p( + ) d= ∫1/(2πσ^2)^d/2 e^-^2/2σ^2( p() + ^T ∇ p()+ 1/2^T ∇^2 p()+ …) d= p() + 0 + σ^2/2 Tr(∇^2 p()) + …= p() + O(σ^2),where Tr(.) denotes the trace of a matrix, and the third step follows from simple component-wise integration. If p() is twice differentiable, then∫ K_aσ^2(,) K_bσ^2(,) p() d = K_(a+b)σ^2(,)( p( b+ a /a + b) + O( σ^2 ) ). Note thatK_aσ^2(,) K_bσ^2(,) =1/(2π aσ^2)^d/2 e^- -^2/2aσ^21/(2π bσ^2)^d/2 e^- -^2/2bσ^2=1/(2π(a+b)σ^2)^d/2 e^- -^2/2(a+b)σ^21/(2πab/a+bσ^2)^d/2 e^- - b + a/a + b^2/2(ab/a+b)σ^2= K_(a+b)σ^2(,)K_ab/a+bσ^2(b + a/a + b,).Therefore, we have∫ K_aσ^2(,) K_bσ^2(,) p() d = K_(a+b)σ^2(,) ∫ K_ab/a+bσ^2(b + a/a + b, ) p() d = K_(a+b)σ^2(,)( p( b+ a /a + b) + O( σ^2 ) ),where the last step follows from Lemma <ref>. In order to prove convergence for the separable model, we need the following results: If p() is Lipschitz continuous, then for a smooth hypersurface ∂ S that divides ℝ^d into S_1 and S_2, and whose curvature has radius lower-bounded by τ > 0,lim_σ→ 01/σ∫_S_1∫_S_2 K_σ^2(_1,_2) p^α(_1) p^β(_2) d_1 d_2= 1/√(2π)∫_∂ S p^α + β() d,where α and β are positive integers. Moreover, for positive integers a,b, and α, β, α', β' such that α + β = α'+β' = γ, we have: lim_σ→ 01/σ∫_S_1∫_S_1[ K_aσ^2(_1,_2) p^α(_1) p^β(_2)- K_bσ^2(_1,_2) p^α'(_1) p^β'(_2) ] d_1 d_2= √(b)-√(a)/√(2π)∫_∂ S p^γ() d.See Appendix <ref>.We now prove the deterministic convergence of 1/n_S^T ^m _S in the following lemma:As n →∞, σ→ 0 such that m2^mσ→ 0 and m^2C^m/nσ^md+1→ 0, we have1/σ1/n_S^T ^m _S→t(m)/√(2π)∫_∂ S p^m+1()d,where t(m) = ∑_r = 0^m-1m-1r (-1)^r (√(r+1) - √(r)). Using (<ref>) and (<ref>), and replacing _R with _S, we have 1/σ1/n_S^T ^m _S = 1/σ∑_k=0^2^m-1 h_k + O( m^2C^m/nσ^md+1). We pair all even-indexed and odd-indexed terms together to rewrite the summation as: ∑_k=0^2^m-1 h_k = ∑_l=0^2^m-1-1 (h_2l + h_2l+1). Now, h_0 and h_1 can be evaluated by repeatedly applying (<ref>) for every Gaussian kernel in the definition from (<ref>). Hence, for the first summation pair, we obtain: h_0 + h_1= ∫_S∫_ℝ^d K_σ^2(,) p^m() p() d d - ∫_S∫_S K_σ^2(,) p^m() p() d d + O(mσ^2) = ∫_S∫_S^c K_σ^2(,) p^m() p() d d + O(mσ^2). For the rest of the terms, we also require the use of (<ref>). However, in this case, we encounter several terms of the form p(θ + (1-θ) ) for some θ∈ [0,1]. Since mσ^2 → 0 and p() is assumed to be Lipschitz continuous, we can approximate such terms by p() or p(). Further, the number of times we have to apply (<ref>) in any h_k is equal to the number of occurrences ofin _k (which is s(k)). Therefore, for 1 ≤ l ≤ 2^m-1-1, we have h_2l + h_2l+1= (-1)^s(2l)[ ∫_S ∫_S K_s(2l)σ^2(,) p^α() p^β() d d- ∫_S ∫_S K_s(2l+1)σ^2(,) p^α'() p^β'() d d] + O(mσ^2), where α, β, α', β' are positive integers such that α + β = α' + β' = m+1. Plugging (<ref>) and (<ref>) into (<ref>), we get: 1/σ1/n_S^T ^m _S= 1/σ∫_S∫_S^c K_σ^2(,) p^m() p() d d+ ∑_r=1^m-1m-1r (-1)^r 1/σ[∫_S ∫_S K_rσ^2(,) p^α() p^β() d d- ∫_S ∫_S K_(r+1)σ^2(,) p^α'() p^β'() d d]+ O(m2^mσ) + O( m^2C^m/nσ^md+1), where we grouped terms based on r = s(2l) in the summation (note that there are m-1r for a given r). Using Lemma <ref>, we conclude that the right hand side of (<ref>) converges as n→∞ and σ→ 0 to 1/√(2π)∫_∂ S p^m+1() d + ∑_r=1^m-1√(r+1)-√(r)/√(2π)∫_∂ S p^m+1() d,which is the desired result. Using the continuous mapping theorem on (<ref>), we can conclude ( 1/σ1/n_S^T ^m _S)^1/m→( t(m)/√(2π)∫_∂ S p^m+1()d)^1/m. Finally, we note that as m →∞, we have( t(m)/√(2π)∫_∂ S p^m+1()d/∫_S p()d)^1/m⟶sup_∈∂ S p(). Therefore, we conclude for the separable model 1/σ^1/mω_m(_S) →sup_∈∂ S p(). §.§ Convergence of bias term for the nonseparable modelFor the nonseparable model, we need to prove convergence of 1/n_A^T ^m _A. This is illustrated in the following lemma:As n →∞, σ→ 0 such that m2^mσ^2 → 0 and m^2C^m/nσ^md→ 0, we have1/n_A^T ^m _A→∫α_A α_A^c p_A() p_A^c() p^m-1() d.Similar to the proof of Lemma <ref>, we use (<ref>) and (<ref>), and replace _R with _A to obtain 1/n_A^T ^m _A = ∑_l=0^2^m-1-1( h_2l + h_2l+1) + O( m^2C^m/nσ^md). Using (<ref>) repeatedly in the definition (<ref>), we get h_0 + h_1= ∫α p_A() p^m() d - ∫( α p_A() )^2 p^m-1() d + O(mσ^2) = ∫α_A α_A^c p_A() p_A^c() p^m-1() d + O(mσ^2), where we used the fact that p() = α_A p_A() + α_A^c p_A^c(). Similarly, for 1 ≤ l ≤ 2^m-1-1, we have h_2l + h_2l+1 = O(mσ^2 ). Putting together (<ref>) and (<ref>) into (<ref>), we get 1/n_A^T ^m _A = ∫α_A α_A^c p_A() p_A^c() p^m-1() d + O(m2^mσ^2) + O( m^2C^m/nσ^md). Taking limits while satisfying the stated rate conditions, we get the desired result.We finally note that as m →∞, we have ( ∫α_A α_A^c p_A() p_A^c() p^m-1() d/∫_A p()d)^1/msup_∈∂ A p(). Therefore, we conclude for the nonseparable model ω_m(_A) →sup_∈∂ A p(). Note that Lemma <ref> for the special case of m = 1 yields 1/n_A^T _A →∫α_A α_A^c p_A() p_A^c() d, which proves Theorem <ref>. §.§ Proof of Theorem <ref>We begin by recalling the definition of the empirical spectral distribution (ESD) of :μ_n(x) := 1/n∑_i=1^n δ(x - λ_i), where {λ_i} are the eigenvalues of . For each x, μ_n(x) is a function of _1, …, _n, and thus a random variable. Note that the fraction of eigenvalues ofbelow a constant t, and its expected value can be computed from the ESD as1/n𝒩_(t)= ∫_0^t μ_n(x) dx,1/n𝒩_(t) = ∫_0^t μ_n(x) dx. Therefore, to understand the behavior of the expected fraction of eigenvalues ofbelow t, we need to analyze the convergence of the expected ESD in the asymptotic limit.The idea is to show the convergence of the moments of μ_n(x) to the moments of a limiting distribution μ(x). Then, by a standard convergence result, μ_n(I)→μ(I) for intervals I. More precisely, let the ⇒ symbol denote weak convergence of measures, then we use the following result that follows from the Weierstrass approximation theorem:Let μ_n be a sequence of probability measures and μ be a compactly supported probability measure. If ∫ x^m μ_n(dx) →∫ x^m μ(dx) for all m ≥ 1, then μ_n ⇒μ. We then use the following result on equivalence of different notions of weak convergence of measures <cit.> in order to prove our result for cumulative distribution functions.μ_n ⇒μ if and only if μ_n(A) →μ(A) for every μ-continuity set A. Therefore, we simply need to analyze the convergence of moments of μ_n(x). Note that the m^th moment of μ_n(x) can be written as:∫ x^m μ_n(x) dx = 1/n∑_i=1^nλ_i^m =1/n^m. We reuse our analysis in Section <ref>, specifically the expansion in (<ref>) to obtain 1/n^m = 1/n1/n^m (-)^m = 1/n^m+1∑_k=0^2^m-1_k. Using the binary representation of k once again similar to (<ref>), we can compute: _0 = ∑_i_1,…,i_m+1[ w_i_1 i_2 w_i_1 i_3 w_i_1 i_4…w_i_1 i_m w_i_1 i_m+1],_1 = ∑_i_1,…,i_m[ w_i_1 i_2 w_i_1 i_3 w_i_1 i_4…w_i_1 i_m w_i_m i_1], ⋮ _k = ∑_i_1,…,i_m[ w_i_1 i_2 w_i_c_1(k) i_3 w_i_c_2(k) i_4…… w_i_c_m-2(k) i_m w_i_c_m-1(k) i_1]. Note that _k has a summation over m indices for k>1, as a result, a factor of 1/n remains in the expectation. Similarly, terms with repeated indices disappear and thus, we have the following for the right hand side of (<ref>) as n→∞: 1/n^m= [ ∫( ∫ K(_1,_2) p(_2) d_2) ……( ∫ K(_1,_m+1) p(_m+1) d_m+1) p(_1) d_1]. Using (<ref>) repeatedly in the equation above, we get:1/n^m = ∫ p^m+1()d + O(mσ^2 ). Therefore, as n →∞ and σ→ 0, we have:∫ x^m μ_n(x) dx →∫ p^m()p()d.From the right hand side of the equation above, we conclude that the m^th moment of the expected ESD ofconverges to the m^th moment of the distribution of a random variable Y = p(), where p() is the probabilty density function of . Moreover, since p_Y(y) has compact support, μ_n(x) converges weakly to the probability density function of p_Y(y). Hence, the following can be said about the expected fraction of eigenvalues of :1/n𝒩_(t) = ∫_0^t μ_n(x)dx ∫_0^t p_Y(y) dy = ∫_p()≤ t p()d.This proves our claim in Theorem <ref>. Note that, to prove the stochastic convergence of the fraction itself rather than its expected value, we would need a condition similar to those in Theorems <ref> and <ref> to hold for each moment. In that case, σ will go to 0 in a prohibitively slow fashion. We believe that this is an artifact of the methods we employ for proving the result. Hence, our conjecture is that the convergence result holds for 1/n𝒩_(t) itself, and we leave the analysis of this statement for future work. § NUMERICAL VALIDATIONWe now present simple numerical experiments[Link to code: <https://github.com/aamiranis/asymptotics_graph_ssl>] to validate our results and demonstrate their usefulness in practice. A key focus in our experiments is to confirm Conjecture <ref>, i.e., the convergence results for the bandwidth estimates also hold for the actual bandwidths. In order to achieve this, we work directly with the bandwidths of the indicators instead of their estimates and numerically validate their convergence for both the separable and nonseparable models. For simulating the separable model, we first consider a data distribution based on a 2D Gaussian Mixture Model (GMM) with two Gaussians: μ_1 = [-1, 0], Σ_1 = 0.25 and μ_2 = [1 ,0], Σ_2 = 0.16, and mixing proportions α_1 = 0.4 and α_2 = 0.6 respectively.The probability density function is illustrated in Figure <ref>.Next, we evaluate the claim of Theorem <ref> on five boundaries, described in Table <ref>.These boundaries are depicted in Figure <ref> and are illustrative of typical separation assumptions such as linear or non-linear and low or high density.For simulating the nonseparable model, we first construct the following smooth (twice-differentiable) 2D probability density function q(x,y) = 3/π[ 1 - (x^2 + y^2) ]^2, x^2 + y^2 ≤ 1 0, x^2 + y^2 > 1 Note that data points (X,Y) can be sampled from this distribution by setting the coordinates X = √(1 - U^1/4)cos(2π V), Y = √(1 - U^1/4)sin(2π V), where U,V ∼Uniform(0,1).We then use q(x,y) to define a nonseparable 2D model with mixture density p(x,y) = α_A p_A(x,y) + α_A^c p_A^c(x,y), where p_A(x,y) = q(x-0.75,y), p_A^c(x,y) = q(x+0.75,y) and α_A = α_A^c = 0.5. The probability density function is illustrated in Figure <ref>. The overlap region or boundary ∂ A for this model is given by ∂ A= { (x,y) : (x-0.75)^2 + y^2 < 1and(x+0.75)^2 + y^2 < 1 }.Further, for this model, we have sup_∂ A p() = 0.2517.In our first experiment, we validate the statements of Theorems <ref> and <ref> by comparing the left and right hand sides of (<ref>) and (<ref>) for corresponding boundaries.This is carried out in the following way: we draw n = 2500 points from each model and construct the corresponding similarity graphs using σ = 0.1. Then, for the boundaries ∂ S_i in the separable model and ∂ A in the nonseparable model, we carry out the following steps: * We first construct the indicator functions _S_i and _A on the corresponding graphs.* We then compute the empirical bandwidth ω(_S_i) and ω(_A) in a manner that takes care of numerical error: we first obtain the eigenvectors of the corresponding , then set ω(_S_i) and ω(_A) to be ν for which energy contained in the graph Fourier coefficients corresponding to eigenvalues λ_j > ν is at most 0.01%, i.e., ω(_S_i)= min{ν| ∑_j:λ_j > ν(_j^T _S_i)^2 ≤ 10^-4} ω(_A)= min{ν| ∑_j:λ_j > ν(_j^T _A)^2 ≤ 10^-4}.The procedure above is repeated 100 times and the mean of ω(_S_i) and ω(_A) are compared with sup_∈∂ S_i p() and sup_∈∂ A p() respectively. The result is plotted in Figure <ref>. We observe that the empirical bandwidth is close to the theoretically predicted value and has a very low standard deviation. This supports our conjecture that stochastic convergence should hold for the bandwidth. To further justify this claim, we study the behavior of the standard deviation of ω(_S_i) and ω(_A) as a function of n in Figure <ref>, where we observe a decreasing trend consistent with our result.For our second experiment, we validate the label complexity of sampling theory-based learning in Conjecture <ref> by reconstructing the indicator function corresponding to ∂ S_3 and ∂ A from a fraction of labeled examples on the corresponding graphs. This is carried out as follows: For a given budget l, we find the set of points L⊂{1,2,…,n} to label of size |L| = l, using pivoted column-wise Gaussian elimination on the eigenvector matrixof <cit.>. This method ensures that the obtained labeled set guarantees perfect recovery for signals spanned by the first l eigenvectors of <cit.>.We then recover the indicator functions from these labeled sets by solving the least squares problem in (<ref>) followed by thresholding. Note that θ is set to the cutoff frequency ω_c(L) of L, which is equal to the l^th eigenvalue of . The mean reconstruction error is defined as E_ mean = No. of mismatches on unlabeled set/Size of unlabeled set. We repeat the experiment 100 times by generating different graphs and plot the averaged E_ mean against the fraction of labeled examples. The result is illustrated in Figure <ref>. We observe that the error goes to zero as the fraction of labeled points goes beyond the respective limit values stated in (<ref>) and (<ref>). This reinforces the intuition that the bandwidth of class indicators and their label complexities are closely linked with the inherent geometry of the data.§ DISCUSSIONS AND FUTURE WORKIn this paper, we provided an interpretation of the graph sampling theoretic approach to semi-supervised learning. Our work analyzed the bandwidth of class indicator signals with respect to the Laplacian eigenvector basis and revealed its connection to the underlying geometry of the dataset. This connection is useful in justifying graph-based approaches for semi-supervised and unsupervised learning problems, and provides a geometrical interpretation of the smoothness assumptions imposed in the bandlimited reconstruction approach.Specifically, our results have shown that an estimate of the bandwidth of class indicators converges to the supremum of the probability density on the class boundaries for the separable model, and on the overlap regions for the nonseparable model. This quantifies the connection between the assumptions of smoothness (in terms of bandlimitedness) and low density separation, since boundaries passing through regions of low data density result in lower bandwidth of the class indicator signals. We numerically validated these results through various experiments.There are several directions in which our results can be extended. In this paper we only considered Gaussian-weighted graphs, an immediate extension would be to consider arbitrary kernel functions for computing graph weights, or density dependent edge-connections such as k-nearest neighbors. Another possibility is to consider data defined on a subset of the d-dimensional Euclidean space.Our analysis also sheds light on the label complexity of graph-based semi-supervised learning problems. We showed that perfect prediction from a few labeled examples using a graph-based bandlimited interpolation approach requires the same amount of labeling as one would need to completely encompass the boundary or region of ambiguity. This quantifies the connection between label complexity of a sampling theory-based approach with the underlying geometry of the problem. We believe that the main potential of graph-based methods will be apparent in situations where one can tolerate a certain amount of prediction error, in which case such approaches shall require fewer labeled data. We plan to investigate this as part of future work.ieeetr§ PROOF OF LEMMA 5The key ingredient required for evaluating the integrals in Lemma <ref> involves selecting a radius R (< τ) as a function of σ that satisfies the following properties as σ→ 0:* R → 0,* R/σ→∞,* R^2/σ→ 0,* ϵ_R/σ→ 0, where ϵ_R := ∫_>R K_σ^2(, )d. A particular choice of R is given by R = √(dσ^2 log(1/σ^2)). Note that R → 0 as σ→ 0. Further, R/σ = √(d log(1/σ^2)),R^2/σ = dσlog(1/σ^2). Hence, R/σ→∞ and R^2/σ→ 0 as σ→ 0. Additionally, substituting the expression for R in the tail bound for the norm of a d-dimensional Gaussian vector gives us: ϵ_R/σ = 1/σ∫_>R K_σ^2(, )d≤1/σ( σ^2 d/R^2)^-d/2 e^-R^2/2σ^2 + d/2= 1/σ( e σ^2 log(1/σ^2))^d/2. Therefore, for d > 1, ϵ_R/σ→ 0 as σ→ 0. Further, it is easy to ensure R < τ for the regime of σ in our proofs.We now consider the proof of equation (<ref>), let I := 1/σ∫_S_1∫_S_2 K_σ^2(_1,_2) p^α(_1) p^β(_2) d_1 d_2. Further, let [S_1]_R indicate a tubular region of thickness R adjacent to the boundary ∂ S in S_1, i.e., the set of points in S_1 at a distance ≤ R from the boundary. Then, we have I = 1/σ∫_[S_1]_R p^α(_1) ∫_S_2 K_σ^2(_1,_2) p^β(_2) d_2 d_1_I_1+ 1/σ∫_[S_1]_R^c p^α(_1) ∫_S_2 K_σ^2(_1,_2) p^β(_2) d_2 d_1_E_1. E_1 is the error associated with approximating I by I_1 and exhibits the following behavior: lim_σ→ 0 E_1 = 0.Note thatE_1≤1/σ( p_ max)^β∫_[S_1]_R^c p^α(_1) ( ∫_S_2 K_σ^2(_1,_2) d_2 ) d_1 ≤1/σ( p_ max)^β∫_[S_1]_R^c p^α(_1) ( ∫_ > R K_σ^2(,) d) d_1 = ϵ_R/σ( p_ max)^β∫_[S_1]_R^c p^α(_1) d_1 ≤ϵ_R/σ( p_ max)^α + β.Using lim_σ→∞ϵ_R/σ = 0, we get the desired result. In order to analyze I_1, we need to define certain geometrical constructions (illustrated in Figure <ref>) as follows:* For each _1 ∈ [S_1]_R, we define a transformation of coordinates as:_1 = _1 + r_1 (_1), where _1 is the foot of the perpendicular dropped from _1 onto ∂ S, r_1 is the distance between _1 and _1, and (_1) is the surface normal at _1 (towards the direction of _1). Since the minimum radius of curvature of ∂ S is τ and R < τ, this mapping is injective.* For each _1 ∈∂ S, let H__1^+ denote the half-space created by the plane tangent on _1 and on the side of S_2. Similarly, let H__1^- denote the half-space on the side of S_1, that is, H__1^- = ℝ^d ∖ H__1^+.* Let W__1^+(x) denote an infinite slab of thickness x tangent to ∂ S at _1 and towards the side of S_2. Let W__1^-(y) denote a similar slab of thickness y on the side of S_1.* Finally, for any , let B(,R) denote the Euclidean ball of radius R centered at .We now consider I_1, the main idea here is to approximate the integral over S_2 by an integral over the half-space H^+__1. Hence, we have: I_1 = 1/σ∫_[S_1]_R p^α(_1) ∫_H^+__1 K_σ^2(_1,_2) p^β(_2) d_2 d_1_I_2+ 1/σ∫_[S_1]_R p^α(_1) ∫_S_2 - H^+__1 K_σ^2(_1,_2) p^β(_2) d_2 d_1_E_2, where E_2 is the error associated with the approximation. Therefore, we have I = I_2 + E_2 + E_1. We now show that as σ→ 0, I_2 →1/√(2π)∫_∂ Sp^α+β()d, and E_2 → 0. lim_σ→ 0 I_2 = 1/√(2π)∫_∂ S p^α + β() d.Using the change of coordinates _1 = _1 + r_1 (_1), we have I_2= 1/σ∫_∂ S∫_0^R p^α(_1 + r_1 (_1)) ( ∫_H^+__1 K_σ^2(_1 + r_1 (_1),_2) p^β(_2) d_2 )|detJ(_1,r_1)| d_1 dr_1, where J(_1,r_1) denotes the Jacobian of the transformation. Now, an arc PQ of length ds at a distance r_1 away from ∂ S gets mapped to an arc P'Q' on ∂ S whose length lies in the interval [ds(1-r_1/τ),ds(1+r_1/τ)]. Therefore, for all points within [S_1]_R, we have ( 1-R/τ)^d-1≤ |detJ(_1,r_1)| ≤( 1+R/τ)^d-1. Further, since p() is Lipschitz continuous with constant L_p, p^α() is also Lipschitz continuous with constant L_p,α. Therefore, for any _1 ∈ [S_1]_R, we have p^α(_1) = p^α(_1) + L_p,αR. This leads to the following simplification for I_2: I_2= ( 1 + O(R^d-1) ) ∫_∂ Sp^α(_1) I_3(_1) d_1+ O(R^d) ∫_∂ S I_3(_1) d_1, where we defined I_3(_1) := 1/σ∫_0^R ∫_H__1^+ K_σ^2(_1 + r_1 (_1),_2) p^β(_2) d_2dr_1. Note that every _2 ∈ H__1^+ can be written as _2 + r_2 (_2), where (_2) = -(_1). Hence, we get I_3(_1)= ∫_ℝ^d-11/(2πσ^2)^d-1/2 e^-_1 - _2^2/2σ^2 p^β(_2 - r_2(_1)) d_2 ×1/σ∫_0^R ∫_0^∞1/√(2πσ^2) e^-(r_1 + r_2)^2/2σ^2 dr_1 dr_2 = (∫_ℝ^d-11/(2πσ^2)^d-1/2 e^-_1 - _2^2/2σ^2 p^β(_2) d_2 + O(R) )×1/σ∫_0^R ∫_0^∞1/√(2πσ^2) e^-(r_1 + r_2)^2/2σ^2 dr_1 dr_2 = (p^β(_1) + O(σ^2) + O(R) )×1/σ∫_0^R ∫_0^∞1/√(2πσ^2) e^-(r_1 + r_2)^2/2σ^2 dr_1 dr_2, where we used Lipschitz continuity of p^β() in the second equality and applied Lemma <ref> to arrive at the last step. Further, using the definition of the Q-function and integration by parts, we note that 1/σ∫_0^R ∫_0^∞1/√(2πσ^2) e^-(r_1 + r_2)^2/2σ^2 dr_1 dr_2= ∫_0^R/σ∫_0^∞1/√(2π) e^-(x + y)^2/2 dxdy = ∫_0^R/σ Q(y)dy = yQ(y) |_0^R/σ - ∫_0^R/σQ'(y) dy= R/σ Q(R/σ)+ 1/√(2π)( 1 - e^-R^2/2σ^2). Therefore, I_3(_1)= (p^β(_1) + O(σ^2) + O(R) ) ×( R/σ Q(R/σ) + 1/√(2π)( 1 - e^-R^2/2σ^2) ). Combining (<ref>) and (<ref>) and using the fact that R/σ→∞ as σ→ 0 (from the definition of R), we getlim_σ→ 0 I_2 = 1/√(2π)∫_∂ S p^α + β() d, which concludes the proof. We now consider the error term E_2 and prove the following result: lim_σ→ 0 E_2 = 0.Let us first rewrite E_2 as follows: E_2 = 1/σ∫_[S_1]_R p^α(_1) I_4(_1) d_1,where we definedI_4(_1) := ∫_S_2 - H^+__1 K_σ^2(_1,_2) p^β(_2) d_2.The key idea is to lower and upper bound I_4(_1) for all _1 using worst case scenarios and evaluate the limits of the bounds. Note that I_4(_1) is largest in magnitude when S_1 or S_2 is a sphere of radius τ, as illustrated in Figures <ref> and <ref>. We now make certain geometrical observations. For any _1 = _1 + r_1 (_1) ∈ [S_1]_R, we observe from Figure <ref> that I_4(_1)≤∫_W__1^-( R^2 - r_1^2/2(τ - r_1)) K_σ^2(_1,_2) p^β(_2) d_2+ ∫_B(_1,R)^c K_σ^2(_1,_2) p^β(_2) d_2 ≤∫_W__1^-(R') K_σ^2(_1,_2) p^β(_2) d_2+ p_ max^βϵ_R. where R' = R^2/2(τ - R). Similarly, from Figure <ref>, we observe that I_4(_1)≥ - [ ∫_W__1^+( R^2 - r_1^2/2(τ + r_1)) K_σ^2(_1,_2) p^β(_2) d_2+ ∫_B(_1,R)^c K_σ^2(_1,_2) p^β(_2) d_2 ] ≥ - [ ∫_W__1^+(R') K_σ^2(_1,_2) p^β(_2) d_2+ p_ max^βϵ_R ]. Substituting these in (<ref>) and using a simplification similar to that of I_2 in (<ref>), we get E_2≤( 1 + O(R^d-1) ) ∫_∂ Sp^α(_1) I_5^-(_1) d_1+ O(R^d) ∫_∂ S I_5^-(_1) d_1 + ϵ_R/σ p_ max^α + β, E_2≥ -( 1 + O(R^d-1) ) ∫_∂ Sp^α(_1) I_5^+(_1) d_1- O(R^d) ∫_∂ S I_5^+(_1) d_1 - ϵ_R/σ p_ max^α + β, where we defined I_5^-(_1) := 1/σ∫_0^R ∫_W__1^-(R') K_σ^2(_1 + r_1 (_1),_2)p^β(_2) d_2dr_1, I_5^+(_1) := 1/σ∫_0^R ∫_W__1^+(R') K_σ^2(_1 + r_1 (_1),_2)p^β(_2) d_2dr_1. Similar to the evaluation of I_3(_1) in (<ref>), we have I_5^+(_1)= (p^β(_1) + O(σ^2) + O(R) )×1/σ∫_0^R ∫_0^R'1/√(2πσ^2) e^-(r_1 + r_2)^2/2σ^2 dr_1 dr_2, I_5^-(_1)= (p^β(_1) + O(σ^2) + O(R) )×1/σ∫_0^R ∫_0^R'1/√(2πσ^2) e^-(r_1 - r_2)^2/2σ^2 dr_1 dr_2. We now evaluate the two 1-D integrals as follows: 1/σ∫_0^R ∫_0^R'1/√(2πσ^2) e^-(r_1 + r_2)^2/2σ^2 dr_1 dr_2= ∫_0^R/σ∫_0^R'/σ1/√(2π) e^-(x + y)^2/2 dxdy = ∫_0^R/σ( Q(y) - Q(y + R'/σ) ) dy = ∫_0^R/σ Q(y) dy +∫_0^R'/σ Q(y)dy - ∫_0^R+R'/σQ(y) dy = R/σ Q(R/σ) + R/σ Q(R'/σ) - R+R'/σ Q(R+R'/σ) 1/√(2π)( 1 - e^-R^2/2σ^2 - e^-R'^2/2σ^2 + e^-(R+R')^2/2σ^2) . Similarly, 1/σ∫_0^R ∫_0^R'1/√(2πσ^2) e^-(r_1 - r_2)^2/2σ^2 dr_1 dr_2= ∫_0^R/σ∫_0^R'/σ1/√(2π) e^-(x - y)^2/2 dxdy = ∫_0^R/σ( Q(y - R'/σ) - Q(y) ) dy = ∫_-R'/σ^0 Q(y)dy + ∫_0^R-R'/σQ(y) dy - ∫_0^R/σ Q(y) dy= R'/σ Q(-R'/σ) + R-R'/σ Q(R-R'/σ) - R/σ Q(R/σ) 1/√(2π)( e^-R'^2/2σ^2 - 1 + e^-(R + R')^2/2σ^2 - e^-R^2/2σ^2). Noting that as σ→ 0, R/σ→∞ and R'/σ→ 0, we conclude that lim_σ→ 0 E_2 = 0.The proof of (<ref>) proceeds in a similar fashion by approximating the inner integral using hyperplanes. Specifically, similar to the proof of (<ref>), we can show that the integral on the left hand side can be written as I + E, where I:= 1/σ∫_[S_1]_R∫_H^-__1[ K_aσ^2(_1,_2) p^α(_1) p^β(_2)- K_bσ^2(_1,_2) p^α'(_1) p^β'(_2) ] d_1 d_2, and E is the residual associated with the approximation that can be shown to go to zero as σ→ 0 (we skip this proof since it is quite similar to the analysis for (<ref>)). In order to evaluate I, we perform a change of coordinates _1 = _1 + r_1 (_1) as before to obtain I= 1/σ∫_∂ S∫_0^R [p^α(_1 + r_1 (_1)) ( ∫_H^-__1 K_aσ^2(_1 + r_1 (_1),_2) p^β(_2) d_2 )-p^α'(_1 + r_1 (_1)) ( ∫_H^-__1 K_bσ^2(_1 + r_1 (_1),_2) p^β'(_2) d_2 ) ]|detJ(_1,r_1)| d_1 dr_1 = ∫_∂ S p^α(_1) I_β(_1) d_1 - ∫_∂ S p^α'(_1) I_β'(_1) d_1 +O( R^d ), where we defined I_β(_1):= 1/σ∫_0^R ∫_H__1^- K_aσ^2(_1 + r_1 (_1),_2)p^β(_2) d_2dr_1, I_β'(_1):= 1/σ∫_0^R ∫_H__1^- K_bσ^2(_1 + r_1 (_1),_2) p^β'(_2) d_2dr_1. By using a change of coordinates for _2 similar to the steps in (<ref>), we obtain I_β(_1)= (p^β(_1) + O(σ^2) + O(R) )×1/σ∫_0^R ∫_0^∞1/√(2π aσ^2) e^-(r_1 - r_2)^2/2aσ^2 dr_1 dr_2,I_β'(_1)= (p^β'(_1) + O(σ^2) + O(R) )×1/σ∫_0^R ∫_0^∞1/√(2π bσ^2) e^-(r_1 - r_2)^2/2bσ^2 dr_1 dr_2. The 1-D integrals can be evaluated as follows: 1/σ∫_0^R ∫_0^∞1/√(2π aσ^2) e^-(r_1 - r_2)^2/2aσ^2 dr_1 dr_2= √(a)∫_0^R/√(a)σ∫_0^∞1/√(2π) e^-(x - y)^2/2 dxdy = √(a)∫_0^R/√(a)σ Q(-y)dy = √(a)∫_0^R/√(a)σ (1 - Q(y)) dy = R/σ - R/σ Q(R/√(a)σ)- √(a)/√(2π)( 1 - e^-R^2/2aσ^2), 1/σ∫_0^R ∫_0^∞1/√(2π bσ^2) e^-(r_1 - r_2)^2/2bσ^2 dr_1 dr_2= R/σ - R/σ Q(R/√(b)σ)- √(b)/√(2π)( 1 - e^-R^2/2bσ^2). Using the fact that α +β = α' + β' = γ, and taking the limit σ→ 0 after putting everything together, we conclude lim_σ→ 0 I = √(b) - √(a)/√(2π)∫_∂ S p^γ () d. Aamir Anis received his Bachelors and Masters of Technology degrees in Electronics and Electrical Communication Engineering from the Indian Institute of Technology (IIT), Kharagpur, India in 2012 and a Ph.D. in Electrical Engineering from the University of Southern California (USC), Los Angeles in 2017. He is the recipient of the Best Student Paper award at the ICASSP 2014 conference held in Florence, Italy. His research interests include graph signal processing with applications in machine learning and multimedia compression. He is currently a Software Engineer at Google Inc., Mountain View, California. Aly El Gamal (S '09-M '15) is an Assistant Professor at the Electrical and Computer Engineering Department of Purdue University. He received his Ph.D. degree in Electrical and Computer Engineering and M.S. degree in Mathematics from the University of Illinois at Urbana-Champaign, in 2014 and 2013, respectively. Prior to that, he received the M.S. degree in Electrical Engineering from Nile University and the B.S. degree in Computer Engineering from Cairo University, in 2009 and 2007, respectively. His research interests include information theory and machine learning.Dr. El Gamal has received a number of awards, including the Purdue Seed for Success Award, the Purdue CNSIP Area Seminal Paper Award, the DARPA Spectrum Challenge (SC2) Contract Award and Phase 1 Top 10 Team Award, and the Huawei Innovation Research Program (HIRP) OPEN Award. He is currenlty a reviewer for the American Mathematical Society (AMS) Mathematical Reviews.A. Salman Avestimehr (S'03-M'08-SM'17) is an Associate Professor at the Electrical Engineering Department of University of Southern California. He received his Ph.D. in 2008 and M.S. degree in 2005 in Electrical Engineering and Computer Science, both from the University of California, Berkeley. Prior to that, he obtained his B.S. in Electrical Engineering from Sharif University of Technology in 2003. His research interests include information theory, the theory of communications, and their applications to distributed computing and data analytics.Dr. Avestimehr has received a number of awards, including the Communications Society and Information Theory Society Joint Paper Award, the Presidential Early Career Award for Scientists and Engineers (PECASE) for “pushing the frontiers of information theory through its extension to complex wireless information networks”, the Young Investigator Program (YIP) award from the U. S. Air Force Office of Scientific Research, the National Science Foundation CAREER award, and the David J. Sakrison Memorial Prize. He is currently an Associate Editor for the IEEE Transactions on Information Theory.AntonioOrtega (F'07) received the Telecommunications Engineering degree from the Universidad Politecnica de Madrid, Madrid, Spain in 1989 and the Ph.D. in Electrical Engineering from Columbia University, New York, NY in 1994.In 1994 he joined the Electrical Engineering department at the University of Southern California (USC), where he is currently a Professor and has served as Associate Chair.He is a Fellow of the IEEE since 2007, and a member of ACM and APSIPA. He has served as associate editor for several IEEE journals, as chair of the Image and Multidimensional Signal Processing (IMDSP) technical committee, and is currently a member of the Board of Governors of the IEEE Signal Processing Society. He was technical program co-chair of ICIP 2008 and PCS 2013.He was the inaugural Editor-in-Chief of the APSIPA Transactions on Signal and Information Processing. He has received several paper awards, including most recently the 2016 Signal Processing Magazine award and was a plenary speaker at ICIP 2013. His recent research work is focusing on graph signal processing, machine learning, multimedia compression and wireless sensor networks.Over 40 PhD students have completed their PhD thesis under his supervision at USC and his work has led to over 300 publications in international conferences and journals, as well as several patents. | http://arxiv.org/abs/1705.09518v2 | {
"authors": [
"Aamir Anis",
"Aly El Gamal",
"Salman Avestimehr",
"Antonio Ortega"
],
"categories": [
"cs.LG"
],
"primary_category": "cs.LG",
"published": "20170526103908",
"title": "A Sampling Theory Perspective of Graph-based Semi-supervised Learning"
} |
1Subaru Telescope, National Astronomical Observatory of Japan, 650 North Aóhoku Place, Hilo, Hawaii, 96720, USA 2Institute of Astronomy, School of Science, University of Tokyo, 2-21-1 Osawa, Maka, Tokyo 181-0015, Japan 3Steward Observatory, University of Arizona, 933 North Cherry Avenue, Rm. N204, Tucson, AZ 85721-0065, USA 4National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 5Institute of Space and Astronomical Science, Japan Aerospace Exploration Agency, 3-1-1, Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan 6Kiso Observatory, Institute of Astronomy, School of Science, University of Tokyo, 10762-30 Mitake, Kiso, Nagano 397-0101, Japan 7Mount Stromlo Observatory, Research School of Astronomy and Astrophysics, Australian National University, Weston Creek P.O., ACT 2611, Australia [8]The principal investigator of the MAGNUM projectIn <cit.>, we described a new method for measuring extragalactic distances based on dust reverberation in active galactic nuclei (AGNs), and we validated our new method with Cepheid variable stars. In this paper, we validate our new method with Type Ia supernovae (SNe Ia) which occurred in two of the AGN host galaxies during our AGN monitoring program: SN 2004bd in NGC 3786 and SN 2008ec in NGC 7469. Their multicolor light curves were observed and analyzed using two widely accepted methods for measuring SN distances, and the distance moduliderived are μ=33.47± 0.15 for SN 2004bd and 33.83± 0.07 for SN 2008ec. These results are used to obtain independently the distance measurement calibration factor, g. The g value obtained from the SN Ia discussed in this paper is g_ SN = 10.61± 0.50 which matches, within the range of 1σ uncertainty, g_ DUST = 10.60, previously calculated ab initio in <cit.>. Having validated our new method for measuring extragalactic distances, we use our new method to calibrate reverberation distances derived from variations of Hβ emission in the AGN broad line region (BLR), extending the Hubble diagram to z≈ 0.3 where distinguishing between cosmologies is becoming possible.§ INTRODUCTION An AGN consists of a hot central engine, thought to be an accretion disk around a black hole, a dust free region of ionized gas surrounding the accretion disk where broad emission lines originate, and a dust torus that lies beyond the BLR. In <cit.>, we used the spectral energy distribution (SED) of the central engine and the dust properties to calculate the light travel time from the central engine to the dust torus in terms of the absolute luminosity of the central engine. We measured the light travel times,Δ t in days, for 17 AGNs, and derived luminosity distances, d, in Mpc, usingd=Δ t× 10^0.2(m_V-A_V-k_V-25+g)where m_V is the mean observed apparent magnitude of the AGN in the V band, A_V is the Galactic extinction in the V band, k_V is the K-correction for the V band, and g=g_ DUST=10.60 is the calculated calibration factor that characterizes the dust sublimation by the radiation field of the central engine <cit.>.We obtained H_0=73± 3 km s^-1 Mpc^-1 in good agreement with H_0=75± 10 km s^-1 Mpc^-1 obtained from empirically calibrated Cepheid variable stars <cit.>.During our AGN monitoring program for the MAGNUM project <cit.>, SNe Ia occurred in two of the monitored AGN host galaxies, SN 2004bd in NGC 3786, and SN 2008ec in NGC 7469. In <ref>, we use our multicolor observations of the SN Ia to derive a luminosity distance, d_ SN, for each SN Ia, which we use in Equation (<ref>) to obtain g_ SN. In <ref> we compare g_ SN with g_ DUST, and also discuss the error budget for the calculation of g_ DUST.Fourteen of the AGNs studied in <cit.> are common in the study by <cit.> to obtain distances from the delay between continuum flaring and an increase in the Hβ emission line flux.They calibrated their relative distances using the surface brightness fluctuation distance of the companion galaxy of NGC 3227, μ =31.86± 0.24. In <ref>, we calibrate their Hβ delays using our distances in <cit.> for the AGNs that we have in common, and we derive g_ Hβ. We present our conclusions in <ref>.§ SUPERNOVAE§.§ Observations Photometric data for SN 2004bd and SN 2008ec were obtained during the MAGNUM AGN monitoring program. The Multiwavelength Imaging Photometer <cit.> on the MAGNUM telescope had a 90 arcsec field of view and obtained images including each host galaxy and its SN. Observations were made in four optical photometric bands U, B, V, and I, from MJD 53,097 to 53,208 for SN 2004bd, and from MJD 54,661 to 54,76 for SN 2008ec.For each SN, the basic characteristics are listed in Table <ref>. §.§ Data Reduction Standard reduction procedures were applied to the images, such as bias subtraction and flat fielding, as in <cit.>. Before proceeding to the photometry, the host galaxy flux was subtracted from each image using template galaxy images made as follows. First, we selected the images observed on nights just before the occurance of SN, which had seeing of one arcsec or better. For SN 2004bd, images of NGC 3786 obtained at MJD 53046.4 were used for all bands. For SN 2008ec, the images obtained at MJD 54649.6, 54651.6, 54653.5, and 54651.6 were used for the U, B, V, and I bands, respectively. Second, we stacked all the images in a band, obtained on the night, in order to improve the signal to noise ratio of the template galaxy image.Any variable AGN components in the template image should not affect the SN photometry, unless the position of the AGN is inside the photometric aperture.However, SN 2004bd was only 5.3 arcsec from the nucleus, which would contaminate the photometry aperture and sky flux measurement area. To avoid AGN effects on the photometric results, we subtracted the AGN component from the template image using GALFIT <cit.>, which models the surface brightness distribution of galaxies or point sources. The template image was subtracted from each observed image after adjusting the seeing size of the template image by Gaussian convolution using the IRAF task GAUSS. The varying AGN component in each observed image remained after the subtraction, so we subtracted it from each observed image using the GALFIT PSF model. We performed aperture photometry with an 8.3 arcsec diameter aperture. Flux in the aperture was calibrated using standard stars listed in <cit.> that were observed on the same night.The resultant light curves in the U, B, V, and I band of SN 2004bd and SN 2008ec are plotted with filled circles in the upper and lower panels, respectively, in Figure <ref>. The photometry following the process above achieved a photometric error around 1% in flux. [The Open Supernova Catalog project (https://sne.space/, arXiv:1605.01054) accumulates the photometric data of SNe in literature, including SN 2004bd and SN 2008ec, and provides their multicolor light curves. However, we used only the data of our own in this paper, in order to avoid possible biases which may arise from using heterogeneous data sets obtained with different instruments, photometric methods, host galaxy flux estimation methods, etc.]§.§ Light Curve Fitting A variety of methods have been developed to correct the intrinsic dispersion of the peak brightness of SN Ia using light curve shapes <cit.>.We selected two methods: the Multicolor Light Curve Shape method <cit.> and the second version of the Spectral Adaptive Light curve Template method <cit.>, to measure the distances of the SNe Ia. These standard and sophisticated methods provide the template light or color index curves and the necessary basic parameters to model the multicolor light curves of any SN Ia, and measure its distance.§.§.§ MLCS2k2 In the MLCS2k2 method, the observed light curve set of any SN Ia in U, B, V, R, and I is fit using the parameter Δ with the template set trained by a sample of nearby SN Ia. The difference from the template curve at the peak magnitude in the V band is represented by Δ.Using the color variation curve, the MLCS2k2 method can also estimate total extinction A_V along the light path to the SN in addition to the peak magnitude. The wavelength independent distance modulus, μ, can be obtainedas μ=μ_λ-A_λ. For instance, the model formula for the V band is given asμ = m_λ(t_0)-M_λ^peak-ζ_λ(α_λ+β_λ/R_V)A_V-P_λΔ-Q_λΔ^2,where A_V is total extinction in the V band, and Δ is differential magnitude from template light curve at the peak in V, which parameterizes the whole light curve model with the coefficients P_λ and Q_λ.Details of parameters ζ_λ, α_λ, and β_λ, concerning the extinction estimation, are discussed in <cit.>.The correction for the Galactic extinction A_V, MW=0.079for SN 2004bd and A_V, MW=0.228 for SN 2008ec, accordingto <cit.>, was applied to the observed light curves priorfitting. The fit converged with a slightly better chi-square valuethan without the correction.We fixed R_V=3.1 following the discussion in <cit.>.Most R_V derived by MLCS2k2 are distributed the range 3.0± 0.1. Setting R_V as a free parameter made our fitting unstable.The best fit model curves for SN 2004bd are shown in the upper panel of Figure <ref> as solid lines. The derived distance modulus is μ=33.53± 0.24 with a total extinction of A_V=0.93± 0.29 mag in the V band. Similarly, the fitted model curves for the SN 2008ec observations are displayed in the lower panel of Figure <ref>. The derived distance modulus is μ=33.88± 0.13 with a total extinction of A_V=1.03± 0.09 in the V band. The systematic errors in these distance moduli are ± 0.21 for SN 2004bd and ± 0.18 for SN 2008ec, originating from several error sources such as uncertainty of model parameters, intrinsic dispersion of SN Ia luminosity, and residuals of the training set light curves from the model curves. These results are listed in the Table <ref>.§.§.§ SALT2 SALT2 is a set of template light curves of SN Ia based on a training set from the spectroscopic monitoring of SN Ia <cit.>.They modeled the light curve set asμ=m^*_λ-M_λ+α_x x_1 -β c,where m^*_λ is a rest-frame magnitude at wavelengthλ, α_x is the slope of light-curvewidth to luminosity relationship which encodes time-variable color asa function of light-curve shape, and β is thecoefficient for constant dust extinction.The values of the parameters obtainedfrom the training set are reported in <cit.>, and we adopttheir results.The SALT2 team provides their fitting code named SNFIT platformedon Python. We used their code, setting the range of wavelength andperiod to 2000Å<λ<9200Å and -20<t<50 days, assimilar as possible to the MLCS2k2 method. We again subtract theGalactic dust extinction from the light curves used for the fit, sothe fitting model gives only the extinction in the SN host galaxy,as for MLCS2k2.The best fit model light curves for SN 2004bd and SN 2008ec fromSALT2 are overdrawn with dashed lines in the upper and lower panels,respectively, in Figure <ref>.The distance moduli obtainedfrom this method are, for SN 2004bd, μ=33.43± 0.19 withA_V=0.82± 0.13, and for SN 2008ec, μ=33.79± 0.08 withA_V=0.65± 0.07. Systematic errors in these distance moduli are ± 0.17 forSN 2004bd and ± 0.16 for SN 2008ec, considering the same errorsources as in MLCS2k2. Our derived apparent magnitude at maximumluminosity in the B-band for SN 2008ec is m_B, max=15.51± 0.02. This is consistent with 15.49± 0.03 in <cit.> and 15.51± 0.03 in <cit.>. Our derived color parameter of 0.24± 0.03 is consitent with 0.21± 0.03 in <cit.>, but not with 0.09± 0.03 in <cit.>. We could not find any comparable resultsfor SN 2004bd in the literature.§.§ SN Ia Distances For each of the two SNe Ia, the distance moduli obtained from thedifferent methods, MLCS2k2 and SALT2, are consistent.In order to insure that our relatively poor sampling of the SN 2004bd light curve has not affected the fitting results, we included the photometric data from <cit.> and obtained μ=33.43± 0.16 using MLCS2k2 and μ=33.37± 0.14 using SALT2, which are consistent with μ=33.43± 0.19 with only our own data. The light curve width parameter and the extinction are also consistent within 1 σ. Therefore, we adopt the results from our homogeneous data for further discussion. With the standard dust extinction law, R_V=3.1 and therefore A_B/A_V=1.34 <cit.>, the total extinction in the V band, A_V, estimated for SN 2004bd by the two methods is consistent within the large errors, while the two values of A_V estimated for SN 2008ec differ by 0.4 mag. This difference might come from including the U band in the color determination for SN 2008ec, because, in both methods, the U band training set uncertainties are large.In order to avoid underestimating the errors, we adopt the simple mean for μ and its error instead of the weighted mean, because the two methods are not independent. They use training sets with many data sets in common. For SN 2004bd, μ=33.48± 0.22, which corresponds to a luminositydistance of d_ SN=49.68± 4.92 Mpc. The systematic errorsare ± 0.19 in μ and ± 4.37 Mpc in d_ SN. For SN 2008ec, μ=33.84± 0.10, corresponding tod_ SN=58.48± 2.80 Mpc. The systematic errors are± 0.17 in μ and ± 4.63 Mpc in d_ SN. Our SN Ia derived distances are consistent with those in theliterature for the parent galaxies derived by other methods, such as the Tully-Fisher relation <cit.>, AGNreverberation <cit.>, and galaxy ring structurediameter <cit.>. The mean and standard deviation of theresults from these studies are μ=33.32± 0.15 for NGC3786, and μ=34.02± 0.56 for NGC7469.§ DUST PROPERTIES OF AGNS Substituting d_ SN into the LHS of Equation (<ref>), we obtain the calibration factor of g_ SN = 9.73 ± 1.16 for SN 2004bd and g_ SN = 11.04 ± 0.56 for SN 2008ec. Taking both random errors and systematic errors into account, the weighted average for our two SNe Ia is g_ SN = 10.61 ± 0.50, and the systematic error in g_ SN, from that ind_ SN, is ± 0.69.The calculated calibration factor g_ DUST=10.60 used in <cit.> is within 1σ of g_ SN. This agreement provides an independent confirmation of the validity of our new method for measuring extragalactic distances described in <cit.>.The standard deviation σ_g_ SN for our two SNe Ia is 0.62,which, in principle, should be the combination of the random andsystematic errors in g_ SN.The errors discussed above give the combined error of √(0.50^2+0.69^2)=0.85, which is greater than σ_g_ SN indicating that the individual random and systematic errors are overestimated by taking the simple mean of the MLCS2k2 and SALT2 methods. Using the weighted mean, and ignoring any covariance from using similar training sets, gives random and systematic errors of 0.34 and 0.49 respectively, and a combined error of 0.60, similar to σ_g_ SN.An estimate of the cosmic variation in our dust-reverberation distances can be obtained by examining the differences between d_ DUST, as given by Equation (<ref>) with g_ DUST=10.60, and d_H =v/H_0 (H_0=73± 3 km s^-1 Mpc^-1) for the 17 AGNs studied in <cit.>.In practice, by substituting d_H into the LHS of Equation (<ref>), we express the cosmic variation of the dust properties in terms of the distribution of g_ DUST, which reflects random errors as well as systematic errors, including possible object to object variations. Our dust reverberation model has three parameters:(1) the dust sublimation temperature, T_ d=1700± 50 K, derived from the H-K, J-K, and J-H color temperatures of the variable near-infrared component in our sample of Seyfert 1 galaxies <cit.>,(2) the power-law index, α_ UV =-0.5 ± 0.2, for the UV to optical continuum emission from the central engine which heats the dust, as found in the SDSS sample of QSOs <cit.>, and(3) the grain size, a, represented by the distribution of f(a)∝ a^-2.75 (0.005 μ m≤ a ≤ 0.20 μ m) that is intermediate between the two opposite extreme distributions of grain size in the literature <cit.>.Using d_H in the LHS of Equation (<ref>), the standard deviation of g_ DUST for the 17 AGNs studied in <cit.> is derived as 0.44. This must include the well estimated errors of ± 0.24 from our Δ t_ DUST measurements, ± 0.1 from intrinsic extinction <cit.>, ± 0.18 from the possible range of T_ d, and ± 0.32 from α.The remaining, necessary error is ± 0.09 and is attributed to the systematic error from the grain size for f(a), much less than the conceivable maximum error ∼± 0.5 <cit.> from assuming the two opposite extreme distributions of a.Thus, the dust properties of all AGNs are not very different. <cit.> claimed that distances measured by <cit.> suffered from the degeneracy between H_0 and dust emissivity, because AGN luminosity was calculated by redshift-based distance. This claim is not correct because <cit.> calculated the luminosity based on the physical model of radiative thermal equilibrium <cit.> which is independent of H_0. The SN distances in this paper confirm that the dust parameters used in <cit.> are reasonable.§ HΒ DISTANCES OF AGNS <cit.> reported a relation between the relative distances for AGN based on the correlation of the BLR size with absolute luminosity. For the 14 AGNs in <cit.> that we have in common, substituting their dust reverberation distances of d_ DUST into the LHS of Equation (<ref>), we use the time delay Δ t_ Hβ between a continuum flare and the increase in the flux in the Hβ emission line from <cit.> to obtain the distribution of g_ Hβ. The mean of g_ Hβ is 13.56, and the deviation is as small as 0.08. Thus, we attribute most of the scatter to cosmic variation in the properties of the BLR.In Figure <ref> we plot their reverberation distances obtained from Equation (<ref>) using Δ t_ Hβ and g_ Hβ=13.56 versus Δ t_ DUST and g_ DUST=10.60. Having determined the mean value of g_ Hβ=13.56, we use Equation (<ref>) to obtain the luminosity distances for all the AGNs with measured Δ t_ Hβ <cit.>. These are plotted against redshift as black squares in Figure <ref> along with our AGNs from <cit.> (red circles) and the galaxies with Cepheid distances <cit.>.§ CONCLUSION The calculation in <cit.> of the calibration factor, g=10.60, in Equation (<ref>), that characterizes the dust sublimation by the radiation field of the AGN central engine isindependently supported by the distances of two SNe Ia which occurred in two different AGN host galaxies during our AGN monitoring program. The agreement between g_ SN and g_ DUST as well as the smallestimated scatter of g_ DUST, after excluding measurement errors (<ref>),indicates that our characterization of the AGN dust is essentially correct and that the dust properties of all AGNs are very similar to our characterization. We have used 14 of our AGNs to evaluate g_ Hβ, and we have used this calibration to produce a Hubble diagram reaching z=0.3, that begins to distinguish between possible cosmologies(<ref>) . With further ground based Hβ observations, this could be extended to z ≈ 1. We thank the staff at the Haleakala Observatories for their help with facility maintenance. This research has been supported partly by the Grants-in-Aid of Scientific Research (10041110, 10304014, 11740120, 12640233, 14047206, 14253001, 14540223, and 16740106) and the COE Research (07CE2002) of the Ministry of Education, Science, Culture and Sports of Japan. [Barvainis(1987)]barv87 Barvainis, R. 1987, , 320, 537 [Bentz et al.(2009)]bent09 Bentz, M. C., Peterson, B. M., Netzer, H., Pogge, R. W., & Vestergaard, M. 2009, , 697, 160 [Cardelli et al.(1989)]card89 Cardelli, J. A., Clayton, C., Mathis, J. S., 1989, , 345, 245 [Cackett et al.(2007)]cack07 Cackett, E. M., Horne, K., & Winkler, H. 2007, , 380, 669 [Collier et al.(1999)]coll99 Collier, S., Horne, K., Wanders, I., & Peterson, B. M. 1999, , 302, L24 [Davis et al.(2007)]dave07 Davis, S. W., Woo, J.-H., & Blaes, O. M. 2007, , 668, 682[Freedman et al.(2001)]free01 Freedman, W. L., Madore, B. F., Gibson, B. K., et al. 2001, , 553, 47 [Freedman and Madore (2010)]free02 Freedman, W. L.,& Madore, B. F. ARA&A,48.673] [Ganeshalingam et al.(2010)]gane10 Ganeshalingam, M., Li, W., Filippenko, A. V., et al. 2010, , 190, 418[Ganeshalingam et al.(2013)]gane13 Ganeshalingam, M., Li, W., & Filippenko, A. V. 2013, , 433, 2240[Guy et al.(2007)]guy07 Guy, J., et al., 2007, å, 466, 11-21 [Guy et al.(2010)]guy10 Guy, J., Sullivan, M., Conley, A., et al. 2010, , 523, A7[Hönig et al.(2014)]hoen14 Hönig, S. F., Watson, D., Kishimoto, M., & Hjorth, J. 2014, , 515, 528 [Hönig et al.(2017)]hoen17 Hönig, S. F., Watson, D., Kishimoto, M., et al. 2017, , 464, 1693[Jha et al.(2007)]jha07 Jha, S., Riess, A. 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Shiromizu (Tokyo: Universal Academy), 235 [Yoshii et al.(2003)]yosh03 Yoshii, Y., Kobayashi, Y., & Minezaki, T., 2003, BAAS, 202, 38.03 [Yoshii et al.(2014)]yosh14 Yoshii, Y., Kobayashi, Y., Minezaki, T., Koshida, S., & Peterson, B. A. 2014, , 784, L11lllllll6 List of Target SupernovaeSN HostR.A.Dec.zad_ DUSTb (Mpc) Reference SN 2004bd NGC 3786 11 39 42.2 +31 54 31.8 0.00893 74.3± 7.8 IAUC 8316, 8317SN 2008ec NGC 7469 23 03 16.6 +08 52 19.8 0.01632 47.7± 1.5 CBET 1437, 1438 Listed references are the first report of the SN detection. aThe heliocentric redshift from the NASA/IPAC Extragalactic Database (NED). bThe luminosity distance of host galaxy based on the dust reverberation of AGN <cit.>.lllcll0pt 6 Results from Light Curve Modeling 2cSN 2004bd 2cSN 2008ec2-35-6 MLCS2k2 SALT2 MLCS2k2 SALT2μ............................. 33.53± 0.24 33.43± 0.19 33.88± 0.13 33.79± 0.08 t_peak,B (MJD)........ 53090.31± 2.59 53095.20± 1.41 54674.00± 0.52 54674.46± 0.11 A_V(mag)................ 0.93± 0.29 0.82± 0.13a 1.03± 0.09 0.65± 0.07a Width parameterb.. 0.07± 0.12 -0.15± 0.03 -0.19± 0.09 -0.07± 0.02 reduced χ^2.............. 0.519004 0.072918 0.247278 0.773540 aThe extinction in the V band is estimated from the B band extinction which SALT2 calculated, adoptingA_B/A_V =1.34 <cit.> bParameters for light curve width of SNIa. Δ in MLCS2k2 and x_1 in SALT2. | http://arxiv.org/abs/1705.09757v2 | {
"authors": [
"Shintaro Koshida",
"Yuzuru Yoshii",
"Yukiyasu Kobayashi",
"Takeo Minezaki",
"Keigo Enya",
"Masahiro Suganuma",
"Hiroyuki Tomita",
"Tsutomu Aoki",
"Bruce A. Peterson"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170527024707",
"title": "Calibration of AGN Reverberation Distance Measurements"
} |
[email protected] Department of Computer Science, Tel-Hai Academic College, Upper Galilee, 12208 [email protected] Department of Mathematics, University of Montana, Missoula, MT 59812, USA Given a prime number p, a field F with char(F)=p and a positive integer n, we study the class-preserving modifications of Kato-Milne classes of decomposable differential forms. These modifications demonstrate a natural connection between differential forms and p-regular forms. A p-regular form is defined to be a homogeneous polynomial form of degree p for which there is no nonzero point where all the order p-1 partial derivatives vanish simultaneously. We define a C_p,m field to be a field over which every p-regular form of dimension greater than p^m is isotropic. The main results are that for a C_p,m field F, the symbol length of H_p^2(F) is bounded from above by p^m-1-1 and for any n ≥⌈ (m-1) log_2(p) ⌉+1, H_p^n+1(F)=0.Decomposable Differential Forms, Quadratic Pfister Forms, Cyclic p-Algebras, Quaternion Algebras, Kato-Milne Cohomology, Linkage [2010] 11E76 (primary); 11E04, 11E81, 12G10, 16K20, 19D45 (secondary) § INTRODUCTION In this paper we study the connection between the Kato-Milne cohomology groups H_p^n+1(F) over a field F with char(F)=p for some prime integer p, and homogeneous polynomial forms of degree p over F. The three main objectives of this work are: * Finding a number n_0 such that for any n ≥ n_0, H_p^n+1(F)=0.* Finding an upper bound for the symbol length of H_p^2(F), which in turn provides an upper bound for the symbol length of pBr(F).* Finding a number s such that any collection of s inseparably linked decomposable differential forms in H_p^n+1(F) are also separably linked.§.§ The Kato-Milne Cohomology Groups Given a prime number p and a field F of char(F)=p, we consider the space of absolute differential forms_F^1, which is defined to be the F-vector space generated by the symbols da subject to the relations d(a+b)=da+db and d(ab)=adb+bda for any a,b ∈ F. The space of n-differential forms _F^n for any positive integer n is then defined by the n-fold exterior power_F^n=^n(_F^1), which is consequently an F-vector space spanned byda_1… da_n, a_i∈ F. The derivation d extends to anoperator d : _F^n →_F^n+1 by d(a_0da_1… da_n)= da_0 da_1… da_n.We define _F^0=F, _F^n=0 for n<0, and _F=⊕_n≥ 0_F^n,the algebra of differential forms over F with multiplication naturally defined by(a_0da_1… da_n)(b_0db_1… db_m)= a_0b_0da_1… da_n db_1… db_m .There exists a well-defined group homomorphism _F^n→_F^n/d_F^n-1, the Artin-Schreier map ℘, which acts on decomposable differential forms as follows:αd β_1/β_1…d β_n/β_n ⟼(α^p-α)d β_1/β_1…d β_n/β_n.The group H_p^n+1(F) is defined to be (℘). By <cit.>, in the case of p=2, there exists an isomorphism H_2^n+1(F)≅⟶I_q^n+1(F)/I_q^n+2(F), given by αd β_1/β_1…d β_n/β_n ⟼ ⟨⟨β_1,…,β_n,α]] I_q^n+2(F)where ⟨⟨β_1,…,β_n,α]] is a quadratic n-fold Pfister form.By <cit.>, when n=1, there exists an isomorphism H_p^2(F)∼⟶ p Br(F), given by αdβ/β ⟼[α,β)_p,F,where [α,β)_p,F is the degree p cyclic p-algebra F ⟨ x,y : x^p-x=α, y^p=β, y x y^-1=x+1 ⟩. In the special case of p=2 and n=1, these cyclic p-algebras are quaternion algebras [α,β)_2,F that can be identified with their norm forms which are quadratic 2-fold Pfister forms ⟨⟨β,α]] (see <cit.>). §.§ C_m and C_p,m Fields A C_m field is a field F over which every homogeneous polynomial form of degree d in more than d^m variables is isotropic (i.e. has a nontrivial zero). It was suggested in <cit.> that if F is a C_m field with char(F) ≠ p then for any n ≥ m, H^n+1(F,μ_p^⊗ n)=0. This fact is known for p=2 because of the Milnor conjecture, proven in <cit.>. It was proven in <cit.> that for any prime p>3, C_m field F with char(F) ≠ p and n ≥⌈ (m-2) log_2(p)+1 ⌉, we have H^n+1(F,μ_p^⊗ n)=0. (The same result holds when p=3 for n≥⌈ (m-3)log_2(3)+3⌉.) The analogous statement for fields F with char(F)=p is that if F is a C_m field then H_p^n+1(F)=0 for every n ≥ m. This is true, as stated in <cit.> and proven explicitly in <cit.>. It follows from the fact that C_m fields F have p-rank at most m, i.e. [F:F^p] ≤ p^m. We consider a somewhat different property of fields that avoids directly bounding their p-rank. We say that a homogeneous polynomial form of degree p over F is p-regular if there is no nonzero point where all the partial derivatives of order p-1 vanish. We denote by u_p(F) the maximal dimension of an anisotropic p-regular form over F. We say F is a C_p,m field if u_p(F) ≤ p^m. We prove that if F is C_p,m then for any n ≥⌈ (m-1) log_2(p) ⌉+1, we have H_p^n+1(F)=0. (See Section <ref> for examples of C_p,m which are not C_m.) Note that when p=2, the notion of a p-regular form coincides with nonsingular quadratic form, and u_p(F) boils down to the u-invariant u(F) of F. In this case, ⌈ (m-1) log_2(p) ⌉+1=m, which recovers the known fact that when u(F) ≤ 2^m, we have H_2^m+1(F) ≅ I_q^m+1(F)/I_q^m+2(F)=0.§.§ Symbol Length in H_p^2(F) By <cit.> (when char(F)=p) and <cit.> (when char(F) ≠ p and F contains a primitive pth root of unity), p Br(F) is generated by cyclic algebras of degree p. The symbol length of a class in p Br(F) is the minimal number of cyclic algebras required in order to express this class as a tensor product of cyclic algebras. The symbol length of p Br(F) is the supremum of the symbol length of all the classes in p Br(F). Recall that when char(F)=p, p Br(F) ≅ H_p^2(F).It was shown in <cit.> that if the maximal dimension of an anisotropic form of degree p over F is d then the symbol length of p Br(F) is bounded from above by ⌈d-1/p⌉-1, providing a characteristic p analogue to a similar result obtained in <cit.> in the case of char(F) ≠ p. As a result, if F is C_m then d ≤ p^m and so this upper bound boils down to p^m-1-1. However, the symbol length of p Br(F) when F is a C_m field with char(F)=p is bounded from above by the p-rank which is at most m (see <cit.>). We show that the forms discussed in <cit.> are actually p-regular forms, which gives the upper bound ⌈u_p(F)-1/p⌉-1 for the symbol length (which coincides with u(F)/2-1 when p=2 as in <cit.>). In particular, if F is C_p,m then the symbol length is bounded from above by p^m-1-1. (This bound is in fact sharp for p=2 as proven in <cit.>.) §.§ Separable and Inseparable Linkage A differential form in H_p^n+1(F) is called “decomposable" if it can be written as αd β_1/β_1∧…∧d β_n/β_n for some α∈ F and β_1,…,β_n ∈ F^×. We say that a collection of decomposable differential forms ω_1,…,ω_m in H_p^n+1(F) are inseparably ℓ-linked if they can be written as ω_i=α_i d β_i,1/β_i,1∧…∧d β_i,n/β_i,n,i ∈{1,…,m}such that β_1,k=…=β_m,k for all k ∈{1,…,ℓ}. We say they are separably ℓ-linked if they can be written in a similar way such that α_1=…=α_m and β_1,k=…=β_m,k for all k ∈{1,…,ℓ-1}. By the identification of decomposable differential forms with quaternion algebras (when p=2 and n=1), cyclic p-algebras (when n=1) and quadratic Pfister forms (when p=2), the notions of inseparable and separable linkage coincide with the previously defined notions of separable and inseparable linkages for these objects. In <cit.> it was proven that inseparable (1-)linkage of pairs of quaternion algebras implies separable (1-)linkage as well. A counterexample to the converse was given in <cit.>. These results extend naturally to Hurwitz algebras (<cit.>) and quadratic Pfister forms (<cit.>). In <cit.> it was shown that when H_2^n+2(F)=0, separable n-linkage and inseparable n-linkage for pairs of quadratic (n+1)-fold Pfister forms are equivalent. In <cit.> it was proven that inseparable (1-)linkage for pairs of cyclic p-algebras of degree p implies separable (1-)linkage as well, and that the converse is not necessarily true. It follows immediately that if two decomposable differential forms in H_p^n+1(F) are inseparably k-linked then they are also separably k-linked. In this paper we generalize this statement for larger collections of forms: every collection of 1+∑_i=ℓ^n 2^i-1 inseparably n-linked decomposable differential forms in H_p^n+1(F) are also separably ℓ-linked. In particular, this means that if three octonion algebras share a biquadratic purely inseparable field extension of F, then they also share a quaternion subalgebra.§ FIELDS WITH BOUNDED U_P-INVARIANT There are examples in the literature of fields with u(F)<2^n+1 and unbounded 2-rank (see <cit.>). In particular, these fields are C_2,m but not C_m. Is there a similar construction of C_p,m fields which are not C_m for prime numbers p>2? The following construction gives an example of a field F which is C_p,0 but clearly not C_0: Let K be a field of characteristic p, L=K(λ_1,…,λ_n) be the function field in n algebraically independent variables (n can also be ∞), and F=L^sep the separable closure of L. Then F is C_p,0 and not C_0.Clearly it is not C_0 because its p-rank is at least n. To show that F is C_p,0 it is enough to show that every p-regular form φ(x_1,…,x_m) of dimension m ≥ 1 over F is isotropic. Let φ(x_1,…,x_m) be a p-regular form of dimension m over F. Since there are no p-regular forms of dimension 1, m>1. Since φ is p-regular, there exists a term with mixed variables and nonzero coefficient. Without loss of generality, assume the power of x_1 in this term is d where 1 ≤ d ≤ p-1. Write φ as a polynomial in x_1 and coefficients in F[x_2,…,x_m]: φ=c_p x_1^p+…+c_1 x_1+c_0. The coefficient c_d is a nonzero homogeneous polynomial form of degree p-d in m-1 variables. Since it is nonzero, we have c_d(a_2,…,a_m) ≠ 0 for some a_2,…,a_m ∈ F, not all zero. Without loss of generality, assume that a_2 ≠ 0, which means we could assume a_2=1. Then c_d(x_2,a_3 x_2,…,a_m x_2) is a nonzero 1-dimensional form. Hence φ(x_1,x_2,a_3 x_2,…,a_m x_2) is a nondiagonal 2-dimensional form of degree p. We shall explain now why this form must be isotropic: Suppose it is anisotropic. Consider the polynomial φ(x_1,1,a_3,…,a_m). This is a polynomial of degree ≤ p and at least d. If its degree is smaller than p then since F is separably closed, the polynomial decomposes into linear factors over F, and then it has a root in F, which means that φ is isotropic. Assume the degree is p. Since it is of degree p, by <cit.> this field extension must be either separable or purely inseparable. It cannot be purely inseparable because it has a nonzero term besides the degree p and 0 terms. Therefore it must be separable, but that contradicts the fact that F is separably closed. One can construct fields F with bounded u_p(F) and infinite p-rank in the following way: Let K be a field of characteristic p with p-rank r. Then for any anisotropic p-regular form φ(x_1,…,x_n) of dimension n, the function field K(φ)=K(x_1,…,x_n : φ(x_1,…,x_n)=0) has p-rank r+n-1. One can also take r=∞ and then the p-rank of K(φ) is ∞ as well.The function field K(φ) of φ is a degree p separable extension of the function field K(x_1,…,x_n-1) in n-1 algebraically independent variables over K. The p-rank of K(x_1,…,x_n-1) is r+n-1 by <cit.>. By <cit.> the p-rank of K(φ) is also r+n-1.Let K be a field of characteristic p and p-rank r, and let M be a positive integer. Then K is a subfield of some field F with p-rank at least r and u_p(F) ≤ M.This F is taken to be the compositum of the function fields of all anisotropic p-regular forms of dimension greater than M. By the previous lemma, the p-rank of F is at least r. Clearly u_p(F) ≤ M. The last corollary provides examples of fields F which are C_p,m by taking M=p^m. The fact that if F is C_m then the field of Laurent series F((λ)) over F is C_m+1 does not hold for C_p,m fields in general. For example, if one takes one of the fields constructed in <cit.> with u(F)=2^m<û(F), then F is C_2,m. However, by <cit.>, u(F((λ)))=2û(F)>2^m+1, hence F((λ)) is not C_2,m+1.§ SYMBOL LENGTH AND P-REGULAR FORMS In this section we describe certain properties of p-regular forms and make a note on the symbol length of classes in pBr(F) when F is a field of characteristic p with bounded u_p(F).Let p be a prime integer and F be a field of characteristic p. Let V=F v_1+…+F v_m be an m-dimensional F-vector space. A map φ : V → F is called a homogeneous polynomial form of degree p if it satisfiesφ(a_1 v_1+…+a_m v_m)=∑_i_1+…+i_m=p c_i_1,…,i_m a_1^i_1… a_m^i_mfor any a_1,…,a_m ∈ F where c_i_1,…,i_m are constants in F. We say that φ is isotropic if there exists a nonzero v in V such that φ(v)=0. Otherwise φ is anisotropic. We say φ is p-regular if there is no v ∈ V ∖{0} for which all the order p-1 partial derivatives of φ vanish. The nonexistence of such points does not depend on the choice of basis. In the special case of p=2, this notion coincides with nonsingularity. In particular, diagonal forms of degree p over F are not p-regular. Given a homogeneous polynomial form φ : V → F, we can consider the scalar extension φ_L of φ from F to L.Given a field extension L/F, if φ_L is a p-regular form for some homogeneous polynomial form φ of degree p over F then φ is p-regular as well.Assume the contrary, that φ is not p-regular, i.e. there exists v ≠ 0 such that all the order p-1 partial derivatives of φ vanish. Since the partial derivatives do not change under scalar extension, all the partial derivatives of φ_L vanish at v, which means that φ_L is not p-regular.Given homogeneous polynomial forms φ : V → F and ϕ : W → F of degree p, we define the direct sum φ⊥ϕ to be the homogeneous polynomial form ψ : V ⊕ W → F defined by ψ(v+w)=φ(v)+ϕ(w) for any v ∈ V and w ∈ W.The form φ⊥ϕ is p-regular if and only if both φ and ϕ are p-regular.If φ is not p-regular, there exists a nonzero v ∈ V such that all the order p-1 partial derivatives of φ vanish. Then all the order p-1 partial derivatives of φ⊥ϕ vanish at the point v ⊕ 0.In the opposite direction, if φ⊥ϕ is not p-regular, then there exists a nonzero v ⊕ w where all the order p-1 partial derivatives of φ⊥ϕ vanish. Without loss of generality, assume v ≠ 0. Then all the order p-1 partial derivatives of φ⊥ϕ vanish at v. Since these derivatives are equal to the derivatives of φ at v, φ is not p-regular.Given a separable field extension L/F of degree p, the norm form N : L → F is a homogeneous polynomial form of degree p. This form is p-regular.By Lemma <ref> it is enough to show that the scalar extension of N to the algebraic closure F of F is p-regular. Now, L ⊗_F F is F×…×F_p times and can be identified with the p × p diagonal matrices with entries in F. Therefore we haveN_F(a_1 e_1+a_2 e_2+…+a_p e_p)=a_1 a_2 … a_p.The latter is clearly p-regular.If φ : V → F is p-regular then for any nonzero scalar c ∈ F, c φ defined by (cφ)(v)=c φ(v) for any v ∈ V is also p-regular. Given a prime number p and a field F with char(F) ≥ p or 0, the homogeneous polynomial form φ(a_1 v_1+a_2 v_2)=α a_1^p-a_1 a_2^p-1+a_2^pover the two-dimensional space V=F v_1+F v_2 is p-regular for any α∈ F.It is enough to note that the partial derivative obtained by differentiating p-1 times with respect to a_2 is (p-1)! a_1 and the partial derivative obtained by differentiating p-2 times with respect to a_2 and once with respect to a_1 is -(p-1)! a_2. Therefore the only point where all the order p-1 partial derivatives vanish is (0,0) and the form is p-regular. In <cit.> it was proven that the symbol length of a p-algebra of exponent p over K is bounded by ⌈d-1/p⌉-1 where d is the maximal dimension of an anisotropic homogeneous polynomial form of degree p over K. Apparently d can be replaced with u_p(F). Let p be a prime integer and let F be a field with char(F) = p and finite u_p(F). Then every two tensor products A=⊗_i=1^m [α_i,β_i)_p,F and B=⊗_i=1^ℓ [γ_i,δ_i)_p,F with (m+ℓ) p ≥ u_p(F)-1 can be changed such that α_1=γ_1.It is enough to show that the homogeneous polynomial form considered in the proof of <cit.> is p-regular. This form φ is the direct sum φ' ⊥ϕ_1 ⊥…⊥ϕ_m ⊥ψ_1 ⊥…⊥ψ_ℓ whereφ' : F × F → F is defined by φ'(a,b)=(∑_i=1^m(α_i)-∑_i=1^ℓ(γ_i)) a^p-a^p-1 b+b^p, for each i ∈{1,…,m}, ϕ_i is β_i N_F[x : x^p-x=α_i]/F, and for each i ∈{1,…,ℓ}, ψ_i is γ_i N_F[x : x^p-x=δ_i]/f. By Lemma <ref> and Remark <ref>, the forms ϕ_1,…,ϕ_m,ψ_1,…,ψ_ℓ are p-regular. By Lemma <ref>, φ' is also p-regular. Consequently, the form φ is p-regular as direct sum of p-regular forms by Lemma <ref>.Let p be a prime integer and let F be a field with char(F) = p and finite u_p(F). Then the symbol length in H_p^2(F) is bounded from above by ⌈u_p(F)-1/p⌉ -1.It follows from Theorem <ref> in the same manner <cit.> follows from <cit.>.Let p be a prime integer and let F be a C_p,m field with char(F) = p. Then the symbol length in H_p^2(F) is bounded from above by p^m-1-1.An immediate result of the previous corollary, given that for a C_p,m field F, u_p(F) ≤ p^m. § SYMBOL LENGTH AND P-RANK By <cit.>, the p-rank of F is an upper bound for the symbol length of H_p^2(F).In fact, it can be shown that if the p-rank of F is a finite integer m then the symbol length of H_p^n+1(F) is bounded from above by mn for any positive integer n. If F is C_m then its p-rank is ≤ m. Suppose F is C_m and that its p-rank is n. Then F=F^p v_1 ⊕…⊕ F^p v_p^n where v_1,…,v_p^n are linearly independent over F^p. Therefore the form φ(a_1,…,a_p^n)=v_1 a_1^p+…+v_p^n a_p^n^p is anisotropic. Since the dimension of φ is p^n, m must be ≥ n. For a C_m field (compared to C_p,m) F, its p-rank (≤ m) provides a better upper bound for the symbol length of H_p^2(F) than the upper bound given in Corollary <ref> (p^m-1-1). The following proposition shows that there exist cases where the symbol length is actually equal to the p-rank: Let K=F((β_1))…((β_n)) be the field of iterated Laurent series in n variables over a perfect field F of characteristic p. Let L/F be a (ℤ/p ℤ)^n-Galois field extension given by L=F[℘^-1(α_1),…,℘^-1(α_n)]. Then the symbol length of the p-algebra D=[α_1,β_1)_p,K⊗…⊗ [α_n,β_n)_p,Kis equal to the p-rank of K.The p-rank of K in the proposition above is n by <cit.>. The p-algebra D is a generic abelian crossed product with maximal (ℤ/p ℤ)^n-Galois subfield K[℘^-1(α_1),…,℘^-1(α_n)], hence it is a division algebra (see <cit.>) and its symbol length is exactly n.In order to construct a Galois extension satisfying the conditions of Proposition <ref>, take α_1,…,α_n to be algebraically independent variables over 𝔽_p. Let F be the perfect closure of 𝔽_p(α_1,…,α_n). Then each field extension F[℘^-1(α_i)] is (ℤ/p ℤ)-Galois and as a set they are mutually linearly independent.Therefore L=F[℘^-1(α_1),…,℘^-1(α_n)] is a (ℤ/p ℤ)^n-Galois extension of the perfect field F.The following example presents a C_m field with p-rank n such that m=2n+1: Let F be the perfect closure of the function field 𝔽_p(α_1,…,α_n) as in remark <ref>. Let K=F((β_1))…((β_n)) be the field of iterated Laurent series in n variables over F. As mentioned above, the p-rank of K is n, and the symbol length of the algebra [α_1,β_1)_p,F⊗…⊗ [α_n,β_n)_p,F is n. The field 𝔽_p(α_1,…,α_n) is a C_n+1 field, and hence so is its perfect closure F. Consequently, K is a C_2n+1 field. § CLASS-PRESERVING MODIFICATIONS OF DECOMPOSABLE DIFFERENTIAL FORMS In this section we study the class-preserving modifications of decomposable differential forms in H_p^n+1(F). These modifications will be used in proving the main results of the following sections. In the special case of p=2 they coincide with the known modifications of quadratic Pfister forms (see <cit.>). Let ω=αd β_1/β_1…dβ_n/β_n be a form in H_p^n+1(F). Then: (a) For any i ∈{1,…,n}, ω=(α+β_i) d β_1/β_1…dβ_n/β_n.(b)For any i ∈{1,…,n} and nonzero f ∈ F[℘^-1(α)], if N_F[℘^-1(α)]/F(f)=0 then ω=0. Otherwise, ω=αd β_1/β_1…∧d(N_F[℘^-1(α)]/F(f) β_i)/N_F[℘^-1(α)]/F(f) β_i∧…dβ_n/β_n.(c) For any i ∈{1,…,n} and γ∈ F, if β_i+γ^p=0 then ω=0. Otherwise, there exists some α' ∈ F such that ω=α' d β_1/β_1…∧d(β_i+γ^p)/β_i+γ^p∧…dβ_n/β_n.(d) For any distinct i,j ∈{1,…,n},ω=αd β_1/β_1∧…∧d β_i/β_i∧…∧d (β_i β_j)/β_i β_j∧…∧d β_n/β_n. (e) For any distinct i,j ∈{1,…,n}, if β_i+β_j=0 then ω=0. Otherwise,ω=αd β_1/β_1∧…∧d (β_i+β_j)/β_i+β_j∧…∧d (β_i^-1β_j)/β_i^-1β_j∧…∧d β_n/β_n. (f) For any distinct i,j ∈{1,…,n} and f ∈ F[℘^-1(α)], if β_i+N_F[℘^-1(α)]/F(f)=0 then ω=0. Otherwise,ω=αd β_1/β_1∧…∧d β_i/β_i∧…∧d ((β_i+N_F[℘^-1(α)]/F(f)) β_j)/(β_i+N_F[℘^-1(α)]/F(f)) β_j∧…∧d β_n/β_n.Parts (a), (b) and (c) are elementary and follow immediately from the equivalent statements for cyclic p-algebras (see <cit.>). Part (d) follows from the linearity of logarithmic differential forms: d (β_i β_j)/β_i β_j=d β_i/β_i+d β_j/β_j.For part (e), if β_i=-β_j then dβ_i ∧ d β_j=0 and so ω=0. Otherwise, it is enough to show thatd β_i/β_i∧d β_j/β_j=d(β_i+β_j)/β_i+β_j∧d(β_i^-1β_j)/β_i^-1β_j.To see this, notice thatd(β_i+β_j)∧ d(β_i^-1β_j)=(β_i^-1+β_i^-2β_j) d β_i ∧ d β_j and divide both sides by (β_i+β_j)(β_i^-1β_j).For (f), recall that for any a ∈ F and b ∈ F^×, [a,b)_p,F is split if and only if b is a norm in the étale extension F[℘^-1(a)]/F, and otherwise it is a division algebra. If β_i+N_F[℘^-1(α)]/F(f)=0 then the algebra [α,β_i)_p,F=0 by the norm condition, and so ω=0. Otherwise, consider the formαd β_1/β_1∧…∧d β_i/β_i∧…∧d ((β_i+N_F[℘^-1(α)]/F(f)) β_j)/(β_i+N_F[℘^-1(α)]/F(f)) β_j∧…∧d β_n/β_n.By (d), it is equal to the sum of ω and the form αd β_1/β_1∧…∧d β_i/β_i∧…∧d (β_i+N_F[℘^-1(α)]/F(f))/β_i+N_F[℘^-1(α)]/F(f)∧…∧d β_n/β_n.If N_F[℘^-1(α)]/F(f)=0 then this form is clearly zero. Otherwise, by (b) it is equal toαd β_1/β_1∧…∧d N_F[℘^-1(α)]/F(f)^-1β_i/N_F[℘^-1(α)]/F(f)^-1β_i∧…∧d (N_F[℘^-1(α)]/F(f)^-1β_i+1)/N_F[℘^-1(α)]/F(f)^-1β_i+1∧…∧d β_n/β_n,which is zero by (c). Part (b) of Lemma <ref> shows that β_n can be replaced by any nonzero element represented by the p-regular form φ:F[℘^-1(α)] → F with f↦β_nN_F[℘^-1(α)]/F(f). In the following proposition we present a p-regular form of larger dimension whose values can also alter the last slot (at the possible cost of changing some of the other inseparable slots). Let α∈ F and β_1,…,β_n ∈ F^×, and write ω=αd β_1/β_1…dβ_n/β_n for the corresponding form in H_p^n+1(F). For each (d_1,…,d_n) ∈{0,1}×…×{0,1}_n times, let V_d_1,…,d_n be a copy of F[℘^-1(α)] and φ_d_1,…,d_n : V_d_1,…,d_n→ F be the homogeneous polynomial form of degree p defined by φ_d_1,…,d_n(f)=N_F[℘^-1(α)]/F(f) ·β_1^d_1·…·β_n^d_n. Write(φ,V)=_[ 0 ≤ d_1,…,d_n ≤ 1; (d_1,…,d_n) ≠ (0,…,0) ] (φ_d_1,…,d_n,V_d_1,…,d_n).If there exists a nonzero v such that φ(v)=0 then ω=0. Otherwise, for every nonzero v ∈ V, there exist β_1',…,β_n-1' ∈ F^× such that :ω=αd β_1'/β_1'…dβ_n-1'/β_n-1'∧d φ(v)/φ(v). To get a feel for the form (φ,V), note that if we set N=N_F[℘^-1(α)]/F then for n=1, (φ,V) = (φ_1,V_1):F[℘^-1(α)] → V with x ↦ N(x)β_1. When n=2, (φ,V):F[℘^-1(α)]^× 3→ F with (x,y,z) ↦ N(x)β_1+N(y)β_2+N(z)β_1β_2. By induction on n. The case of n=1 holds by the analogy to cyclic p-algebras. Assume it holds for a certain n-1. We shall show it holds also for n. The vector space V decomposes as V_0 ⊕ V_1 where V_0=_0 ≤ d_1,…,d_n-1≤ 1 V_d_1,…,d_n-1,0 and V_1=_0 ≤ d_1,…,d_n-1≤ 1 V_d_1,…,d_n-1,1. The latter decomposes as V_1=V_0,…,0,1+V_1'. There is a natural isomorphism τ : V_1' → V_0 identifying each V_d_1,…,d_n-1,1 with V_d_1,…,d_n-1,0. Under this homomorphism, we have φ(v)=β_n φ(τ(v)) for any v ∈ V_1'.Let v be a nonzero vector in V. Then v=v_0+v_1'+v_0,1 where v_0 ∈ V_0, v_1' ∈ V_1' and v_0,1∈ V_0,…,0,1 (≅ F[℘^-1(α)]).Note that φ(v)=φ(v_0)+φ(v_1')+φ(v_0,1)=φ(v_0)+β_n (φ(τ(v_1))+N_F[℘^-1(α)]/F(v_0,1)). Write v_1=v_1'+v_0,1.If v_1'=0 and v_0,1=0 then the statement follows immediately from the induction hypothesis. If v_1'=0 and v_0,1≠ 0 then the statement follows from the induction hypothesis and Lemma <ref> (e).Assume v_1' ≠ 0. If φ(τ(v_1'))=0 then by the induction hypothesis, ω=0. Otherwise, by the induction hypothesis, there exist β_1',…,β_n-2' such thatαd β_1/β_1…dβ_n-2/β_n-2∧d β_n-1/β_n-1=αd β_1'/β_1'…dβ_n-2'/β_n-2'∧d φ(τ(v_1'))/φ(τ(v_1')), and so αd β_1/β_1…dβ_n-2/β_n-2∧d β_n-1/β_n-1∧ dβ_n/β_n=αd β_1'/β_1'…dβ_n-2'/β_n-2'∧d φ(τ(v_1'))/φ(τ(v_1'))∧d β_n/β_n. By Lemma <ref> (f), if φ(τ(v_1'))+N_F[℘^-1(α)]/F(v_0,1)=0 then ω=0, and otherwise we haveω=αd β_1'/β_1'…dβ_n-2'/β_n-2'∧d φ(τ(v_1'))/φ(τ(v_1'))∧d ((φ(τ(v_1'))+N_F[℘^-1(α)]/F(v_0,1)) β_n)/(φ(τ(v_1'))+N_F[℘^-1(α)]/F(v_0,1)) β_n.Consequentlyαd β_1/β_1…dβ_n-2/β_n-2∧d β_n-1/β_n-1∧ dβ_n/β_n=αd β_1/β_1…dβ_n-2/β_n-2∧d β_n-1/β_n-1∧d (φ(v_1))/φ(v_1).If v_0=0, this completes the picture. Assume v_0 ≠ 0. By the assumption, there exist β_1”,…,β_n-2” such thatαd β_1/β_1…dβ_n-2/β_n-2∧d β_n-1/β_n-1=αd β_1”/β_1”…dβ_n-2”/β_n-2”∧d φ(v_0)/φ(v_0), and so αd β_1/β_1…dβ_n-2/β_n-2∧d β_n-1/β_n-1∧ dφ(v_1)/φ(v_1)=αd β_1”/β_1”…dβ_n-2”/β_n-2”∧d φ(v_0)/φ(v_0)∧d φ(v_1)/φ(v_1). By Lemma <ref> (e), if φ(v_0)+φ(v_1)=0 then ω=0, and otherwiseω=αd β_1”/β_1”…dβ_n-2”/β_n-2”∧d (φ(v_0)^-1φ(v_1))/φ(v_0)^-1φ(v_1)∧d (φ(v_0)+φ(v_1))/φ(v_0)+φ(v_1)and since φ(v)=φ(v_0)+φ(v_1), this proves the statement.Using the same setting as Proposition <ref>, writeV_1=⊕_0 ≤ d_1,…,d_n-1≤ 1 V_d_1,…,d_n-1,1.Then for every nonzero v_1 ∈ V_1, assuming ω≠ 0, we haveω=αd β_1/β_1…∧dβ_n-1/β_n-1∧d φ(v)/φ(v). The vector space V_1 decomposes as V_1' ⊕ V_0,…,0,1 where V_1' is the direct sum of all V_d_1,…,d_n-1,1 with (d_1,…,d_n-1) ≠ (0,…,0). Take τ to be the natural isomorphism from V_1' to V_0. Let v_1 be a nonzero element in V_1. It can therefore be written as v_1=v_1'+v_0,1 where v_1' ∈ V_1' and v_0,1∈ V_0,…,0,1 (≅ F[℘^-1(α)]). By the previous propositionαd β_1/β_1…dβ_n-2/β_n-2∧d β_n-1/β_n-1=αd β_1'/β_1'…dβ_n-2'/β_n-2'∧d φ(τ(v_1'))/φ(τ(v_1')),for some β_1',…,β_n-2' ∈ F^×. Consequently,αd β_1/β_1…dβ_n-2/β_n-2∧d β_n-1/β_n-1∧d β_n/β_n=αd β_1'/β_1'…dβ_n-2'/β_n-2'∧d φ(τ(v_1'))/φ(τ(v_1'))∧d β_n/β_n.By Lemma <ref> (f) we have ω=αd β_1'/β_1'…dβ_n-2'/β_n-2'∧d φ(τ(v_1'))/φ(τ(v_1'))∧d (φ(τ(v_1'))+N_F[℘^-1(α)]/F(v_0,1))β_n/(φ(τ(v_1'))+N_F[℘^-1(α)]/F(v_0,1))β_nand soαd β_1/β_1…d β_n-1/β_n-1∧d β_n/β_n=αd β_1/β_1…dβ_n-1/β_n-1∧d φ(v_1)/φ(v_1). Using the same setting as Proposition <ref>, let ℓ be an integer between 1 and n.WriteW=⊕_[ 0 ≤ d_1,…,d_n ≤ 1; (d_ℓ,…,d_n) ≠ (0,…,0) ] V_d_1,…,d_n.Then for every nonzero v ∈ W, assuming ω≠ 0, there exist β_ℓ',…,β_n-1' ∈ F such thatω=αd β_1/β_1…∧d β_ℓ-1/β_ℓ-1∧d β_ℓ'/β_ℓ'∧…dβ_n-1'/β_n-1'∧d φ(v)/φ(v). The vector space W decomposes as W_n ⊕ W_n-1⊕…⊕ W_ℓ such that for each k between ℓ and n,W_k=⊕_[ 0 ≤ d_1,…,d_k-1≤ 1; d_k=1, d_k+1=…=d_n=0 ] V_d_1,…,d_n.Let v be a nonzero vector in W.Then v can be written accordingly as v=v_n+…+v_ℓ where each v_k belongs to W_k. For each k ∈{ℓ,…,n}, by the previous lemmaαd β_1/β_1∧…∧d β_k-1/β_k-1∧d β_k/β_k= αd β_1/β_1∧…∧d β_k-1/β_k-1∧d φ(v_k)/φ(v_k).Thereforeαd β_1/β_1∧…∧d β_n/β_n= αd β_1/β_1∧…∧d β_ℓ-1/β_ℓ-1∧d φ(v_ℓ)/φ(v_ℓ)∧…∧d φ(v_n)/φ(v_n).Then by Proposition <ref> (e) we can change the last term to be d(φ(v_ℓ)+…+φ(v_n))/φ(v_ℓ)+…+φ(v_n) at the cost of possibly changing the slots ℓ to n-1.§ LINKAGE OF DECOMPOSABLE DIFFERENTIAL FORMSGiven a field F of char(F)=p, a positive integer n and an integer ℓ∈{1,…,n}, every collection of 1+∑_i=ℓ^n 2^i-1 inseparably n-linked decomposable differential forms in H_p^n+1(F) are separably ℓ-linked as well.Write m=2^n-1+…+2^ℓ-1. Let {α_i d β_1/β_1∧…∧d β_n/β_n : i ∈{0,…,m}} be the collection of m+1 inseparably n-linked decomposable differential forms in H_p^n+1(F) under discussion. The number of n-tuples (d_1,…,d_n) with (d_ℓ,…,d_n) ≠ (0,…,0) is m. We denote arbitrarily the elements of the set {β_1^d_1·…·β_n^d_n : 0 ≤ d_1,…,d_n ≤ 1, (d_ℓ,…,d_n) ≠ (0,…,0)} by γ_1,…,γ_m (with possible repetitions).Recall that the norm of an element x+λ y in F[λ : λ^p-λ=α] is x^p-x y^p-1+y^p α. Therefore by Corollary <ref>, for every i ∈{0,…,m} and any choice of x_i,1,y_i,1,…,x_i,m,y_i,m∈ F with (x_i,1,…,y_i,m) ≠ (0,…,0) we can change the form α_i d β_1/β_1∧…∧d β_n/β_n to α_i d β_1/β_1∧…∧d β_ℓ-1/β_ℓ-1∧d b_i,ell/b_i,ell∧…∧d b_i,n-1/b_i,n-1∧d δ_i/δ_i where δ_i=γ_1 (x_i,1^p-x_i,1 y_i,1^p-1+α_i y_i,1^p)+…+γ_m (x_i,m^p-x_i,m y_i,m^p-1+α_i y_m,1^p)for some b_i,ℓ,…,b_i,n-1∈ F.Therefore, in order to show that the forms in the collection are separably ℓ-linked, it is enough to show that the following system of m equations in 2m(m+1) variables has a solution:α_0+δ_0 =α_1+δ_1 α_0+δ_0 =α_2+δ_2⋮α_0+δ_0 =α_m+δ_mIf we take x_0,i=x_1,i=…=x_m,i, y_0,i=1 and y_i,i=0 and y_i,j=1 for all i,j ∈{1,…,m} with i ≠ j, then the ith equation in this system becomes a linear equation in one variable x_0,i (whose coefficient is γ_i). This system therefore has a solution.Given a field F of char(F)=2, a positive integer n and an integer ℓ∈{1,…,n}, every collection of1+∑_i=ℓ^n 2^i-1 inseparably n-linked quadratic (n+1)-fold Pfister forms are also separably ℓ-linked. Follows Theorem <ref> and the identification of quadratic Pfister forms with decomposable differential forms appearing in <cit.>. By the identification of octonion algebras with their 3-fold Pfister norm forms (<cit.>), we conclude that if three octonion F-algebras share a biquadratic purely inseparable field extension of F then they also share a quaternion subalgebra. (Recall that when char(F)=2, an octonion algebra A over F is of the form Q+Qz where Q=[α,β)_2,F is a quaternion algebra, z^2=γ and z ℓ=ℓ^σ z for every ℓ∈ Q where σ is the canonical involution on Q, α∈ F and β,γ∈ F^×. In particular, when A is a division algebra, F[√(β),√(γ)] is a subfield of A and a biquadratic purely inseparable field extension of F.)It is not known to the authors if the sizes of the collections mentioned in Theorem <ref> are sharp in the sense that larger collections of inseparably n-linked differential forms in H_p^n+1(F) need not be separably ℓ-linked. Even the very special case of three inseparably linked quaternion algebras would be very interesting, in either direction. Are every three inseparably linked quaternion algebras also separably linked? § VANISHING COHOMOLOGY GROUPS One can use Proposition <ref> to show that for fields F with finite u_p(F) the cohomology groups H_p^n+1(F) vanish from a certain point and on. See section <ref> for the definition of u_p(F). Given a field F with char(F)=p, for any n with (2^n-1)p≥ u_p(F), H^n+1_p(F)=0.Let ω=αd β_1/β_1…dβ_n-1/β_n-1∧d β_n/β_n be an arbitrary decomposable form in H_p^n+1(F). Consider the direct sum Φ of the form φ from Proposition <ref> and the two-dimensional formψ(x,y)=α x^p-x^p-1 y+y^p. Note that by Lemmas <ref>, <ref> and <ref>, Φ is p-regular. If (2^n-1) p+1 ≥ u_p(F) then (Φ)=(2^n-1) p+2, hence Φ is isotropic, which means that there exist (x_0,y_0,v_0) ≠ 0 such that ψ(x_0,y_0)+φ(v_0)=0. If v_0=0 then it means α∈℘(F), and therefore ω is trivial. If φ is isotropic then ω is trivial by Proposition <ref>. Otherwise, φ(v_0) ≠ 0. By proposition <ref>, ω=αd β_1'/β_1'∧…∧d β_n-1'/β_n-1'∧d φ(v_0)/φ(v_0) for some β_1',…,β_n-1' ∈ F. If x_0=0 then Φ(0,y_0,v_0)=y_0^p+φ(v_0)=0, and so φ(v_0)=(-y_0)^p, which means ω is trivial. If x_0 ≠ 0 then Φ(1,y_0/x_0,v_0/x_0)=0 as well. By Proposition <ref>, ω=αd β_1'/β_1'∧…∧d β_n-1'/β_n-1'∧d φ(v_0/x_0)/φ(v_0/x_0) for some β_1',…,β_n-1' ∈ F. Then, by Lemma <ref> (a), we can add φ(v_0/x_0) to α, but then the coefficient is congruent to Φ(1,y_0/x_0,v_0/x_0) modulo ℘(F), and therefore ω is trivial.If F is a C_p,m field with char(F)=p then for all n ≥⌈ (m-1) log_2(p) ⌉+1, H_p^n+1(F)=0.Since F is a C_p,m field, u_p(F) ≤ p^m. If n ≥⌈ (m-1) log_2(p) ⌉+1 then2^n ≥ p^m-1+1, and so 2^n · p ≥ p^m+p, which means that 2^n · p-p ≥ p^m, and therefore (2^n-1) p+1 ≥ p^m+1 ≥ u_p(F)+1.If we take m=1 then ⌈ (m-1) log_2(p) ⌉+1=1, and therefore H_p^1+1(F)=H_p^2(F)=0 for C_p,1 fields F. The group H_p^2(F) is isomorphic to pBr(F), which is generated by symbol algebras [α,β)_p,F. Another way of seeing that pBr(F) is trivial for C_p,1 is to consider an arbitrary generator A=[α,β)_p,F=F ⟨ x,y :x^p-x=α, y^p=β, y x y^-1=x+1 ⟩ and to show that it must be split.If F[x] is not a field then this algebra is split, so suppose this is a field. Then consider the norm form N : F[x] → F. For any choice of f ∈ F[x]^×, the element z=f y satisfies z^p=N(f) β and z x z^-1=x+1, so A=[α,N(F) β)_p,F. The element w=x+z satisfies w^p-w=α+N(f) β (see <cit.>). Now, consider the homogeneous polynomial form φ :F × F × F[x] → F(a,b,f) ↦α a^p-a^p-1 b+b^p+N(f) β.This form is p-regular by Lemma <ref>, Remark <ref> and Lemma <ref>. It is isotropic because its dimension is p+2 and F is C_p,1, i.e. there exist a_0,b_0 ∈ F and f_0 ∈ F[x] not all zero such that φ(a_0,b_0,f_0)=0. The case of a_0=b_0=0 is impossible because then N(f_0)=0 which means that F[x] is not a field, contrary to the assumption. If a_0=0 then the element t=b_0+f_0 y satisfies t^p=0, which means A is split. If a_0 ≠ 0 then the element t=x+b_0/a_0+1/a_0 f_0 y satisfies t^p-t=0, which again means A is split. Recall that for p=2, the notion of u_p(F) coincides with the u-invariant of F. Then the statement recovers the known fact that if n satisfies 2^n+1 > u(F) then H_2^n+1(F)=0.§ BIBLIOGRAPHYamsalpha | http://arxiv.org/abs/1705.09553v2 | {
"authors": [
"Adam Chapman",
"Kelly McKinnie"
],
"categories": [
"math.RA",
"11E76 (primary), 11E04, 11E81, 12G10, 16K20, 19D45 (secondary)"
],
"primary_category": "math.RA",
"published": "20170526124055",
"title": "Kato-Milne Cohomology and Polynomial Forms"
} |
=1unsrt #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/ 0 School of Physics and Astronomy, University of Glasgow,Glasgow, G12 8QQ, UKTop quark properties measurements at the LHC Mark OwenOn behalf of the ATLAS and CMS Collaborations December 30, 2023 ============================================================ Highlights of measurements of the properties of the top quark at the LHC are presented. The measurements probe a range of the properties of the top quark, including the structure of the Wtb vertex, the top-Z coupling and the top-quark mass. The results are compared to Standard Model predictions and in some cases limits on physics beyond the Standard Model are also extracted in the context of effective field theory models. The measurements use data collected by the ATLAS and CMS experiments during pp collisions at a centre-of-mass energy of 8 or 13 TeV. § INTRODUCTIONThe top quark is the heaviest fundamental particle discovered to date and it decays before it has a chance to hadronise. These characteristics not only allow for precision tests of the Standard Model (SM), but also open a potential window to physics beyond the SM. A selection of recent measurements from the ATLAS <cit.> and CMS <cit.> collaborations are discussed below[For a full list of top properties measurements, please see the public websites of the collaborations: <http://cms-results.web.cern.ch/cms-results/public-results/publications/TOP/index.html>, <http://cms-results.web.cern.ch/cms-results/public-results/preliminary-results/TOP/index.html> and <https://twiki.cern.ch/twiki/bin/view/AtlasPublic/TopPublicResults>. ]. The measurements use both top-quark pair and single-top quark production modes. For the analyses using production, two decay modes with low background rates are utilised: the lepton-plus-jets decay mode, where one W boson decays leptonically and the other decays into a pair of quarks and the dilepton decay mode, where both W bosons decays leptonically. § THE WTB VERTEX In the SM the top quark is predicted to decay almost exclusively into a W boson and b-quark. The decay time of the top quark is shorter than the characteristic time for hadronisation and this means the top quark provides a unique window to observe the properties of a bare quark. The decay products of the top quark can therefore be used to probe the nature of the Wtb vertex. §.§ W boson polarisation The W bosons produced in top decays can be either left-handed, right-handed or longitudinally polarised. The corresponding fractions (, and 0) are well predicted in the SM, however the presence of new physics in the Wtb vertex could result in fractions different to the SM predictions. Experimentally, these fractions can be accessed by measuring the helicity angle θ^* between the charged lepton or down-type quark and the direction of the top quark in the rest frame of the W boson. The distribution for the cosine of the helicity angle depends on the polarisation fractions according to:1/Γd Γ/d cosθ^* = 3/8(1 - cosθ^*)^2+ 3/4(sin^2 θ^*)0 + 3/8(1 + cosθ^*)^2 . The ATLAS and CMS collaborations have both published recent measurements of the W boson polarisation fractions using the 8 TeV LHC data <cit.>. Both experiments select events with one high transverse momentum () electron or muon and at least four high jets. Kinematic fit techniques are then used to fully reconstruct the system and the angle θ^* between the charged lepton and the direction of the top quark is then reconstructed in the rest frame of the W boson. The data are fitted to the reconstructed cosθ^* distributions in order to extract the measured polarisation fractions. Figure <ref> shows the templates in the ATLAS analysis for the left-handed, right-handed and longitudinal polarisation states. There is clear discriminating power between the different polarisation states. The cosθ^* distribution measured by CMS is shown in Figure <ref> and the data are seen to be in good agreement with the SM predictions. The largest uncertainties in the measurements arise from the Monte Carlo (MC) modelling of top quark events and the jet energy scale (JES). The results are compared to previous LHC results and the SM prediction in Figure <ref>, where good agreement between the SM and the measurements can be seen. §.§ Top quark polarisation Electroweak single-top quark production at the LHC is dominated by t-channel exchange and the top quarks produced are predicted to be highly polarised, in particular along the direction of the spectator-quark momentum <cit.>. The polarisation (P) is related to the angle between a top-quark decay product and the top-quark spin axis (θ_l) according to:1/Γd Γ/d cosθ_l = 1/2(1 + α P cosθ_l),where α is the spin analysing power of the decay product, which for charged leptons is α(ℓ^±) = ± 0.998 <cit.>.ATLAS has recently measured <cit.> a set of angular asymmetries that are sensitive to the top-quark polarisation and six independent W boson spin observables <cit.>. The measurements use events with one high electron or muon and exactly two jets (one of which must be identified as originating from a b-quark). Selection requirements are imposed to reject the background from W+jets and events. The measured angular forward-backward asymmetriesA_FB = N(cosθ > 0) - N(cosθ < 0)/N(cosθ > 0) + N(cosθ < 0)for two angles θ_l and θ_l^N are related to the top polarisation (P) and the W boson spinobservable ⟨ S_2⟩ according to: P = 2 A^ℓ_FB/α and ⟨ S_2⟩ = -4/3 A^N_FB. The values of the observables extracted from the asymmetries are P = 0.97± 0.12 and ⟨ S_2⟩ = 0.06 ± 0.05. Good agreement is seen between the data and the SM predictions. The CMS experiment has previously measured the polarisation in single top events, finding P=0.52±0.22 <cit.>. The measurementagrees within two standard deviations with the SM prediction of 0.9. §.§ Constraints on the Wtb vertex If the energy scale of new physics is not directly accessible in top quark production at the LHC, then the impacts of new physics can be parameterised in the effective operator formalism and the most general Wtb Langrangian can be written as:L_Wtb = - g/√(2) bγ^μ ( +)tW^-_μ- g/√(2) b iσ^μνq_ν/ ( + )tW^-_μ + h.c.The termsandare the left- and right-handedvector and tensor couplings, respectively.[ In Equation <ref>, g is the weak coupling constant,and q_ν are the mass and the four-momentum of the W boson, respectively, ≡ (1∓γ^5)/2 are the left- and right-handed projection operators, and σ^μν =[γ^μ, γ^ν]/2. ] In the SM at tree-level, V_L is the CKM matrix element V_tb and the anomalous couplings g_L, V_R and g_R are all zero. The W boson and top-quark polarisation measurements discussed in the previous sections have been used to place limits on the anomalous couplings.The single top polarisation measurement is mainly sensitive to the imaginary part of g_R. The ATLAS measurements of the angular asymmetries for θ_l and θ^N_l are used in conjunction with analytical expressions <cit.> to extract limits on Im g_R. The correlation between the two asymmetry measurements (-0.05) is accounted for in the limit setting procedure. The limits set at the 95% confidence level are Im ∈ [-0.18, 0.06]. The CMS experiment has designed a dedicated analysis to search for anomalous couplings in single-top events <cit.>, where multivariate classifiers are used to separate the SM single-top events from potential contributions from non-zero anomalous couplings. No significant excess is seen and limits are set on different combinations of couplings. Figure <ref> shows the limits set in the and plane.The measurement of the W boson polarisation fractions by ATLAS has been used to set limits on the real parts of the anomalous couplings using the EFTfitter tool <cit.>.Figure <ref> shows the limits set on the and couplings, under the assumptions =1 and =0. The different precision measurements sensitive to the Wtb vertex are complementary. In the interpretations done by the collaborations to date, it has always been necessary to set at least one of the couplings to the SM values. This motivates future combinations of these precision measurements, in order to obtain constraints that are free of SM assumptions and to exploit the complementary sensitivity of the different measurements.§ PRODUCTION OF TOP QUARKS IN ASSOCIATION WITH VECTOR BOSONS The large integrated luminosity delivered by the LHC allows the possibility to study the rare production of top-quark pairs in association with either a Z or W boson (+V). The production of +Z is particularly interesting, since it probes the top-Z coupling. ATLAS and CMS have both measured the +Z and +W cross-sections using the 13 TeV data <cit.>. The ATLAS measurement uses the 2015 dataset (corrsponding to a luminosity of 3.2 fb^-1), while the CMS measurement uses the data collected during the first half of 2016, which corresponds to an integrated luminosity of 12.9 fb^-1. The measurements are limited by statistics and hence this report will focus on the more precise CMS measurement.The +V processes can produce final states with multiple-leptons and jets originating from b-quarks. Both experiments select events with either two leptons (electrons or muons) with the same-sign charge, three leptons or four leptons. For the dilepton channel, CMS selects events with at least 2 jets, at least one of which is identified as being likely to have originated from a b-quark (referred to as a b-jet) and then uses a multivariate technique to separate the signal from the background. To maximise the signal significance, the dilepton events are further categorised according to the dilepton charge and the number of jets and b-jets. The trilepton events are required to have at least 2 jets and events where a same-flavour opposite-sign charge (SFOS) lepton pair has an invariant mass close to the Z boson mass are rejected. The events are then categorised according to the number of jets and b-jets into twelve disjoint signal regions. The tetralepton events are required to have one SFOS lepton pair consistent with a Z boson and to contain at least two jets. In the μμμμ, eeee and μμ ee channels, events where the second SFOS lepton pair is consistent with a Z boson are rejected to reduce the background from ZZ events. Events are categorised according to whether or not they contain at least one b-jet.The background predictions use data-based methods for the the backgrounds where at least one lepton originates from non-prompt sources (leptons from heavy-flavour hadron decay, misidentified hadrons, muons from light-meson decay in flight, or electrons from unidentified photon conversions), while other background sources are estimated using simulation. The modelling of the backgrounds is checked in control regions. Figure <ref> shows the number of events in each lepton flavour channel for a control region that selects events with three leptons and at most one jet. This region is dominated by WZ events and good agreement is seen between the data and the background prediction.The +W and +Z cross-sections are extracted by making a combined fit to all the different signal regions. The data observed in the trilepton signal region with at least four jets are shown in Figure <ref>, where good agreement is seen between the data and the SM prediction. The measured cross-sections are shown in Figure <ref> for both the ATLAS and CMS analyses. Agreement is seen between the data and the SM predictions. The measurements are dominated by the statistical uncertainty and hence measurements utilising larger datasets are eagerly awaited.§ THE TOP QUARK MASS The top quark mass is a fundamental parameter in the SM and precisely measuring its value is vital for establishing the consistency of the SM <cit.>. ATLAS has recently performed a measurement using top-quark pair events in the dilepton channel <cit.>. The events are selected from the 2012 data taken at √(s)=8 TeV and the dataset corresponds to an integrated luminosity of 20.2 fb^-1. The selection requires exactly two leptons (either electrons or muons) and at least two jets, where at least one of the jets is required to be a b-jet. Additional requirements on the missing transverse momentum, the invariant mass of the lepton pair and the scalar sum of the of the selected jets and leptons are applied to reduce the background from Z+jets events. The dilepton channel has the drawback that the prescence of two neutrinos makes the full reconstruction of the system challenging. The two jets with the highest probability to originate from b-quarks are taken as originating from the two top quarks. There are then two possible assignments of the leptons and b-jets. The combination that leads to the lowest average invariant mass of the two lepton-b-jet pairs () is selected. The top mass can then be extracted from the distribution. A phase-space restriction on the average of the two lepton-b-jet pairs (p_T,ℓ b) is used to obtain the smallest total uncertainty in . The selected requirement is p_T,ℓ b > 120 GeV and effectively selects a region where the systematic uncertainties are reduced.The top-quark mass is extracted by performing a template fit to the distribution. The signal templates are constructed by fitting Monte Carlo samples generated with different top quark masses with the sum of a Landau function and a Gaussian distribution. Figure <ref> shows the templates for three different values, demonstrating the sensitivity of the distribution to the top-quark mass. The figure also shows the data compared to the template with the best fit value of the top-quark mass and good agreement is seen between the data and the fitted template. The top-quark mass is measured to be = 172.99 ± 0.41 ± 0.74 GeV, where the first uncertainty is the statistical uncertainty and the second is the total systematic uncertainty. The systematic uncertainty is dominated by the understanding of the jet energy scale (0.54 GeV), the MC modelling of top-quark pair events (0.35 GeV) and the jet energy scale for jets originating from b-quarks (0.30 GeV). The measurement is the most precise measurement of the top-quark mass in dilepton events to date.The CMS collaboration has recently made the first measurement of the top-quark mass using the 13 TeV LHC data <cit.>. The analysis uses events with one muon and at least four jets (of which two must be b-jets) and the strategy closely follows the lepton+jets analysis that used the run 1 data <cit.>. A kinematic fit is applied to the selected events, where the fit constrains the W boson mass to 80.4 GeV <cit.> and the top and anti-top masses to be the same. The goodness-of-fit probability (gof) of the kinematic fit is required to be at least 0.2 in order to remove poorly reconstructed events.The top mass reconstructed from the kinematic fit (m_top^fit) is used along with the reconstructed W boson mass in a likelihood-fit to simultaneously extract the top-quark mass and the jet energy scale factor (JSF). The JSF allows the overall energy scale of jets to be constrained using the reconstructed W boson mass. In the likelihood-fit, events are weighted by their gof in order to reduce the impact of poorly reconstructed events. The reconstructed top mass distribution is shown in Figure <ref>, which shows the excellent mass resolution and good agreement between the data and the simulation. The top quark mass is measured to be = 172.62 ± 0.38 ± 0.7 GeV, where the first uncertainty is statistical and the second is the total systematic uncertainty. The fitted JSF is in agreement with 1, as shown in Figure <ref>. The largest systematic uncertainties are from the jet energy scale (0.51 GeV) and the MC modelling of top-quark events (0.4 GeV). While the total uncertainty does not reach the precision achieved in run 1 <cit.>, the understanding of the 13 TeV data is still at an early stage compared to run 1 and improvements in the uncertainties in the future can be anticipated.§ SUMMARY The heavy mass of the top quark provides the opportunity to make precision measurements of a `bare' quark. Recent measurements of W boson polarisation fractions in top-quark pair events and of top-quark polarisation in single-top quark events have probed the structure of the Wtb vertex. The measurements are all in agreement with the SM, with no signs of new physics. The high energy and large datasets provided by the LHC allow to measure the rare +Z and +W processes. The recent CMS measurement with 13 TeV data shows evidence for both processes and the larger datasets expected in the years ahead will allow for precision tests of the top-Z coupling. The top-quark mass has been measured to high precision with the LHC data. The recent dilepton measurement from ATLAS utilises a kinematic phase space selection to reduce the systematic uncertainties and CMS has made the first run 2 top mass measurement. Future prospects include a combination of the LHC run 1 results and precise measurements with the run 2 data.§ REFERENCES 99ATLASdet ATLAS Collaboration, JINST 3 S08003 (2008).CMSdet CMS Collaboration, JINST 3 S08004 (2008).Aaboud:2016hsq ATLAS Collaboration,Eur. Phys. J. C77 (2017) 264,arXiv:1612.02577 [hep-ex].Khachatryan:2016fky CMS Collaboration,Phys. Lett. B762 (2016) 512,arXiv:1605.09047 [hep-ex].STpol1 G. Mahlon and S. Parke,Phys. Lett. B476 (2000) 323, arXiv:hep-ph/9912458].STpol2 R. Schwienhorst, Q.-H. Cao, C.-P. Yuan and C. Mueller,Phys. Rev. D 83 (2011) 034019, arXiv:1012.5132 [hep-ph].STpol3 A. Brandenburg, Z. G. Si and P. Uwer,Phys. Lett. B539 (2002) 235, arXiv:hep-ph/0205023].Aaboud:2017aqpATLAS Collaboration,JHEP1704, 124 (2017), arXiv:1702.08309 [hep-ex].STpol4 J. A. Aguilar-Saavedra and J. Bernabu,Phys. Rev. D93 (2016) 011301, arXiv:1508.04592 [hep-ph].Khachatryan:2015dzz CMS Collaboration,JHEP1604 (2016) 073,arXiv:1511.02138 [hep-ex].STpol5 J. A. Aguilar-Saavedra and J. Bernabu,Nucl. Phys. B840 (2010) 349, arXiv:1005.5382 [hep-ph].STpol6 J. A. Aguilar-Saavedra and S. Amor dos Santos,Phys. Rev. D89 (2014) 114009, arXiv:1404.1585 [hep-ph].Khachatryan:2016sib CMS Collaboration,JHEP1702 (2017) 028,arXiv:1610.03545 [hep-ex].EFTfitter N. Castroet al.,Eur. Phys. J. C76 (2016) 432, arXiv:1605.05585 [hep-ex].Aaboud:2016xve ATLAS Collaboration,Eur. Phys. J. C77 (2017) 40,arXiv:1609.01599 [hep-ex].CMS-PAS-TOP-16-017 CMS Collaboration,CMS-PAS-TOP-16-017 (2016), <https://cds.cern.ch/record/2205283>.Baak:2014ora M. Baaket al. [Gfitter Group],Eur. Phys. J. C74 (2014) 3046,arXiv:1407.3792 [hep-ph].Aaboud:2016igd ATLAS Collaboration,Phys. Lett. B761 (2016) 350,arXiv:1606.02179 [hep-ex].CMS-PAS-TOP-16-022 CMS Collaboration,CMS-PAS-TOP-16-022 (2017), <https://cds.cern.ch/record/2255834>.Khachatryan:2015hba CMS Collaboration,Phys. Rev. D93 (2016)072004,arXiv:1509.04044 [hep-ex].pdg K. A. Olive,Chin. Phys. C 40 (2016) 100001. | http://arxiv.org/abs/1705.09089v1 | {
"authors": [
"Mark Owen"
],
"categories": [
"hep-ex"
],
"primary_category": "hep-ex",
"published": "20170525082241",
"title": "Top quark properties measurements at the LHC"
} |
Comment on “Are two nucleons bound in lattice QCD for heavy quark masses? - Sanity check with Lüscher's finite volume formula -” Frank Winter December 30, 2023 ================================================================================================================================We develop two-dimensional Brownian dynamics simulations to examinethe motion of disks under thermal fluctuations and Hookean forces.Our simulations are designed to beexperimental-like, since the experimental conditions define the availabletime-scales which characterize the solution of Langevin equations.To define the fluid model and methodology,we explain the basics of the theory of Brownian motion applicable to quasi-twodimensional diffusion of optically-trapped microspheres.Using the data produced by the simulations,we propose an alternative methodology to calculate diffusion coefficients. We obtain that, using typical input parameters in video-microscopy experiments,the averaged values of the diffusion coefficient differ from the theoretical one less than a 1%. § INTRODUCTION Brownian motion, i.e., the random movement of objects immersed in a fluid,was theoretically described by Einstein more than a century ago <cit.> from a microscopic perspective,demonstrating the molecular structure of the fluid <cit.>.The Einstein's classical approach neglected hydrodynamics memory and inertia effects,since they appear at very short-time scales,something experimentally available only very recently <cit.>. This assumption theoretically implies that the particle velocity cannot be defined and the trajectories of Brownian particles are fractal <cit.>. Therefore, the study of Brownian motion is determined by the available experimental set-up, which defines the detected time-resolution of the stochastic jumps.The standard experimental methodology to study Brownian motion is to mix a small concentration of micro-nanospheres with a certain fluid. The suspension sample is deposited into a glass cell which is insertedin an optical instrument, such as a video-microscopy or an interferometry set-up,where the trajectories of the beads can be recorded for ulterior analysis.A common practice to facilitate the study of particles' motion is using optical tweezers <cit.>.This technique exerts a restoring force under the object,allowing the experimentalist to move the particle inside the fluid in a quasi-twodimensional plane.Many optical tweezers set-ups allow to trap several objects,but single-particle tracking is usually employed to improve spatial and temporal resolution.During the last decades, optical traps have permitted to developa wide variety of experiments in colloidal motion. To cite some examples: about the effectscaused by confinement <cit.>,the hydrodynamic interaction between particles <cit.>,discovering resonances from hydrodynamic memory at short-time scales <cit.>,regarding micro-rheology <cit.>, or even to produce Brownian Carnot engines <cit.>,along with many other applications <cit.>.In spite of the evident benefits of optical tweezers in the research of colloidal physics,this experimental methodology generates an external force under the bead which can modify the dynamics of the particle <cit.>. This force can be also itself modified by the thermo-physical properties of the surrounding complex fluid <cit.>.Therefore, a good dynamical characterization of the external harmonic potentialdetected by stochastic particle motion is needed to correctly measure the values of the diffusion coefficient.Here, we study by experimental-like computer simulations the diffusion of a single sphereobserved in a two-dimensional plane under optical tweezers, i.e.,we investigate the Brownian motion of a disk under harmonic potentials. In this work, we show how the solution of the Fokker-Planck equation <cit.> allows us to propose an iterative approach as an alternative methodologyto calculate the diffusion coefficient of the disk. Our objective in this work is to emulate the dynamics of a trapped single-bead in avideo-microscopy experiment by means of Brownian dynamics simulations.These simulations are designed to be experimental-like using typical input parametersbut without the limitations which appear in an experimentalset-up, like image analysis miscalculations or confinement effects in the bead's diffusion. § BROWNIAN DYNAMICS SIMULATIONS We develop Brownian dynamics simulations,which are a simplification of Stokestian dynamics,but neglecting hydrodynamic interactions (HI) between particles <cit.>. Our model of colloidal fluid is designed to be compared with video-microscopy (VM) experiments,where we can observe real-time motion of the colloids in two dimensionsand where we can storage the particles' position for defined temporal steps, according tothe frame-rate of the camera.The software simulates a suspension of micro-spheres in a Newtonian fluid in a 2D or pseudo-2D configuration of sedimented micro-particles <cit.>. In analogy with a image-based VM lab,we are able to change external parameters, expressed in physical units,such as the concentration of particles in the suspension,the viscosity of the fluid, the focal distance, the size of the spheres or the temperature of the bath. Our simulation model has been implemented using an open-source Java-based software named “Easy Java/JavaScript Simulations” (EjsS) <cit.>,allowing to create visual simulations of physical systems based on ordinary differential equations.The equations described in this section have been resolved numerically by EjsSusing a Euler-Richardson algorithm —alternativealgorithms are available but they provided the same results.This simulation methodology has been successfullydeveloped and tested in more complex systemscomposed of many Brownian particles under different internal and external forces <cit.>.Theoretically, the movement of particles under thermal fluctuations is studied by means of the Langevin equation <cit.>,which is the Newton's second law equation including a stochastic force F_B:m v̇ = F_H+F_B+F_E+F_Dfor a sphere of radius a and mass m immersed in a medium of viscosity η and density ρ. In eq. (<ref>), v is the velocity vector, v̇≡ dv/dt,F_H are the hydrodynamic forces, F_E are externalforces over the particles in the fluid, and F_D are hard-disk forces that avoid the particles from overlapping. We neglect full hydrodynamic interactions, v̇∼ 0, and, therefore,the hydrodynamic contribution is reduced to F_H = -γ v, i.e., the Stokes drag of an isolated particle, where γ = 6πη a is the friction coefficient.This inertial term can be neglected by evaluating the Reynolds number, Re, which compares inertial and viscous forces. For a bead of radius a immersed in a fluid of viscosity η and density ρ, we have Re= ρ v a / η.For micro-particles in water-like fluids, Re is low enough, allowing us to exclude hydrodynamic interactions when the experimental time-scale is not very low —in the order of the microsecond.<cit.> When using HI, the lubrication forces prevent the particles to overlap.If not, hard-disk forces have to be included. In this work, we use single-particle configuration, and the hard-disk forces are not necessary.The Brownian or stochastic force is characterized by ⟨F_B⟩ =0 and by ⟨F_B(0) F_B(t) ⟩ = 2 k_B T γ δ(t) where k_B is the Boltzmann's constant and δ(t) is the unit tensor <cit.>.To implement this force, we use a random vector 𝐧 with values in the interval [-1,1] generated by the Box-Muller transformation <cit.>. Then, we have:F_B= √(2d k_B T γ / dt) 𝐧where here d is the dimension, and dt will be the time step in the simulations.Regarding the external forces,we use a restoring Hooke-like one: F_E = -κr, where κ is the trap stiffness andassuming that the trapped disk is centered in its initial position. We do not include any difference between x and y coordinates,and we can define the harmonic potential in one-dimension x asU_κ = (1/2) κ x^ 2. This potential is the harmonic approximationfor a trapping potential generated by the focused laser beam of the optical tweezers,which is valid for the central region of the potential well <cit.>.Under this conditions, we have the following overdamped Langevin equation for the trapped disk:ṙ = γ^-1 (-κ_r r+F_B)where the stochastic term is given by eq. (<ref>).Finally, we make the Stokes-Einstein relation for self-diffusion coefficient explicit <cit.>:D = k_B T γ^-1This diffusion coefficient, D, is the quantity we want to obtain by analyzing the data ofthe simulations.The most simple version of micro-rheology consists in estimating the value of the fluid's viscosity, η,by calculating D from the analysis of particle motion in the fluid.§ HARMONIC POTENTIALS IN STOCHASTIC MOTION The solution of the complete set of Langevin equations for a spherical particle in harmonic potentials with no-slip boundaries in a Newtonian fluid and with hydrodynamic effects <cit.>only depends on several timescales: τ_f ≡ρ_f a^2/η, τ^*_p ≡ m^*/γ and τ_κ≡γ/κ. where ρ_f is the density of the fluid, m_p and the mass of the particle andm^*≡ m_p+m_f/2 is a modified mass influenced by hydrodynamics,where m_f is the mass of the displaced fluid.The two first time scales are related to fluid vortex propagation and inertial time-scales,whereas τ_κ measures the ratio between the Stokes friction coefficient and the optical trap constant, κ. The most applied statistic magnitude in the analysis of Brownian motion is the mean-square displacement (MSD),defined by:MSD(t) ≡⟨Δr^2(t)⟩≡⟨ (r(t)-r(0))^2⟩The MSD for a micro-sized particle immersed in a Newtonian fluid,in time scales lower than τ_κ, behaves as MSD(t) = 2dD t, where d is the dimension on the MSD. This expression defines the diffusive behavior and allows to calculate the diffusion coefficient D. At even lower timescales, the ballistic regime is predominant initiating a temporal power-law behavior,where MSD(t) ∼ t^2.At higher times, when t∼τ_κ, the optical trap is dominant,and the MSD shows a plateau equal to MSD(t > τ_κ)=2k_B T/κ.The one-dimensional MSD of a harmonically trapped bead in a Newtonian fluid, without inertia effects, is <cit.>:⟨Δ x^2(t) ⟩= 2 k_B T/κ (1-e^-κ t/γ)A standard method to obtain the self-diffusion coefficient is to fit eq. (<ref>) to the MSD data calculated from the disk's positions.An important statistical quantity is the probability distribution of the particles' jumps, Δr,also called Gaussian propagator or van Hove autocorrelation function <cit.>:ρ_D (Δr, τ)= 1/(4 π D τ)^d/2 exp(-Δr^2/4Dτ)for a fixed and constant time-lapse τ between particles' jumps in d dimensions. If we assume there is no difference in the diffusion process by using τ (lapse time) or t (absolute time),the moments of the distribution can be obtained from the propagator byδr^n(t) ≡< |Δr(t)|^n> = ∫ |Δr|^n ρ_D(Δr,t) d^d (Δr). And then, the MSD is calculated using n=2, obtaining δr^2(t) ≡MSD(t) = 2d D t.The diffusion coefficient can be extracted from the one-dimensional variance of the Gaussian distribution (<ref>) since σ_x^ 2 =2Dτ. A similar approach can be applied to the trap stiffness through Boltzmann statistics <cit.> on the particles' positions. The average motion of a trapped particle can be described by means of the probability density distribution ρ(x,t) which obeys the Fokker-Planck equation <cit.>. This can be written <cit.> in a simple one-dimensional form asdρ(x)/dx = (k_B T)^-1 F(x)ρ(x).If we use an optical trap modeled as a restoring force with spring constant κ, the solution is a Gaussian function on the coordinate x: ρ_κ (x) = (κ/2π k_B T)^1/2exp(-1/2κ x^2/k_B T)The variance of the distribution allows to obtain the stiffness of the trap by σ_κ^ 2 = k_B T/κ. An example of these one-dimensional distributions can be seen in Fig. <ref>. In a more general case of stochastic motion confined by harmonic potentials,the solution of the Fokker-Plank equation leads to a more complicated distribution <cit.>.In that situation, the probability of transition of a trapped colloidal particle fromthe position x_0 to x in lapse-time τ is:ρ_FP(x_0, x, τ) =1/√(2πα(τ))exp[ - (x - x_0 e^-λτ)^2/2 α(τ)]where:λ ≡κ / γ α(τ)≡k_B T/κ(1 - e^-2λτ)Note that λ can be written in terms of the diffusion coefficient as λ = κ D/k_B T by using eq. (<ref>) and (<ref>). For short values of the time-step, τ≪ 1/λ,this distribution is identical to the standard Gaussian propagator, (<ref>). On the other hand, for long values of the temporal step, it approximates to distribution of particle positions, (<ref>). For intermediate times, the distribution depends of a memory factor which appears in the initial position for every jump of the particle.To obtain that memory factor, e^-λτ, first we need to know the diffusion coefficient, D,which is the quantity we want to estimate from the disk's movement. It is important to observe that, when τ≪ 1/λ, we obtain α(t) = 2Dt. Eq. (<ref>) only differs from the standard MSD, eq. (<ref>), by a factor 2 in the spring constant (κ→κ/2). This will allow us to define a relaxed MSD through the temporal jumps with memory effect,which we will identify with eq. (<ref>).§ RESULTS We use the model previously explained, where the disk'spositions are obtained following the overdamped Langevin equation, eq. (<ref>), to analyze the diffusion of a trapped simulated disk, always under standard experimental conditions, when we the increase the stiffness of applied the restoring force. The input data are those of a typical experiment using commercial optical tweezers:trapped particle of diameter d= 1.9 μm, at temperature T=295.5 K,time-lapse τ=2.5 ms (400 images per second in the video-microscopy set-up),during a total time of 50 s.The internal time step in the simulations, dt, is fixed to dt=10^-4 s.The typical stiffnesses of the traps are κ∼ 0.02-2 μN/m. The surrounding fluid is water (η=0.95 mPa.s) and the beadis located far enough from the influence of nearby walls <cit.>.Using these experimental input values, we are in the intermediate situation of the Fokker-Planck solution described in the former section. Indeed, 1/λ∼ 9-840 ms is obtained in the interval of typical κ,not far from the input time-step, τ=2.5 ms. Here lies the importance of using experimental-like simulations,since Brownian motion theoretical explanations depend on the time scale of observation. Under this conditions, the theoretical diffusion coefficient is D = 0.239 μm^2/s according to eq. (<ref>). To have into account this effect, we develop an alternative iterative method to obtain the diffusion coefficient.From a step n to a total number of steps N, we define a relaxed mean-squared displacement, MSD^*(t):MSD^*(t) ≡⟨Δ x_n,λ^2 ⟩≡1/N-n∑_i (x_n+i - x_i e^-λ n τ)^2where this MSD should verify eq. (<ref>).By using here that σ_κ^2 ≡ k_B T/κ, we can define a function ξ(nτ):ξ(nτ) ≡ - log( 1 - ⟨Δ x_n,λ)^2 ⟩/σ_κ^2) = 2λ n τThe function ξ(nτ) grows linearly with time with slope 2 λ,something which allows to obtain λ from a linear regression.By calculating κ independently from the distribution ρ_κ (r) (<ref>) (as in Fig. <ref>)or by the plateau in the MSD,and applying an iterative method until obtaining a stable λ value.In Fig. <ref>, we show an example of one-dimensional standard MSD calculation, eq. (<ref>)and the relaxed MSD^* defined by eq. (<ref>),for several values of the input trap stiffness κ_o. Both quantities have a similar behavior,displaying a plateau at higher times when reaching the time-scale τ_k.However, the MSD^* shows a correction on this plateau, which increases when the restoring force is more intense. In the Inset of Fig. <ref>, the linear behavior of the defined function ξ(t), eq. (<ref>),can be seen, allowing to obtain λ and, consequently, D. In Table <ref>, we show a complete set of the values for self-diffusion coefficientsin the range of the trap stiffnesses used in video-microscopy experiments.We summarize the values for the self-diffusion coefficientscalculated using the standard MSD and eq. (<ref>), D, and through the iterative method, D^*. The data shown is an average of the calculations for coordinates x and y,which have been analyzed separately after simulations. We also show the κ values obtained from thesimulated data of the position of the disk to be compared with the input data.This comparison between κ_o and κ allows to understand the reach of the experimental-like simulations, which are statistically limited,as it should be when measuring with a standard video-microscopy set-up.The coefficient of diffusion values do not depend on the κ value,allowing to calculate an average value for D from the data of Table <ref>. The averages provide ⟨ D^* ⟩ =0.241 ± 0.008 μm^2/sand ⟨ D ⟩ =0.238 ± 0.007 μm^2/s. Both values nicely agree with the theoretical value, with a relative error lower than 1%. § CONCLUSIONS We have developed experiment-like Brownian dynamics simulations of two-dimensional disks under harmonic potentials to evaluate the reach of experiments of microsphere diffusion with optical tweezers,which are modeled as external Hookean forces. These type of computational studies, based on emulating experimental set-ups,are quite useful to design time- and cost-efficient experimental procedures. After summarize the basic theory of Brownian motion applied to the relevant time-scale,we observe that the solution of the Fokker-Planck equation to this stochastic system allows us to modify the standard definition of the mean-square displacement by including a memory termin the initial position of the disk's jumps.Based on this relaxed MSD, we propose an alternative method to obtain the self-diffusion coefficient. The values calculated through that method are compared to the self-diffusion coefficients obtained using the standard mean-square displacement. Under the experimental conditions, the averaged values of the diffusion coefficients obtained from both methods return values which differfrom the theoretical less than 1%.§ ACKNOWLEDGMENTS We want to thanks F. Ortega for joined investigations with optical tweezers,J.A. 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Residual Expansion Algorithm: Fast and Effective Optimization for Nonconvex Least Squares Problems Daiki Ikami Toshihiko Yamasaki Kiyoharu Aizawa The University of Tokyo, Japan {ikami, yamasaki, aizawa}@hal.t.u-tokyo.ac.jp =============================================================================================================================================== We propose the residual expansion (RE) algorithm: a global (or near-global) optimization method for nonconvex least squares problems. Unlike most existing nonconvex optimization techniques, the RE algorithm is not based on either stochastic or multi-point searches; therefore, it can achieve fast global optimization. Moreover, the RE algorithm is easy to implement and successful in high-dimensional optimization. The RE algorithm exhibits excellent empirical performance in terms of k-means clustering, point-set registration, optimized product quantization, and blind image deblurring. § INTRODUCTION Many problems in computer vision and machine learning can be formulated as optimization problems. If we can formulate a problem as a convex optimization, we can solve it by convex optimization techniques such as gradient-based methods. However, most optimization problems are nonconvex and often have many local minima. In these cases, convex optimization techniques can find only local minima. Global optimization of nonconvex problems is an NP-hard problem in most cases. Therefore, heuristic methods are often used to find a global (or near-global) optimum. There are two major approaches: good initialization and stochastic optimization. The former is fast and effective if we can obtain a good initial guess <cit.>; however, many optimization problems do not provide this. The latter is random search or multiple-point search, which is represented by simulated annealing (SA) <cit.>, particle swarm optimization (PSO) <cit.>, and genetic algorithms (GA) <cit.>. Although these methods are effective with low-dimensional optimization problems, it is difficult to obtain good solutions with high-dimensional ones. Moreover, these approaches often require excessive computation time to obtain a good solution. In this paper, we propose a fast and effective optimization method for nonconvex least squares (LS) problems such as k-means clustering and point-set registration. First, we propose a novel measure of convergence called RE convergence: this represents how far we can expand data points along their residual directions under convergence. fig:RE_fig1 shows k-means results and expanded data points. fig:RE_fig1_1 depicts convergence on expanded data while fig:RE_fig1_2 shows a case that is not converged. We presume that RE convergence is associated with global convergence. In fact, we can prove that the solution that is stable on a large expansion is the global optimum in the case of a one-dimensional quartic minimization problem. Additionally, we propose a heuristic algorithm to find a solution that is stable on the large expansion, which we term the residual expansion (RE) algorithm. This algorithm is based on neither multiple-point search nor random search, and thus fast computation can be achieved. Our contribution is as follows: * We propose a novel concept of convergence, RE convergence. We show the relationship between RE convergence and the global optimum. * We propose the RE algorithm, which can be applied for any nonconvex LS problem. We show that the RE algorithm is fast, effective, and easy to implement. * We show the RE algorithm's excellent performance for various nonconvex LS problems such as k-means clustering, point-set registration, optimized product quantization, and blind image deblurring. § RELATED WORKS §.§ Nonconvex least squares problems We focus on nonconvex LS problems, of which many exist. In this paper, we study the following four important problems in computer vision and machine learning. §.§.§ K-means clustering K-means clustering is one of the most popular clustering methods with various applications such as quantization <cit.>, feature learning <cit.>, and segmentation <cit.>. K-means clustering assigns data vectors x_1,…,x_n∈ℝ^d to the nearest representative clusters. The optimization problem can be formulated as min_C,Z1/2‖X-CZ‖_F^2 z_ij = {0,1}, ‖z_i‖_1 = 1, where X=[x_1,…,x_n]∈ℝ^d× n is a data matrix, C=[c_1,…,c_k] ∈ℝ^d× k is a matrix of cluster centroids, and Z = [z_1,…,z_n]∈ℝ^k× n is an assignment matrix. The most popular optimization method is Lloyd's algorithm <cit.>, which has an update step (fix Z and update C) and an assignment step (fix C and update Z). Hartigan's algorithm <cit.> achieves better clustering than Lloyd's algorithm because the set of local minima of Hartigan's algorithm is a subset of those of Lloyd's algorithm <cit.>. For good initialization, k-means++ <cit.> is often used because of its efficiency and effectiveness. §.§.§ Point-set registration Point-set registration is a fundamental problem in computer vision. Here we consider a rigid 3D-point-set registration problem: Given source point sets X=[x_1,…,x_n]∈ℝ^3× n and target point sets Y=[y_1,…,y_m] ∈ℝ^3× m, we estimate the best rigid transformation parameters. In this paper, we consider the following optimization problem with a point-to-point cost function: min_R,t,Z1/2‖RX+t1^⊤-YZ‖_F^2z_ij = {0,1}, ‖z_i‖_1 = 1, R^⊤R=I, where R∈(3) is a rotation matrix, t∈ℝ^3 is a translation vector, and Z = [z_1,…,z_n]∈ℝ^k× n is an assignment matrix. I is an identity matrix and 1 is a vector of all ones. The iterative closest point (ICP) algorithm <cit.> is a well-known alternating optimization method: it fixes Z and updates R, t, and then fixes R, t and updates Z. To obtain a global minimum, some studies adopt stochastic optimization, such as GA <cit.>, PSO <cit.>, and SA <cit.>. Recently, Yang proposed Go-ICP <cit.>, which guarantees global optimality by using the branch-and-bound algorithm. However, it requires significant computation time. §.§.§ Optimized product quantization Optimized product quantization (OPQ) <cit.>, which is an extension of product quantization (PQ), is an efficient fast approximate nearest neighbor search method. The optimization problem in OPQ is described by min_R,C,Z1/2∑_i=1^N‖x_i-R[ [ C^(1)z_i^(1);⋮; C^(M)z_i^(M) ]]‖_2^2z_ij^(m) = {0,1}, ‖z_i^(m)‖_1 = 1,R^⊤R=I, where X,C,Z have the same meaning as in Section 2.1.1 and R is a rotation matrix. The optimization problem of eq:OPQ can be solved by alternating optimization of R, C, and Z <cit.>. Ge also proposed a parametric optimization method that assumes the data follows a parametric Gaussian distribution <cit.>. §.§.§ Blind image deblurring Blind image deblurring has long been a challenging problem in computer vision. From a blurred image B∈ℝ^h× w, we estimate an original image I∈ℝ^h× w and blur kernel k∈ℝ^k× k by minimizing the following cost function: min_I,k1/2‖I⊗k-B‖_F^2 + γ_I R_I(I) + γ_k R_k(k), where R_I(I) and R_k(k) are the regularization terms, and ⊗ denotes the convolution operator. For R_I(I), L0-norm (or approximately L0-norm) <cit.>, or L1/L2 functions <cit.> are proposed to enforce the sharp edges of the original image. For R_k(k), L2-norm <cit.> or L1-norm <cit.> are often used. We refer to the paper <cit.> for a recent comparative study of blind image deblurring. We can minimize eq:BlindImageDeconv by alternating optimization of I and k. For fast and effective optimization, a coarse-to-fine strategy <cit.> is generally employed. §.§ Nonconvex optimization techniques Most nonconvex optimization techniques are based on stochastic optimization, including GA <cit.>, PSO <cit.>, and SA <cit.>. These methods generally do not work well or require significant computation time for high-dimensional optimization problems. Several studies <cit.> have employed these nonconvex optimization techniques to our target problems described in sec:nonconvexLS; however, these methods are not often used in practice. Our approach is related to graduated nonconvexity (GNC) <cit.>, which first solves a simplified optimization problem and then gradually transforms the problem into the original nonconvex problem. The basic concept of graduated optimization methods is extinguishing local minima by using a convexified objective function, and then gradually changing the objective function to the original function. We refer readers to <cit.> for a recent survey of graduated optimization. In contrast to GNC, our approach is explicitly to escape from poor local minima by using a largely expanded objective function and then gradually transforming it into the original function, as described in sec:REalgorithm. § RESIDUAL EXPANSION CONVERGENCE First, we describe RE convergence, which indicates how we can expand data along their residual directions. RE convergence is a measure of the depth of convergence, and our proposed algorithm is based on this concept. We show a relationship between the global optimum and RE convergence. We discuss a nonconvex least squares (LS) optimization problem whose objective function is given by E() = 1/2‖y-f()‖_2^2. Our definitions are as follows. Let ^* be a local minimum point of eq:LS. We define the α-expanded objective function E_α(): E_α() = 1/2‖ŷ-f()‖_2^2. where ŷ is constructed by expanding y in the residual direction with a magnitude of α as ŷ = y+α(y-f(^*)), We call the operation that constructs the α-expanded objective function residual expansion (with α). ^* is called α RE convergence if there exists a constant α≥0 such thatis still a local optimum of E_α(). In particular, the maximum (or supremum) constant is called the RE constant[If ^* is always a local minimum of E_α(^*) under all α≥0, we define the RE constant as ∞. ]. Our hypothesis is that a solution with a larger α-RE constant is closer to the global optimum solution. This hypothesis holds in the case of quartic minimization, as presented in section 3.1.1. §.§ Unconstrained and differentiable problems We consider one of the simplest cases: unconstrained and differentiable LS problems. Given a local optimum ^*, we can obtain first- and second-order derivatives of the α-expanded objective function E_α() at ^* as ∇ E_α(^*) = (1+α)J^⊤(^*)(y-f(^*)) = 0, ∇^2 E_α(^*) = J^⊤(^*)J(^*) + (1+α)S(^*). where J is a Jacobian matrix and S(^*) is S(^*) = ∑_i∇^2f_i(^*)(y_i-f_i(^*)). eq:RE_derivative1 means that ^* is always a stationary point of E_α(). Therefore, ^* is a local minimum of E_α() if and only if ∇^2 E_α() is a positive semi-definite (PSD) matrix. If S is not a PSD matrix, there is a α≥ 0 which satisfies the fact that ∇^2 E_α() is not a PSD matrix. fig:REConvergence shows examples of α-expanded objective functions. Residual expansion elevates the objective function around ^*, and if α is sufficiently large then it ceases to be a local minimum. One-dimensional quartic minimization: Here we consider a quartic minimization problem—in particular, one that can be formulated as an LS problem: E(θ) = 1/2((y_1-θ^2)^2+(y_2-θ)^2). We consider the case where eq:QuarticMinimization has two local minima θ_1 and θ_2. The following theorem then holds: Let θ_1 and θ_2 be local minima points of eq:QuarticMinimization with RE constants of α_1 and α_2, respectively. θ_1 is the global minimum point if α_1 > α_2 and θ_2 is the global minimum point otherwise. Please refer to the supplementary materials. §.§ General relationship between the α RE convergence and the global optimum It is not obvious when our hypothesis, i.e., that a solution with a larger RE constant is closer to the global optimum, is valid. Unfortunately, we can easily find a counterexample in k-means clustering, as shown in fig:CounterExample. However, our algorithm, which aims to find a solution with a large RE constant, works well from an empirical perspective in many nonconvex LS problems. § RESIDUAL EXPANSION ALGORITHM In this section, we propose the RE algorithm, which aims to find a solution with a large RE constant. Since it is difficult to find the solution with the largest RE constant exactly, we employ a heuristic strategy. The RE algorithm has two steps: parameter updating and residual expansion. We show an intuitive illustration of the algorithm in fig:REalgorithm. For the residual expansion step, we expand data along their residual direction. This results in elevating the objective function around the current solution as in fig:REConvergence. For the parameter-updating step, instead of minimizing the original function eq:LS, we minimize the following expanded objective function in each iteration: E_t() = 1/2‖ŷ^(t) -f()‖_2^2, where ŷ^(t) is an expanded data vector: ŷ^(t) = y+α^(t)r^(t), r^(t) =p^(t)(y-f(^(t)))+(1-p^(t))r^(t-1). where α and 0<p≤ 1 are expansion and momentum parameters, respectively. Note that, if p=1, eq:REalgorithm1 is an exactly α^(t)-expanded objective function on ^(t). The momentum parameter is important for achieving good performance and ensuring that the RE algorithm does not to diverge, as described later. The RE algorithm iterates through parameter updating by minimizing eq:REalgorithm1 and residual expansions by eq:REalgorithm2_1 and eq:REalgorithm2_2. We use a large α^(0) initially to find a solution with a large RE constant. Then we decrease α^(t) gradually to achieve convergence, analogous to a temperature parameter in SA. We summarize the RE algorithm in alg:REalgorithm. §.§ Parameter setting for convergence The RE algorithm has two parameters, α and p, for each iteration. We decrease α^(t) to 0 for convergence. when α=0, there is no residual expansion and RE algorithm is guaranteed to converge if the original LS problem has a convergence-guaranteed algorithm. However, inadequate parameters of α and p cause unstable optimization. We consider the norm of r^(t+1). We can obtain ‖r^(t+1)‖_2^2= ‖ p(y-f(^(t+1))) + (1-p)r^(t)‖_2^2 ∼ (1-p-α p)^2‖r^(t)‖_2^2. We use f(^(t+1))∼ŷ^(t) for the last approximation of eq:parameterSetting. eq:parameterSetting suggests (1-p-α p)^2≤1 to make the RE algorithm stable. A good way to determine these values of α and p is described in sec:ADMM. §.§ Advantages of the RE algorithm Our algorithm consists of two steps of parameter updating and residual expansion. Parameter updating is based simply on a typical LS problem approach. Therefore, if there is a source code which minimizes eq:LS, for example, by alternative optimization or gradient methods, then we can implement our algorithm by adding a residual expansion step to the existing code, which can be done in a few lines of code. Moreover, the computational complexity of residual expansion is generally less than that of parameter updating. Therefore, we can achieve faster optimization than most nonconvex optimization techniques based on multi-point search or random search, such as SA and GA. As described in sec:RelatedNonconvex, GNC is a similar approach to ours; however GNC often does not apply for LS problems. Our algorithm can be applied for any nonconvex LS problem provided that there is a method for finding a local optimum, such as Lloyd's algorithm for k-means clustering and ICP algorithms for point-set registration. § ALTERNATE DIRECTION METHOD OF MULTIPLIERS FOR LEAST SQUARES PROBLEMS In this section, we apply the alternate direction method of multipliers (ADMM) <cit.> to solve eq:LS. We show that ADMM is a special case of the RE algorithm for LS problems. Moreover, ADMM suggests a modified RE algorithm for regularized LS problems. We introduce an auxiliary variable z = y-f() and rewrite eq:LS as a constrained optimization problem: min_z,1/2‖z‖_2^2 z=y-f(). We can construct the augmented Lagrangian function of eq:LS_constrained as =0mu =0mu =0mu L_t(,z,) = 1/2‖z‖_2^2 + ^⊤(z-y+f()) + μ^(t)/2‖z-y+f()‖_2^2 . We take the alternating direction approach for solving eq:LR_LagrangeFunction and then obtain update rules as ^(t+1) = _ L_t(, z^(t), ^(t)) z^(t+1) = _z L_t(^(t+1), z, ^(t)) λ^(t+1)=λ^(t)+μ^(t)(z^(t+1)-y+f()^(t+1)) §.§ Relation to the RE algorithm We can simplify eq:ADMM_update1_1, eq:ADMM_update1_2 and eq:ADMM_update1_3 as =0mu =0mu =0mu ^(t+1)=_μ^(t)/2‖y+(1-μ^(t)/μ^(t))z^(t)-f()‖_2^2, z^(t+1)=(1/1+μ^(t))z^(t)+(μ^(t)/1+μ^(t))(y-f(^(t+1))). Details of the derivation are described in the supplementary material. This is a special case of the RE algorithm of eq:REalgorithm2_1 and eq:REalgorithm2_2 with α^(t) = (1-μ^(t))/μ^(t), p^(t) = μ^(t)/(1+μ^(t)). There are two main advantages to using ADMM. First, we can choose only μ instead of parameters α and p in the general RE algorithm. eq:relationADMMandRE_1 and eq:relationADMMandRE_2 always satisfy (1-p-α p)^2<1, which is a condition necessary for avoiding divergence to infinity, as described in sec:parameterSetting, and this update achieves good performance in experiments. Second, ADMM can treat regularized LS optimization problems, such as blind image deblurring (eq:BlindImageDeconv). We will describe this in the next section. §.§ Regularized least squares problems We consider a regularized LS problem as follows: E() = 1/2‖y-f()‖_2^2 + γ R(). When we apply the RE algorithm in a straightforward manner, we can obtain the following objective function in each iteration: E_t() = 1/2‖ŷ^(t)-f()‖_2^2+ γ R(). In the case of ADMM, from eq:ADMM_update2_1, the objective function is as follows: E_t() = μ^(t)/2‖ŷ^(t)-f()‖_2^2+ γ R(). We can find that the difference between eq:RegularizedLS2 and eq:RegularizedLS3 is simply the coefficient of the squared term. We find that minimizing eq:RegularizedLS3 achieves better performance than minimizing eq:RegularizedLS2. We summarize the RE algorithm based on ADMM in alg:REalgorithm2. § IMPLEMENTATION DETAILS We used the RE algorithm based on ADMM (alg:REalgorithm2) unless otherwise stated. In the update of(line 2 in alg:REalgorithm2), we perform alternating optimization with a single iteration; for example, with k-means clustering, the cluster centers and assignments are updated only once. The four problems we treat in this paper can be minimized by alternating optimization. For the parameter μ, we adapt μ^(t+1)=min(ρμ^(t),1), where ρ=exp(-log(μ^(0))/T) to satisfy μ^(T)=1. Therefore, we only need to determine the two parameters μ^(0) and T in our method. § EXPERIMENTAL RESULTS We evaluate the performance of the RE algorithm on four nonconvex LS problems: k-means clustering, 3D point set registration, OPQ, and single blind image deblurring. All experiments were executed on an Intel Core i5-4200U CPU (1.60 GHz) with 8GB of RAM, and were implemented in MATLAB[Our codes will be available if the paper is accepted.] except for Go-ICP <cit.>. For Go-ICP and its comparison experiment, we used the publicly available code[http://iitlab.bit.edu.cn/mcislab/yangjiaolong/go-icp/] implemented in C++. §.§ K-means clustering We compared our method with k-means++ <cit.>, which is a good initialization method, and Hartigan's algorithm <cit.>. For Hartigan's algorithm, we first used Lloyd's algorithm <cit.> with k-means++ initialization for fast computation. We reported the total time of Lloyd's algorithm and Hartigan's algorithm. For the other method, we used Lloyd's algorithm for optimization. We used random initialization for the RE algorithm. For error measurement, we used the objective function value of eq:kmeans and reported relative error from the value of k-means++ (therefore, the relative error of k-means++ is always 1). §.§.§ Synthetic data experiments We start with two synthetic datasets as shown in fig:KmeansSynthetic. We repeated each method 50 times from different initializations and report the average relative errors. tbl:KmeansSynthetic1 shows the results of our method with different μ^(0) and T. We found that larger T achieved better performance. We also found that smaller μ^(0) achieved better performance in dataset B; however, larger μ^(0) achieved better performance in dataset A. This indicates that the best setting μ^(0) is different for different data distributions. Intuitively, dataset B requires a larger residual expansion (in other words, small μ^(0)) to escape from a poor local minimum, while dataset A requires a smaller residual expansion. We show comparison results in tbl:KmeansSynthetic2. We repeated each method 50 times from different initializations. K-means++ worked well with dataset B. On the other hand, Hartigan's algorithm can improve the results of k-means++ in dataset A; however, this does not work in dataset B. The RE algorithm worked best in both cases, even though it was initialized by random seeding. Moreover, the RE algorithm with T=30 achieved comparable speed to k-means++ with better performance for dataset A. §.§.§ Real-world data experiments We used two real-world datasets for comparison: the cloud dataset[https://archive.ics.uci.edu/ml/datasets/Cloud] and the COIL20 dataset <cit.>. We performed experiments in the same manner as in sec:kmeansSynthetic. tbl:KmeansRealworld1 shows comparative results. In the cloud dataset, k-means++ achieves faster and better clustering than random seeding. The RE algorithm with T=30 achieved better clustering than k-means++ with about 1.8 times the computational cost. The RE algorithm with T=1000 performed best, and found the near-global optimum in every case. For the COIL20 dataset, although k-means++ and Hartigan's algorithm did not work well, the RE algorithm significantly outperformed the other methods. §.§ Point set registration We compared the RE algorithm with the ICP algorithm and Go-ICP <cit.>. Go-ICP is known as a method that can achieve global optimization. We used the bunny model from the Stanford3D dataset[http://graphics.stanford.edu/data/3Dscanrep/], as in fig:Bunny. For the target model, we used a partial point set as in fig:Bunny2. In the experiments, point sets were normalized within a cube of [-1,1]^3. We made a rotation matrix R_gt from a random rotation axis and the rotation angle ϕ. The target point set was constructed by this rotation matrix, and we added Gaussian noise with a standard deviation of σ=0.03. We performed 50 tests with different random rotation axes at each rotation angle ϕ=π/3, 5π/12, π/2. For measurement of the error, we used the objective value eq:ICP and regarded the results as successful if the objective error was less than 1 (this value is approximately twice of the average objective value using Go-ICP, as in fig:GoICPResults). We first show the comparison results between the RE algorithm and the ICP algorithm as in tbl:ICPResults1. The RE algorithm with T=30 achieved a better success rate with almost the same number of iterations as the ICP algorithm. Using a large T can improve the results to a small extent. We also compare our method to Go-ICP <cit.>. Go-ICP has two steps: the ICP algorithm and the branch-and-bound algorithm. We compared the original Go-ICP and RE + Go-ICP, which has the two steps of ICP with the RE and branch-and-bound algorithms. fig:GoICPResults plots all 50 results in ϕ=5π/12. Note that this comparison was implemented entirely in C++. Go-ICP always found the global optimum solution; however, it required significant computation. RE + Go-ICP reduced computational cost while achieving global optimization. §.§ Optimized product quantization We show that the RE algorithm is successful in OPQ optimization problems. We compare our method with the alternating optimization method <cit.>. For a dataset, we used SIFT 1M <cit.>, which contains 100,000 128-dimensional SIFT descriptors for training. We set the subspace number M=8 and cluster number k=256, which are often used in the field of approximate nearest neighbor search. For error measurement, we used the objective function value of eq:OPQ. For our method, we set μ^(0)=0.5. We plot the objective function value versus iteration number in fig:OPQObjVsIter. We performed five repetitions using different initializations and report the average objective values obtained. Our method improved the objective function value; moreover, it achieved rapid convergence in the cases of T=30 and T=100. The RE algorithm elevates the objective function around the current solution; in other words, it transforms the gradient for the current solution into a steeper gradient, potentially causing rapid convergence. §.§ Blind image deblurring We evaluated our method with single blind image deblurring. There are many formulations for blind image deblurring. In this paper, we followed Pan 's formulation <cit.>, which can be minimized by alternating optimization. We compared three methods as follows: a coarse-to-fine strategy <cit.> and RE algorithms based on alg:REalgorithm and alg:REalgorithm2. We used the uniform blurred text images from the dataset provided by Lai <cit.>, which contains five latent images and four blurring kernels for a total of 20 blurred images. For all methods, we used the same objective function parameters, such as the regularization coefficients. For our method, we set μ^(0) = 0.2 and T=100. We show the results in tbl:DeblurResults1. Our method significantly outperforms Pan's method <cit.> and is successful for a significantly blurred image, as in fig:DeblurFig. We found that alg:REalgorithm2 is superior to alg:REalgorithm in the cases of {image #3, kernel #4} and {image #5, kernel #4}. Note that these results are obtained by minimizing the same objective function, however using different optimization methods. Therefore our method likely improves upon other methods which use different objective functions <cit.>. § CONCLUSION We proposed the RE algorithm, which is a novel global optimization algorithm for nonconvex LS problems. This method is based on a novel measurement of global convergence called RE convergence. We presented theoretical analysis of RE convergence and empirical results showing excellent performance of the RE algorithm for various optimization problems. There remain many open questions in both theoretical and empirical aspects. We can guarantee that the solution with the largest RE constant is the global optimum in limited cases. To which problems this applies remains unknown. We plan to investigate the applicability of the RE algorithm to other nonconvex optimization problems, including non-LS problems. § ACKNOWLEDGEMENT This research is partially supportted by CREST (JPMJCR1686) and KAKENHI（15K12025） ieee | http://arxiv.org/abs/1705.09549v1 | {
"authors": [
"Daiki Ikami",
"Toshihiko Yamasaki",
"Kiyoharu Aizawa"
],
"categories": [
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"primary_category": "cs.CV",
"published": "20170526120850",
"title": "Residual Expansion Algorithm: Fast and Effective Optimization for Nonconvex Least Squares Problems"
} |
Hasso Plattner Institute (HPI), University of Potsdam, Germany P.O. Box 900460 PotsdamD-14480 haojin.yang, christian.bartz, [email protected] [email protected] Neural Networks (BNNs) can drastically reduce memory size and accesses by applying bit-wise operations instead of standard arithmetic operations. Therefore it could significantly improve the efficiency and lower the energy consumption at runtime, which enables the application of state-of-the-art deep learning models on low power devices. BMXNet is an open-source BNN library based on MXNet, which supports both XNOR-Networks and Quantized Neural Networks. The developed BNN layers can be seamlessly applied with other standard library components and work in both GPU and CPU mode. BMXNet is maintained and developed by the multimedia research group at Hasso Plattner Institute and released under Apache license. Extensive experiments validate the efficiency and effectiveness of our implementation. The BMXNet library, several sample projects, and a collection of pre-trained binary deep models are available for download at <https://github.com/hpi-xnor> <ccs2012> <concept> <concept_id>10010147.10010178.10010224</concept_id> <concept_desc>Computing methodologies Computer vision</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10010520.10010521.10010542.10010294</concept_id> <concept_desc>Computer systems organization Neural networks</concept_desc> <concept_significance>500</concept_significance> </concept> <concept> <concept_id>10011007.10011006.10011072</concept_id> <concept_desc>Software and its engineering Software libraries and repositories</concept_desc> <concept_significance>500</concept_significance> </concept> </ccs2012> [500]Software and its engineering Software libraries and repositories [500]Computer systems organization Neural networks [500]Computing methodologies Computer visionBMXNet: An Open-Source Binary Neural Network Implementation Based on MXNet Haojin Yang, Martin Fritzsche, Christian Bartz, Christoph Meinel December 30, 2023 ============================================================================§ INTRODUCTION In recent years, deep learning technologies achieved excellent performance and many breakthroughs in both academia and industry. However the state-of-the-art deep models are computational expensive and consume large storage space. Deep learning is also strongly demanded by numerous applications from areas such as mobile platforms, wearable devices, autonomous robots and IoT devices. How to efficiently apply deep models on such low power devices becomes a challenging research problem. The recently introduced Binary Neural Networks (BNNs) could be one of the possible solutions for this problem.Several approaches <cit.> introduce the usage of BNNs. These BNNs have the capability of decreasing the memory consumption and computational complexity of the neural network. This is achieved by on the one hand storing the weights, that are typically stored as 32 bit floating point values, as binary values, by binarizing the floating point values with the sign function, to be of either {0, 1} or {-1, 1}, and storing several of them in a single 32 bit float or integer. Computational complexity, on the other hand, is reduced by usingandfor performing matrix multiplications used in convolutional and fully connected layers. Most of the publicly available implementations of BNN do not store the weights in their binarized form <cit.>, nor useand <cit.> while performing the matrix multiplications in convolutional and fully connected layers.The deep learning library Tensorflow <cit.> tries to decrease the memory consumption and computational complexity of deep neural networks, by quantizing the 32 bit floating point weights and inputs into 8 bit integers. Together with the minimum and maximum value of the weight/input matrix, 4× less memory usage and also decreased computational complexity is achieved, as all operations only need to be performed on 8 bit values rather than 32 bit values.BMXNet stores the weights of convolutional and fully connected layers in their binarized format, which enables us to store 32/64 weights in a single 32/64 bit float/integer and use 32× less memory. During training and inference we binarize the input to each binary convolution and fully connected layer in the same way as the weights get binarized, and perform matrix multiplication using bit-wise operations ( and ). Our implementation is also prepared to use networks that store weights and use inputs with arbitrary bit widths as proposed by Zhou et al. <cit.>.The deep learning library MXNet <cit.> serves as a base for our code. MXNet is a high performance and modular deep learning library, that is written in C++. MXNet provides Bindings for other popular programming languages like Python, R, Scala and Go, and is used by a wide range of researchers and companies.§ FRAMEWORK BMXNet provides activation, convolution and fully connected layers that support quantization and binarization of input data and weights. These layers are designed as drop-in replacements for the corresponding MXNet variants and are called ,and . They provide an additional parameter, , which controls the bit width the layers calculate with.A Python example usage of our framework in comparison to MXNet is shown in <ref> and <ref>. We do not use binary layers for the first and last layer in the network, as we have confirmed the experiments of <cit.> showing that this greatly decreases accuracy.The standard block structure of a BNN in BMXNet is conducted as: QActivation-QConv/QFC-BatchNorm-Pooling as shown in Listing <ref>. language=Python, escapeinside=(*@@*),basicstyle=,keywordstyle=,stringstyle=,commentstyle=, breaklines=true highlight_gray .25 [language=Python, caption=LeNet,label=listing:lenet] def get_lenet():data = mx.symbol.Variable('data')# first conv layerconv1 = mx.sym.Convolution(...)tanh1 = mx.sym.Activation(...)pool1 = mx.sym.Pooling(...)bn1 = mx.sym.BatchNorm(...)# second conv layerconv2 = mx.sym.Convolution(...)bn2 = mx.sym.BatchNorm(...)tanh2 = mx.sym.Activation(...)pool2 = mx.sym.Pooling(...)# first fullc layerflatten = mx.sym.Flatten(...)fc1 = mx.symbol.FullyConnected(..)bn3 = mx.sym.BatchNorm(...)tanh3 = mx.sym.Activation(...)# second fullcfc2 = mx.sym.FullyConnected(...)# softmax losslenet = mx.sym.SoftmaxOutput(...)return lenet.25 [language=Python, caption=Binary LeNet,label=listing:blenet] def get_binary_lenet():data = mx.symbol.Variable('data')# first conv layerconv1 = mx.sym.Convolution(...)tanh1 = mx.sym.Activation(...)pool1 = mx.sym.Pooling(...)bn1 = mx.sym.BatchNorm(...)# second conv layer(*@ba1 = mx.sym.QActivation(...) @*)(*@conv2 = mx.sym.QConvolution(...) @*)bn2 = mx.sym.BatchNorm(...)pool2 = mx.sym.Pooling(...)# first fullc layerflatten = mx.sym.Flatten(...)(*@ba2 = mx.symbol.QActivation(..) @*)(*@fc1 = mx.symbol.QFullyConnected(..) @*)bn3 = mx.sym.BatchNorm(...)tanh3 = mx.sym.Activation(...)# second fullcfc2 = mx.sym.FullyConnected(...)# softmax losslenet = mx.sym.SoftmaxOutput(...)return lenet§.§ Quantization The quantization on bit widths ranging from 2 to 31 bit is available for experiments with training and prediction, using low precision weights and inputs. The quantized data is still stored in the default 32 bit float values and the standard MXNet dot product operations are applied.We quantize the weights following the linear quantization as shown by <cit.>. Equation <ref> will quantize a real number input in the range [0,1] to a number in the same range representable with a bit width of k bit. quantize(input, k) = round((2^k - 1) * input) /2^k - 1§.§ Binarization The extreme case of quantizing to 1 bit wide values is the binarization. Working with binarized weights and input data allows for highly performant matrix multiplications by utilizing the CPU instructionsand .§.§.§ Dot Product withandFully connected and convolutional layers heavily rely on dot products of matrices, which in turn require massive floating point operations. Most modern CPUs are optimized for these types of operations. But especially for real time applications on embedded or less powerful devices (cell phones, IoT devices) there are optimizations that improve performance, reduce memory and I/O footprint, and lower power consumption <cit.>.To calculate the dot product of two binary matrices A∘ B, no multiplication operation is required. The element-wise multiplication and summation of each row of A with each column of B can be approximated by first combining them with theoperation and then counting the number of bits set to 1 in the result which is the population count <cit.>. language=C++, escapeinside=(*@@*),basicstyle=,keywordstyle=,stringstyle=,commentstyle=, breaklines=true[language=Python, caption=Baseline xnor GEMM Kernel,label=listing:xnorgemm] void xnor_gemm_baseline_no_omp(int M, int N, int K,BINARY_WORD *A, int lda,BINARY_WORD *B, int ldb,float *C, int ldc) for (int m = 0; m < M; ++m)for (int k = 0; k < K; k++)BINARY_WORD A_PART = A[m*lda+k]; for (int n = 0; n < N; ++n)C[m*ldc+n] += __builtin_popcountl((*@∼@*)(A_PART (*@∧@*) B[k*ldb+n]));We can approximate the multiplication and addition of two times 64 matrix elements in just a few processor instructions on x64 CPUs and two times 32 elements on x86 and ARMv7 processors. This is enabled by hardware support for theandoperations. They translate directly into a single assembly command. The population count instruction is available on x86 and x64 CPUs supporting SSE4.2, while on ARM architecture it is included in the NEON instruction set.An unoptimized GEMM (General Matrix Multiplication) implementation utilizing these instructions is shown in <ref>. The compiler intrinsicis supported by both gcc and clang compilers and translates into the machine instruction on supported hardware.is the packed data type storing 32 (x86 and ARMv7) or 64 (x64) matrix elements, each represented by a single bit.We have implemented several optimized versions of the xnor GEMM kernel. We leverage processor cache hierarchies by blocking and packing the data, use unrolling techniques and OpenMP for parallelization. §.§.§ Training We carefully designed the binarized layers (utilizingandcount operations) to exactly match the output of the built-in layers of MXNet (computing with BLAS dot product operations) when limiting those to the discrete values -1 and +1. This enables massively parallel training with GPU support by utilizing CuDNN on high performance clusters. The trained model can then be used on less powerful devices where the forward pass for prediction will calculate the dot product with the xnor and popcount operations instead of multiplication and addition.The possible values after performing anandmatrix multiplication (m × n)A∘(n × k)B are in the range [0,+n] with the step size 1, whereas a normal dot product of matrices limited to discrete values -1 and +1 will be in the range [-n,+n] with the step size 2. To enable GPU supported training we modify the training process. After calculation of the dot product we map the result back to the range [0,+n] to match the xnor dot product, as in Equation <ref>. output_xnor_dot = output_dot + n/2 §.§.§ Model Converter After training a network with BMXNet, the weights are stored in 32 bit float variables. This is also the case for networks trained with a bit width of 1 bit. We provide a model converter[<https://github.com/hpi-xnor/BMXNet/tree/master/smd_hpi/tools/model-converter>] that reads in a binary trained model file and packs the weights ofandlayers. After this conversion only 1 bit of storage and runtime memory is used per weight. A ResNet-18 network with full precision weights has a size of 44.7 MB. The conversion with our model converter achieves 29× compression resulting in a file size of 1.5 MB (cf. Table <ref>).§ EVALUATIONIn this section we report the evaluation results of both efficiency analysis and classification accuracy over MNIST <cit.>, CIFAR-10 <cit.> and ImageNet <cit.> datasets using BMXNet.§.§ Efficiency AnalysisAll the experiments in this section have been performed on Ubuntu 16.04/64-bit platform with Intel 2.50GHz×4 CPU with popcnt instruction (SSE4.2) and 8G RAM.In the current deep neural network implementations, most of the fully connected and convolution layers are implemented using GEMM.According to the evaluation result from <cit.>, over 90% of the processing time of the Caffe-AlexNet <cit.> model is spent on such layers.We thus conducted experiments to measure the efficiency of different GEMM methods. The measurements were performed within a convolution layer, where we fixed the parameters as follows: filter number=64, kernel size=5×5, batch size=200, and the matrix sizes M, N, K are 64, 12800, kernel_w× kernel_h× input Channel Size, respectively. Figure <ref> shows the evaluation results.The colored columns denote the processing time in milliseconds across varying input channel size;anddenote the xnor_gemm operator in 32 bit and 64 bit;denotes the 64 bit xnor_gemm accelerated by using the OpenMP[<http://www.openmp.org/>] parallel programming library;further accumulated the processing time of input data binarization. From the results we can determine thatachieved about 50× and 125× acceleration in comparison to Cblas(Atlas[<http://math-atlas.sourceforge.net/>]) and naive gemm kernel, respectively. By accumulating the binarization time of input data we still achieved about 13× acceleration compared with Cblas method.Figures <ref> and <ref> illustrate the speedup achieved by varying filter number and kernel size based on the naive gemm method. §.§ Classification AccuracyWe further conduted experiments with our BNNs on the MNIST, CIFAR-10 and ImageNet datasets. The experiments were performed on a work station which has an Intel(R) Core(TM) i7-6900K CPU, 64 GB RAM and 4 TITAN X (Pascal) GPUs.By following the same strategy as applied in <cit.> we always avoid binarization at the first convolution layer and the last fully connected layer.Table <ref> depicts the classification test accuracy of our binary, as well as full precision models trained on MNIST and CIFAR-10.The table shows that the size of binary models is significantly reduced, while the accuracy is still competitive. Table <ref> demonstrates the validation accuracy of our binary, partially-binarized and full precision models trained on ImageNet.The ResNet implementation in MXNet consists of 4 ResUnit stages, we thus also report the results of a partially-binarized model with specific full precision stages. The partially-binarized model with the first full precision stage shows a great accuracy improvement with very minor model size increase, compared to the fully binarized model.§ EXAMPLE APPLICATIONS§.§ Python ScriptsThe BMXNet repository <cit.>contains python scripts that can train and validate binarized neural networks. The script smd_hpi/examples/binary_mnist/mnist_cnn.py will train a binary LeNet <cit.> with the MNIST <cit.> data set. To train a network with the CIFAR-10 <cit.> or ImageNet <cit.> data set there is a python script based on the ResNet-18 <cit.> architecture. Find it at smd_hpi/examples/binary-imagenet1k/train_cifar10/train_[dataset].py. For further information and example invocation see the corresponding README.md§.§ Mobile Applications§.§.§ Image ClassificationThe Android applicationand iOS applicationcan classify the live camera feed based on a binarized ResNet-18 model trained on the ImageNet dataset.§.§.§ Handwritten Digit DetectionThe iOS applicationcan classify handwritten numbers based on a binarized LeNet model trained on the MNIST dataset. § CONCLUSIONWe introduced BMXNet, an open-source binary neural network implementation in C/C++ based on MXNet. The evaluation results show up to 29× model size saving and much more efficient xnor GEMM computation. In order to demonstrate the applicability we developed sample applications for image classification on Android as well as iOS using a binarized ResNet-18 model. Source code, documentation, pre-trained models and sample projects are published on GitHub <cit.>. ACM-Reference-Format | http://arxiv.org/abs/1705.09864v1 | {
"authors": [
"Haojin Yang",
"Martin Fritzsche",
"Christian Bartz",
"Christoph Meinel"
],
"categories": [
"cs.LG",
"cs.CV",
"cs.NE"
],
"primary_category": "cs.LG",
"published": "20170527205210",
"title": "BMXNet: An Open-Source Binary Neural Network Implementation Based on MXNet"
} |
On the Capacity of Fractal Wireless Networks With Direct Social Interactions Ying Chen, Rongpeng Li, Zhifeng Zhao, and Honggang Zhang York-Zhejiang Lab for Cognitive Radio and Green Communications College of Information Science and Electronic Engineering Zhejiang University, Zheda Road 38, Hangzhou 310027, China Y. Chen, Z. Zhao, R. Li, and H. Zhang are with College of Information Science and Electronic Engineering, Zhejiang University. Email: {21631088chen_ying, lirongpeng, zhaozf, honggangzhang}@zju.edu.cn This paper is financially supported by the Program for Zhejiang Leading Team of Science and Technology Innovation (No. 2013TD20), the Zhejiang Provincial Technology Plan of China (No. 2015C01075), and the National Postdoctoral Program for Innovative Talents of China (No. BX201600133). December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================ The capacity of a fractal wireless network with direct social interactions is studied in this paper. Specifically, we mathematically formulate the self-similarity of a fractal wireless network by a power-law degree distribution P(k), and we capture the connection feature between two nodes with degree k_1 and k_2 by a joint probability distribution P(k_1,k_2). It is proved that if the source node communicates with one of its direct contacts randomly, the maximum capacity is consistent with the classical result Θ(1/√(nlog n)) achieved by Kumar <cit.>. On the other hand, if the two nodes with distance d communicate according to the probability d^-β, the maximum capacity can reach up to Θ(1/log n), which exhibits remarkable improvement compared with the well-known result in <cit.>. Capacity, Fractal Networks, Social Interactions, Self-Similarity, Throughput, Complex Networks§ INTRODUCTIONThe research on the capacity of a wireless network has aroused intense interests in recent years. However, little attention has been paid to fractal wireless networks. As a vital property of networks, fractal behavior has been discovered in many wireless networking scenarios <cit.>. For example, the coverage boundary of the wireless cellular networks shows a fractal shape, and the fractal feature can inspire the design of the handoff scheme in mobile terminals <cit.>. Moreover, many significant networks in the real world exhibit the fractal characteristics naturally, such as the protein homology network, the undirected WWW and so on <cit.>. In the theory of fractal wireless networks, self-similarity belongs to one of the most important characteristics. Self-similarity of fractal wireless networks requires the node degree distribution P(k) to remain unchanged when the network grows. In order to meet this requirement, power-law degree distribution P(k)∼ k^-γ must be followed in a fractal wireless network. Actually, the power-law distribution lays the foundation for the capacity analysis of fractal wireless networks.In 2000, Gupta and Kumar proved that the throughput in wireless networks can only reach Θ(1/√(nlog n)) when the network size is n <cit.>, where the symbol Θ refers to the order of magnitude and T(n)=Θ (h(n)) means that the two functions T(n) and h(n) have the same rate of growth. Afterwards, H. R. Sadjadpour et al. discovered that the maximum capacity can be improved in scale-free networks <cit.><cit.><cit.>. However, to our best knowledge, there have been no theoretical researches about the capacity of fractal wireless networks with direct social interactions, which is exactly what we study in this paper.In the remainder of this article, we introduce the foundations of a fractal wireless network and discussthenetwork model in Section II. Then the maximum throughput is derived in Section III and the results are discussed in Section IV. Finally, we draw a conclusion in Section V.§ BACKGROUND AND MODELS §.§ Foundations of A Fractal NetworkFig.1 illustrates the topology in depth of a general fractal wireless network, in which we assume that a node v_i is going to transmit a data packet to another node. In other words, the node v_i is the source node. The nodes v_j1 and v_j2 are directly connected with the node v_i and are regarded as the direct social contacts of v_i. Then v_k is chosen among the direct social contacts to communicate with v_i and it is known as the destination node. Usually, a node has many direct social contacts, and the degree of one node refers to the number of its direct social contacts.In order to characterize fractal wireless networks, it is essential to introduce the concept of renormalization through the box-covering algorithm <cit.>. Renormalization is a technique to examine the internal relationship by using a box to cover several nodes and virtually replacing the whole box by a new node. Besides, if there exists a link between two nodes in two boxes respectively, then the two corresponding nodes evolved from the boxes will be connected. Mathematically, the network could be minimallycovered by N_B(l_B) boxes of the same length scale l_B under the renormalization, where l_B is the size of boxes measured by the maximum path length between any pair of nodes inside the box and N_B(l_B) is the minimum value among all possible situations. If the network is a fractal wireless network, we have the following relations, namely <cit.>:{[ N_B(l_B)∼ n ·l_B^ - d_Bk_B(l_B)/k_hub∼l_B^ - d_gn_h(l_B)/k_B(l_B)∼l_B^ - d_e, ]. where n is the number of nodes in the network. A hub indicates the node of the largest degree inside each box, while k_B(l_B) and k_hub denote the degree of the box and the hub respectively. n_h(l_B) refers to the number of links between the hub of a box and nodes in other connected boxes. The three indexes d_g, d_B and d_e denote the degree exponent, the fractal exponent, and the anti-correlation exponent, respectively.In addition, for fractal wireless networks, the degree exponent d_g and the fractal exponent d_B are both finite. The anti-correlation exponent d_e reveals the repulsion effect between the hubs while large anti-correlation exponent tends to result in fractal wireless networks.On the other hand, the joint probability distribution P(k_1,k_2) gauges the possibility to have a connection between two nodes with degree k_1 and k_2. In fractal wireless networks, P(k_1,k_2) can be expressed as <cit.>:P(k_1,k_2)∼ k_1^-(γ-1)· k_2^-ϵ (k_1>k_2), where γ is the degree distribution exponent and ϵ is the correlation exponent. It is proved that there exist certain relations among the aforementioned key parameters: γ= 1 + d_B/d_g and ϵ= 2 + d_e/d_g <cit.>, which suggest that γ and ϵ in a fractal wireless network are larger than 1 and 2 respectively. A reasonable explanation for the condition k_1>k_2 is that nodes with larger degrees have a higher priority in contacts selection, so they choose nodes of smaller degrees and set up connections with them.§.§ Network ModelIn order to clarify the capacity of a fractal wireless network with direct social interactions clearly and orderly, we assume that the fractal social network is deployed in a unit area square and the n nodes are uniformly distributed, as displayed in Fig. 2. The Protocol Model in <cit.> is adopted as the measurement of a successful transmission. A transmission is successful if and only if |X_i-X_j|⩽ r(n) and |X_k-X_i|≥ (1+Δ)|X_i-X_j|. X_i and X_j refer to the transmitter node and the receiver node respectively. X_k denotes any other transmitter sharing the same channel with X_i and Δ is the protection factor of interference zone. To eliminate the isolated nodes, the transmission range r(n) has to exceed some threshold. It is proved that r(n) must reach Θ(√(log n/n)) to guarantee the connectivity of the network <cit.>. In Fig. 2, thesolid-line circle with the radius of r(n) displays the transmission range.A TDMA (Time Division Multiple Access) scheme is designated as the MAI (multiple access interference) avoidance method in the time domain. The network is cut into a number of smaller squares with side length C_1r(n). The interference units refer to those cells containing at least two nodes closer than (2+Δ)r(n) respectively[1], and cells, which simultaneously transmit, should not be the interference units with each other. Therefore, the squares signed with stars, whose distances are at least T squares from each other, are permitted to transmit at the same time, where T≥ (2+Δ)/C_1.Since the selection of the destination node affects the capacity of a fractal wireless network as well, in this paper, the destination node is chosen in two different ways. In the first case, the destination node is selected according to the uniform distribution 1/q. That is to say, the source node communicates with one of its direct social contacts randomly, and all the contacts have the same opportunities to communicate with the source node. In another case, it is considered to be reasonable that the destination node is selected according to the power-law distribution d^-β<cit.>, where d refers to the distance between the source node and the destination node, and β indicates that the source node communicates with different nodes with different preferences<cit.>. In other words, the direct social contacts closer to the source node have more opportunities to communicate with it. On the other hand, it is noteworthy that β is closely related with the configuration of the fractal wireless networks. Accordingly, the parameter β can be noted as β=β(γ,ϵ), which indicates that β is a function of the degree distribution exponent γ and the correlation exponent ϵ. The routing scheme in the space domain is pretty simple. When the source node is about to send a data packet, it chooses one node from its neighbor squares closest to the destination node to relay the packet. The packet eventually reaches the destination node after multiple hops. The red line with arrows denotes one possible routing path in Fig. 2. All the squares in red have the same hops x from the source node and the total number of them is 4x. The radius of the two dotted-line circles are used as the distances between the nodes in red squares and the source node instead of their real distances. 𝐃𝐞𝐟𝐢𝐧𝐢𝐭𝐢𝐨𝐧 1. The elementary symmetric polynomial σ_p,N(Q), 1≤ p≤ N <cit.> of variables Q=(q_1,q_2,…,q_N) is noted as[σ_p,N(Q)=σ_p,N(q_1,q_2,…,q_N); =∑_ 1≤ i_1≤ i_2≤… i_p≤ Nq_i_1q_i_2⋯ q_i_p. ] 𝐃𝐞𝐟𝐢𝐧𝐢𝐭𝐢𝐨𝐧 2. The elementary symmetric polynomial σ^k_p,N-1(Q), 1≤ p≤ N-1 of variables Q=(q_1,q_2,…,q_N) except q_k is noted asσ^k_p,N-1(Q)=σ_p,N-1(q_1,q_2,…,q_k-1,…,q_k+1,…,q_N).𝐋𝐞𝐦𝐦𝐚 1. Let the set Q = {q_1,q_2, ··· ,q_N} contains N≥ 2 non-negative real numbers. If q is finite, then we have <cit.>σ_1,N(Q)σ_q,N(Q)/(q+1)σ_q+1,N(Q)=Θ(N/N-q). § THE UPPER BOUND OF THE CAPACITY In this section, we will follow the aforementioned properties of fractal wireless networks and try to clarify the specific derivation procedure of the maximum capacity with direct social interactions. In addition, we also study the impact of a particular destination selection rule on the maximum achievable throughput by taking account of two different cases, which include uniformly and power-law distributed destinations. §.§ The Case of Uniformly Distributed DestinationsIn the first case, we take the uniform distribution of the destination node into consideration. In other words, we assume that the source node selects one of its direct social contacts as the destination node randomly, so each contact node has the same possibility to communicate with the source node. In this situation, we give the result of the maximum capacity in Theorem 1 and proof it subsequently.𝐓𝐡𝐞𝐨𝐫𝐞𝐦 1. For a fractal wireless network with n nodes satisfying the conditions below: 1) the direct social contacts are selected according to the joint probability distribution P(k_1,k_2) = k_1^ - (γ- 1)k_2^ - ϵ/M_γ ,ϵ; 2) the degree of each node follows the power-law degree distribution P(k) = k^ - γ/∑_k = 1^n k^ - γ; 3) the destination node is chosen by the uniform distribution P(v_t = v_k|v_k∈C) = 1/q.Then the maximum capacity λ_max of the fractal wireless network with direct social interactions isλ _max = Θ(1/√(n ·log n)). In order to get the result in Theorem 1, we first present the relationship between the capacity and the average number of hops E[X], so we can solve the problem by finding out E[X]. Secondly, we give the expression of E[X]. Thirdly, We divide E[X] into two separate cases E_1 and E_2 with a boundary q_0, which indicates a relatively large degree. E_1 is the average number of hops when q ≤q_0, where the degree q of the source node is a finite integer, meanwhile E_2 is the average number of hops when q > q_0, where q is considered to be infinite. After E_1 and E_2 are obtained respectively, the capacity derivation is achieved by backtracking. Next, we will follow the proof sketch above, and then have the following lemmas.𝐋𝐞𝐦𝐦𝐚 2. Assume that λ is the data rate for every node, X is the number of hops from the the source node to the destination node. E[X] denotes the expectation of X for any transmission between a pair of source and destination nodes. W is the bandwidth assigned to the whole network. Then we have λ⩽λ_max=Θ(1/log n· E[X]). Proof: The total traffic which the network has to deal with is nλ E[X] while the network can handle W/T^2C_1^2r(n)^2 simultaneously. So the inequality nλ E[X] ≤W/T^2C_1^2r^2(n) must be satisfied. Recalling that the order of r(n) is Θ(√(log n/n)) in Section II, then we can get Eq. (<ref>). ▪ 𝐋𝐞𝐦𝐦𝐚 3. Let the source node v_i locate in the central square whose degree is q, where q=1,2,…,n. v_k denotes the destination node whose degree is q_k. The variables q_1,q_2, ··· ,q_N in Q = (q_1^ - ϵ,q_2^ - ϵ, ··· ,q_N^ - ϵ) denote the degrees of N potential direct socialcontacts whose degrees are smaller than q. Then the average number of hops can be given as E[X] = ∑_q = 1^n q^ - γ/∑_b = 1^n b^ - γ·∑_x=1^1/r(n)x∑_l = 1^4x∑_v_k∈s_lq_k^ - ϵσ _q - 1,N - 1^k(Q)/q·σ _q,N(Q). Proof: Let P(k=q) denote the probability when the degree of the source node is q, while E[X|source v_i,k = q] is the average number of hops under the condition that the source node v_i has q direct social contacts, then E[X] can be written asE[X] = ∑_q=1^n P(k=q) E[X|source v_i,k = q]. Let P(X=x) denote the probability of x hops ranging from 1 to 1/r(n). The event X=x is true if and only if v_klocates in the red squares s_l(l=1,2,… 4x) and it is selected as the destination node v_t. Therefore, E[X|sourcev_i,k = q] can be expanded as[E[X|source v_i,k = q] = ∑_x=1^1/r(n)x · P(X = x); = ∑_x=1^1/r(n)x ·∑_l = 1^4x∑_v_k∈s_lP(v_t = v_k). ] Let 𝐂 be the set of all direct social contacts. v_t=v_k implies that v_k is chosen as the destination node v_t after it is selected as a direct social contact. In other words,P(v_t=v_k)=P(v_k∈𝐂)· P(v_t=v_k|v_k∈𝐂). The set {v_i_1,v_i_2,…,v_i_q} contains q direct social contacts of the source node. Taking all possible combinations of nodes into consideration, the probability that the source node has q direct social contacts is[P(|𝐂|=q)=∑_1≤ i_1≤ i_2≤… i_q≤ NP(𝐂={v_i_1,v_i_2,…,v_i_q}); =∑_1≤ i_1≤ i_2≤… i_q≤ N(q^-(γ-1))^qq^-ϵ_i_1q^-ϵ_i_2⋯ q^-ϵ_i_q/(M_γ,ϵ)^q, ] where N is the number of nodes whose degrees are less than q and the source node selects direct social contacts only among thosenodes. N grows as fast as n becauseN=N_<q=n·∑^q-1_b=1b^-γ/∑^n_b=1b^-γ=Θ(n). The probability that the set 𝐂 consists of q particular nodes isP(𝐂 = {v_i_1,v_i_2, ···v_i_q}) = q_i_1^ - ϵq_i_2^ - ϵ··· q_i_q^ - ϵ/∑_1 ≤i_1≤···≤i_q≤ Nq_i_1^ - ϵq_i_2^ - ϵ··· q_i_q^ - ϵ. Consequently, we obtain the probability that v_k is selected as a direct social contact and simplify it with the elementary symmetric polynomials in Definition 1 and 2, namely:[ P(v_k∈C) = q_k^ - ϵ∑_1 ≤i_1≤···≤i_q - 1≤ Nq_i_1^ - ϵq_i_2^ - ϵ··· q_i_q - 1^ - ϵ/∑_1 ≤i_1≤···≤i_q≤ Nq_i_1^ - ϵq_i_2^ - ϵ··· q_i_q^ - ϵ; = q_k^ - ϵσ _q - 1,N - 1^k(Q)/σ _q,N(Q). ] Then we have the complete expression of Eq. (<ref>) after expanding Eq. (<ref>) with Eq. (<ref>) - Eq. (<ref>). ▪ 𝐋𝐞𝐦𝐦𝐚 4. When the degree of the source node is not greater than q_0, i.e. q ≤q_0, the average number of hops E_1 isE_1 =Θ(r(n)^ - 1). Proof: According to the meaning of E_1, it should be given asE_1 = ∑_q = 1^q_0q^ - γ/∑_b = 1^n b^ - γ·∑_x=1^1/r(n)x∑_l = 1^4x∑_v_k∈s_lq_k^ - ϵσ _q - 1,N - 1^k(Q)/q·σ _q,N(Q) . All situations of selecting q-1 nodes from 𝐂 can be parted into two categories according to the condition whether v_k is chosen. If it is chosen, we have to select the other q-2 nodes from 𝐂 except v_k. Otherwise we select the other q-1 nodes in 𝐂 except v_k, that is to say,[σ _q - 1,N - 1^k(Q) = σ _q - 1,N(Q) - q_k^ - ϵσ _q - 2,N - 1^k(Q); = σ _q - 1,N(Q) - q_k^ - ϵ(σ _q - 2,N(Q) - q_k^ - ϵσ _q - 3,N - 1^k(Q)). ] Since every term in the equation above is positive, then we haveσ _q - 1,N(Q) - q_k^ - ϵ·σ _q - 2,N(Q) < σ _q - 1,N - 1^k(Q) < σ _q - 1,N(Q). A transformation of the Lemma 1 suggests thatσ _1,N(Q)σ _q - 1,N(Q)/q ·σ _q,N(Q) = Θ (N/N - q + 1) = Θ (1). Moreover, the probability that the degree of the source node is not greater than q_0 isP(q ≤q_0) = ∑_q = 1^q_0q^ - γ/∑_b = 1^n b^ - γ= Θ (1). Therefore, we get the upper bound of E_1 according to Eq. (<ref>) - Eq. (<ref>).[ E_1 < ∑_x=1^1/r(n)x∑_l = 1^4x∑_v_k∈s_lq_k^ - ϵσ _q - 1,N(Q)/q·σ _q,N(Q); ≡q_k^ - ϵ/q·σ _1,N(Q)∑_x=1^1/r(n)x∑_l = 1^4x∑_v_k∈s_l 1 . ] where the symbol ≡ indicates the same order of magnitude on the two sides of an equation.All the n nodes are distributed uniformly in the unit area, and the side length of each square is C_1r(n), so the summation term in Eq. (<ref>) can be solved as[ ∑_x=1^1/r(n)x∑_l = 1^4x∑_v_k∈s_l1; ≡∑_x=1^1/r(n)x · 4x ·C_1^2r^2(n)· n · 1; ≡ n · r(n)^2∑_x=1^1/r(n)x^2; ≡Θ(n· r(n)^ - 1). ] The q_k^ - ϵ term in Eq. (<ref>) can be replaced with its mean in the upper bound for convenience,E[q_k^ - ϵ] = ∑_b = 1^q - 1P(k = b) ·b^ - ϵ= ∑_b = 1^q - 1b^ - (γ+ ϵ )/∑_b = 1^n b^ - γ≡Θ (1). On the other hand,σ _1,N(Q) = ∑_j = 1^N q_j^ - ϵ≡ N ·∫_1^q - 1u^ - ϵu^ - γ/∑_b = 1^n b^ - γ du ≡Θ (n), By combining Eq. (<ref>) - Eq. (<ref>) together, we have the upper bound of E_1, that isE_1 =O(r(n)^ - 1). Similarly, the lower bound of E_1 is[ E_1 > ∑_x=1^1/r(n)x∑_l = 1^4x∑_v_k∈s_lq_k^ - ϵσ _q - 1,N(Q) - q_k^ - 2ϵσ _q - 2,N(Q)/q·σ _q,N(Q); = upper bound - ∑_x=1^1/r(n)x∑_l = 1^4x∑_v_k∈s_lq_k^ - 2ϵσ _q - 2,N(Q)/q·σ _q,N(Q) . ] It turns out that the second term in the lower bound is∑_x=1^1/r(n)x∑_l = 1^4x∑_v_k∈s_lq_k^ - 2ϵσ _q - 2,N(Q)/q·σ _q,N(Q)≡Θ(n^-1· r(n)^ - 1). The order in Eq. (<ref>) is negligible compared with the upper bound in Eq. (<ref>), so the order of E_1 is solved. ▪ 𝐋𝐞𝐦𝐦𝐚 5. When the degree of the source node is greater than q_0, i.e. q>q_0, the average number of hops E_2 isE_2 =Θ(r(n)^ - 1). Proof: Similar to the case E_1, E_2 is given asE_2 = ∑_q = q_0 + 1^n q^ - γ/∑_b = 1^n b^ - γ·∑_x=1^1/r(n)x∑_l = 1^4x∑_v_k∈s_lq_k^ - ϵσ _q - 1,N - 1^k(Q)/q·σ _q,N(Q) . Since N is large enough and the degrees of q direct social contacts are independent and identically distributed, the law of large numbers can work here. Let X_i_j = q_i_j^ - ϵ,Y_i_j = logX_i_j and Y denotes the mean of Y_i_j , then we have[ q_k^ - ϵσ _q - 1,N - 1^k(Q)/q·σ _q,N(Q)≡∑_1 ≤i_1≤···≤i_q≤ N,∃ m,i_m = k∏_j = 1^qX_i_j/q·∑_1 ≤i_1≤···≤i_q≤ N∏_j = 1^qX_i_j;≡∑_1 ≤i_1≤···≤i_q≤ N,∃ m,i_m = kexp (∑_j = 1^qY_i_j)/q·∑_1 ≤i_1≤···≤i_q≤ Nexp (∑_j = 1^qY_i_j);≡∑_1 ≤i_1≤···≤i_q≤ N,∃ m,i_m = kexp (qY )/q·∑_1 ≤i_1≤···≤i_q≤ Nexp (qY );≡( [ N - 1; q - 1 ])/q·( [ N; q ]) = 1/N = Θ (n^-1). ] Besides, the probability that the degree of the source node is greater than q_0 is P(q > q_0) = ∑_q = q_0 + 1^n q^ - γ/∑_b = 1^n b^ - γ= Θ (1). Then Eq. (<ref>) can be simplified by Eq. (<ref>) - Eq. (<ref>), namely:E_2≡∑_x=1^1/r(n)x∑_l = 1^4x∑_v_k∈s_l1/n≡Θ(r(n)^ - 1). Therefore, the order of E_2 is obtained. ▪Now we can prove Theorem 1, from Lemma 4 and Lemma 5, we haveE[X] = E_1 + E_2 = Θ(r(n)^ - 1). Together with Lemma 2, we can obtain the results in Theorem 1. ▪§.§ The Case of Power-law Distributed DestinationsIn the second case, we assume that the probability that the source node communicates with one of its direct social contacts is proportional to d^-β, where d refers to the distance between the source node and the destination node, and β indicates that the closer social contacts have more opportunities to communicate with the source node. Different from the case of uniformly distribued destinations, fractal wireless networks achieve another maximum throughput in this situation, which is clarified in Theorem 2 and proofed in the remainder of the subsection.𝐓𝐡𝐞𝐨𝐫𝐞𝐦 2. For a fractal wireless network with n nodes satisfying the conditions below: 1) the direct social contacts are selected according to the joint probability distribution P(k_1,k_2) = k_1^ - (γ- 1)k_2^ - ϵ/M_γ ,ϵ; 2) the degree of each node follows the power-law degree distribution P(k) = k^ - γ/∑_k = 1^n k^ - γ; 3) the destination node is chosen by the power-law distribution P(v_t = v_k|v_k∈C) = d^ - β/∑_j = 1^q d_j^ - β.Then the maximum capacity λ_max of the fractal wireless network with direct social interactions isλ _max = {[ Θ(1/√(n ·log n)),0≤β (γ ,ϵ ) ≤2;Θ(1/√(n^3 - β·logn^β- 1)),2 < β (γ ,ϵ ) ≤3;Θ (1/log n),β (γ ,ϵ ) > 3. ]. Proof: The proof of Theorem 2 is pretty similar to Theorem 1, so we just give some key lemmas in the derivation procedure. 𝐋𝐞𝐦𝐦𝐚 6. Let the source node v_i locate in the central square whose degree is q, where q=1,2,…,n. v_k denotes the destination node whose degree is q_k and its distance from the source node is d_k. The variables q_1,q_2, ··· ,q_N in Q = (q_1^ - ϵ,q_2^ - ϵ, ··· ,q_N^ - ϵ) denote the degrees of N potential direct social contacts whose degrees are smaller than q. Let D = (d_1^ - β,d_2^ - β, ··· d_q^ - β), and d_j (1≤ j≤ q) in D denotes the distance between the jth social contact and the source node. Then the average number of hops can be given as E[X] = ∑_q = 1^n q^ - γ/∑_b = 1^n b^ - γ·∑_x=1^1/r(n)x∑_l = 1^4x∑_v_k∈s_lq_k^ - ϵσ _q - 1,N - 1^k(Q)/σ _q,N(Q)·d_k^ - β/σ _1,q(D). 𝐋𝐞𝐦𝐦𝐚 7. When the degree of the source node is not greater than q_0, i.e. q ≤q_0, the average number of hops E_1 isE_1 = {[ Θ(r(n)^ - 1), 0≤β≤2;Θ (r(n)^β- 3), 2 < β≤3;Θ (1), β> 3. ]. 𝐋𝐞𝐦𝐦𝐚 8. When the degree of the source node is greater than q_0, i.e. q>q_0, the average number of hops E_2 isE_2 = {[ Θ (r(n)^ - 1),0≤β≤2;Θ (r(n)^β- 3), 2 < β≤3;Θ (1), β> 3. ]. Combined with Lemma 2, we can obtain the result in Theorem 2. ▪ § DISCUSSIONSAfter mathematically obtaining the maximum achievable capacity in Section III, we display the results in an intuitive way. Fig. 3 illustrates the simulation diagrmas of the maximum capacity in two different cases. By Theorem 2, we find that the capacity varies for different values of β. When β has a relatively small value, there is no obvious increase in the throughput compared with the classical result achieved by Kumar in <cit.>. Particularly, when β=0, the situation is reduced to the one with uniformly distributed destinations as stated in Section 3.1. As β increases to 2∼ 3, the throughput leads to an exponential growth. After it exceeds 3, the throughput reaches up to Θ (1/log n), which exhibits remarkable improvement compared with Θ (1/√(nlog n)) in <cit.>. This increase can be explained by the reduction in the average number of hops for each transmission between the source node and the destination node, which is depicted in Fig. 4. As shown in Fig. 4, when β varies between 0 and 2, the average number of hops does not decrease distinctly compared with the case where the destination nodes are chosen uniformly. When β exceeds 2, the source node prefers to communicate with the closer direct social contacts, which leads to the exponentially growth of the maxium throughput. When β is greater than 3, only Θ (1) average hops are taken for each transmission, which raises the maximum capacity up toΘ (1/log n) finally.§ CONCLUSION AND FUTURE WORKSIn this paper, we study the capacity of fractal wireless networks with direct social interactions. It has been proved that if the source node communicates with one of its direct social contacts randomly, the maximum capacity in Theorem 1 corresponds with the classical result Θ(1/√(nlog n)) achieved by Kumar <cit.>. On the other hand, if the two nodes with distance d communicate according to the probability d^-β, the maximum capacity in Theorem 2 isλ _max= {[ Θ (1/√(n ·log n)), 0≤β(γ,ϵ)≤2;Θ (1/√(n^3 - β·logn^β- 1)), 2 < β(γ,ϵ)≤3;Θ (1/log n), β(γ,ϵ)> 3. ]. We notice that β has a straight-forward effect on the maximum capacity of fractal wireless networks with direct social interactions. Besides, as β might be influenced by the properties of fractal wireless networks, the maximun capacity is related to the degree distribution exponent γ and the correlation exponent ϵ. Although they are not included in the results explictly, it is reasonable that they have influence on the capacity of the fractal wireless networks. In other words, the degree distribution exponent γ and the correlation exponent ϵ would determine the maximum throughput to some extent. What is the explict relationship between β and the parameters γ and ϵ? How do β and ϵ shape the final capacity? We leave all these open issues in the future works.IEEEtran | http://arxiv.org/abs/1705.09751v1 | {
"authors": [
"Ying Chen",
"Rongpeng Li",
"Zhifeng Zhao",
"Honggang Zhang"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170527013330",
"title": "On the Capacity of Fractal Wireless Networks With Direct Social Interactions"
} |
Center for Computational Relativity and Gravitation, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY 14623, USACenter for Computational Relativity and Gravitation, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY 14623, USACenter for Astrophysics and Planetary Science, Cornell University, Ithaca, New York 14853, USACenter for Relativistic Astrophysics and School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USACenter for Computational Relativity and Gravitation, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY 14623, USADepartment of Physics, Princeton University, Jadwin Hall, Princeton, NJ 08544, USA Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto M5S 3H8, CanadaCenter for Relativistic Astrophysics and School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USAGravitational Wave Physics and Astronomy Center, California State University Fullerton, Fullerton, California 92834, USACanadian Institute for Theoretical Astrophysics, University of Toronto, Toronto M5S 3H8, Canada Department of Physics, University of Toronto, Toronto M5S 3H8, CanadaCenter for Computational Relativity and Gravitation, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY 14623, USATheoretical Astrophysics 350-17, California Institute of Technology, Pasadena, CA 91125, USAMax Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Potsdam-Golm, GermanyCenter for Relativistic Astrophysics and School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USACenter for Relativistic Astrophysics and School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USACenter for Astrophysics and Planetary Science, Cornell University, Ithaca, New York 14853, USACanadian Institute for Theoretical Astrophysics, University of Toronto, Toronto M5S 3H8, CanadaCenter for Relativistic Astrophysics and School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USACenter for Computational Relativity and Gravitation, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY 14623, USAGravitational Wave Physics and Astronomy Center, California State University Fullerton, Fullerton, California 92834, USAMax Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Potsdam-Golm, GermanyCanadian Institute for Theoretical Astrophysics, University of Toronto, Toronto M5S 3H8, Canada Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Mühlenberg 1, 14476 Potsdam-Golm, Germany Canadian Institute for Advanced Research, 180 Dundas St. West, Toronto, ON M5G 1Z8, CanadaTheoretical Astrophysics 350-17, California Institute of Technology, Pasadena, CA 91125, USACenter for Relativistic Astrophysics and School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, USATheoretical Astrophysics 350-17, California Institute of Technology, Pasadena, CA 91125, USA Caltech JPL, Pasadena, California 91109, USACenter for Astrophysics and Planetary Science, Cornell University, Ithaca, New York 14853, USA Theoretical Astrophysics 350-17, California Institute of Technology, Pasadena, CA 91125, USACenter for Computational Relativity and Gravitation, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, NY 14623, USAWe present and assess aBayesian method to interpret gravitational wave signals from binary black holes.Our method directly comparesgravitational wave data to numerical relativity simulations.This procedure bypassesapproximations used in semi-analytical models for compact binary coalescence.In this work,we use only the full posterior parameter distribution for generic nonprecessing binaries, drawing inferences away from the set of NR simulations used, via interpolation of a single scalar quantity (the marginalized log-likelihood,) evaluated by comparing data to nonprecessing binary black hole simulations. We also compare the data to generic simulations, and discuss the effectiveness of this procedure for generic sources.We specifically assess the impact of higher order modes, repeating our interpretation with both l≤2 as well as l≤ 3 harmonic modes.Using the l≤ 3 higher modes, we gain more information from the signal and can better constrain the parameters of the gravitational wave signal.We assess and quantify several sources of systematic error that our procedure could introduce, including simulation resolution and duration; most are negligible.We show through examples that our method can recover the parameters for equal mass, zero spin; GW150914-like; and unequal mass, precessing spin sources.Our study of this new parameter estimation method demonstrates we can quantify and understand the systematic and statistical error.This method allows us to use higher order modes from numerical relativity simulations to better constrain the black hole binary parameters. A Parameter Estimation Method that Directly Compares Gravitational Wave Observations to Numerical RelativityY. Zlochower December 30, 2023 ============================================================================================================§ INTRODUCTIONOn September 14, 2015 gravitational waves (GW) were detected for the first time at the Laser Interferometer Gravitational Wave Observatory (LIGO) in both Hanford, Washington and Livingston, Louisiana <cit.>.The LIGO Scientific Collaboration and Virgo Collaboration (LVC) concluded that the source of the GW signal was a binary black hole (BBH) system with masses m_1=26.2^+5.2_-3.8 and m_2=29.1^3.7_-4.4 that merged into a more massive black hole (BH) with mass m_f=62.3^+3.7_-3.1 <cit.>.These parameters were estimated by comparing the signal to state-of-the-art semi-analytic models <cit.>.However, in this mass regime, LIGO issensitive to the last few cycles of coalescence, characterized by a strongly nonlinear phase not comprehensively modeled by analytic inspiral or ringdown models.In <cit.>, the LVC reanalyzed GW150914 with an alternative method that compares the data directly to numerical relativity (NR), which include aspects of the gravitational radiation omitted by the aforementioned models.This additional information led to a shift in some inferred parameters(e.g., the mass ratio) of the coalescing binary. In this work, weassess the reliability and utility of this novel parameter estimation method in greater detail.For clarity and relevance, we apply this method to synthetic data derived from black hole binaries qualitatively similar to GW150914. Previous work <cit.> demonstrated by example that this method could access information about GW sources using higher order modes that was not presently accessible by other means.In this work, we demonstrate the utilityof this method with a larger set of examples, showing we recover (known) parameters of a synthetic source more reliably when higher order modes are included. More critically,we presenta detailed study of the systematic and statistical parameter estimation errors of this method.This analysis demonstrates that these sources of error are under control allowing us to identify source parameters and conduct detailed investigations into subtle systematic issues, such asthe impact of higher order modes on parameter estimation.For simplicity and to best leverage the most exhaustively explored region of binary parameters, our analysis emphasizessimulations of nonprecessing black hole binaries as in <cit.>, particularly simulations with mass ratios andspins that are highly consistent with GW150914.The paper is outlined as follows.Section <ref> lists the simulations used in the study (both for our template bank and synthetic sources), describes our method of choice with regards to waveform extraction, and briefly describes the method (see Section III in <cit.>).Section <ref> describes the diagnostics used in our assessment of the systematics, illustrating each with concrete examples.Section <ref> describes several sources of error and their relative impact on our results. Section <ref> presents 3 end-to-end runs, q=1 zero spin; q=1.22 anti-aligned (GW150914-like); and q=1.23 short precessing, including both l≤2 and l≤ 3 (for the GW150914-like) results.Section <ref> summarizes our findings. Appendix <ref> includes more end-to-end studies that use intrinsically different sources to explore more of the parameter space using our method. For context,the same method used to analyze GW150914 has also been applied to synthetic data using numerical relativity simulations <cit.>. § METHODS AND INPUTS§.§ Numerical relativity simulations A numerical relativity (NR) simulation of a coalescing compact binary can be completely characterized by its intrinsic parameters, namely its individual masses and spins.We parameterize the binary using the mass ratio q=m_1/m_2 with the convention q≥ 1 (m_1≥ m_2) and the dimensionless spin parametersχ_i=S_i/m_i^2.where i=1,2 indexes the component black holes in the binary.With regard to spin, we define another dimensionless parameter that is a combination of the spins <cit.>:χ_ eff=(S_1/m_1+S_2/m_2)·L̂/M.Figure <ref> illustrates our NR template bank, with each simulation represented as a point in the χ_ eff,q plane.Finally we quantify the duration of each simulation signal by a dimensionless parameter Mω_0, corresponding to the dimensionless starting binary frequency measured at infinity. For a given simulation, the GW strain h(t,r,n̂) can be characterized by a spin-weighted spherical harmonic decomposition at large enough distance: h(t,r,n̂)=Σ_l≥2Σ_m=-l^lh_lm(t,r)_-2Y_lm(n̂). In this expression, n̂ is characterized by polar angles ι,-ϕ_ ref; see <cit.>. For the majority of sources,the (2,± 2) mode dominates the summation and can adequately characterize the observationally-accessible radiation in any direction to a relative good approximation; however, other higher modes can often contribute in a significant way to the overall signal <cit.>. More exotic sources(i.e. high mass ratio and/or precessing, high spins) have significant power in higher modes <cit.>. §.§ Simulations usedIn this work, we use a wide parameter-range of NR simulations similar to the set used in <cit.>.We use all of the 300 public and 13 non-public SXS simulations for a total of 313 <cit.>.From the RIT group, we use all 126 public and 281 non-public simulations to bring the total contribution up to 407 <cit.>.Wealso usea total of 282 simulations provided from the GT group <cit.>.Including all the contributions from these three groups, we have a total NR template bank of 1002 simulations.Figure <ref> shows all the NR simulations in the 2D parameter space of χ_ eff, as defined in Eq. (<ref>), vs 1/q i.e. the mass ratio. All these simulations have already been published and were produced by one of three familiar procedures, see Appendix A in <cit.> for more details for each particular group.From these simulations, we selected 12 simulations to focus on as candidate synthetic sources.Table <ref> shows the specific simulations used, specifying the mass ratio (q>1), component spins of each BH, and total mass.To simplify the process of referring to these heterogeneous simulations, in the last column we assign a shorthand label to each one.These candidates have a variety of mass ratios and spins including zero, aligned, and precessing systems from different NR groups. The first three simulations (RIT-1a,-1b, and -1c) haveidentical initial conditions/parameters, carried out with different simulation numerical resolution. In many of the validation studies, RIT-1a is used; this is a GW150914-like simulation with comparable masses and anti-aligned spins. We use this simulation for its relative simplicity (higher order modes start to become important at the total mass we'll scale the simulation to, namely 70M_⊙) and to relate it to our similar work done on the real event GW150914.In this paper, we present 3 end-to-end studies of our parameter estimation method using data from synthetic sources. We use: a zero spin q=1.0 NR simulation (SXS-1) to show that the method recovers the parameters for the most basic source, an aligned spin GW150914-like simulation (SXS-0233) to show that higher order modes and therefore NR is needed to optimally recover the parameters even with aligned spin cases, and a precessing source (SXS-0234v2) to show our method arrives at reasonable conclusions for any heavy, comparable-massbinary system with generic spins.§.§ Extracting asymptotic strain from ψ_4(r,t) From our large and heterogeneous set of simulations, we need to consistently and reproducibly estimater h_lm(t). Many general methods for strain estimation exist; see the review in <cit.>. The method adopted here must be robust, using the minimal subset of all groups' output; function with all simulations, precessing or not; and rely on only knowledge of asymptotic properties, not (gauge-dependent) information about dynamics.For these reasons, we implemented our own strain reconstruction and extrapolation algorithm, which as input requires only ψ_4,lm(t) on some (known) code extraction radius. This method combines two standard tools –perturbative extrapolation <cit.> and the fixed-frequency integration method <cit.> – into a single step.Specifically, we extract r h(t) at infinity from ψ_4(r,t) at finite radius using a perturbative extrapolation technique based on Eq. (29) in <cit.>, implemented in the fourier domain and using a low-frequency cutoff<cit.>. Specifically, if f_ min is identified as the minimum frequency content for the mode,we construct the gravitational wave strain from ψ_4 at a single finite radius fromr h̃_lm(f) = ψ̃_4,lm/(iω)^2(1 - 2 M/r) [1 - (ℓ-1)(ℓ+2)/2 r1/iω+ (ℓ-1)(ℓ+2)(ℓ^2+ℓ-4)/8 r^21/(i ω)^2] + ψ̃_4,l+1,m/(iω)^22i a/(ℓ+1)^2√((ℓ+3)(ℓ-1)(ℓ+m+1)(ℓ-m+1)/(2l+1)(2l+3)) [iω - ℓ(ℓ+3)/r] - ψ̃_4,l-1,m/(i ω)^22i a/(ℓ)^2√((ℓ+1)(ℓ-2)(ℓ+m)(ℓ-m)/(2l-1)(2l+1))[i ω - (ℓ-2)(ℓ+1)/r]where the effective frequency is implemented asiω = i 2πsign(f) max(|f|, f_ min)and where a is an estimate for the final black hole spin. This method nominally introduces an obvious obstacle to practical calculation: the last two terms manifestly require an estimate of a and are tied to a frame in which the final black hole spin is aligned with our coordinate axis. In practice, the two spin-dependent terms are small and can be safely omitted in most practical calculations; moreover, each group provides a suitable estimate for the final state.We will clearly indicate when these terms are incorporated into our analysis in subsequent discussion. When implementing this procedure numerically, we first clean ψ_4,lm using pre-identified simulation-specific criteria to eliminate junk radiation at early and late times, tapering the start and end of the signal to avoid introducing discontinuities.For example, for many simulations and for all modes, any content in ψ_4,lm prior to t ≤ r + t_0was set to zero, for some suitable t_0 (fixed for all modes); subsequently, to eliminate the discontinuity this choice introduces, each mode was multiplied by a Tukey window chosen to cover 5% of the remaining waveform duration. Similarly, all data after a mode-dependent time t_e was set to zero, where the time t_e was identified via the first time (after the time where |ψ_4,22| is largest)where r |ψ_4,lm| fell below a fixed, mode-independent threshold.To smooth discontinuity, a cosine taper was applied at the end, with duration the larger of either 15 M or 10% of the remaining post-coalescence duration, whichever is larger.The Fourier transform implementation includes additional interpolation/resampling and padding.First, particularly to enable non-uniform time-sampling, each mode is interpolated and resampled to a uniform grid, with spacing set by the time-sampling rate of the underlying simulation. In carrying out this resampling, the waveform is padded to cover a duration 2T+100 M, where T is the remaining duration of the (2,2) mode after the truncation steps identified above. To simplify subsequent visual interpretation and investigation, the padding is aligned such that the peak of the (2,2) mode occurs near the center of the interval (t=0).Finally, the characteristic frequency M f_ min, (l,m) is identified from the starting frequency of eachψ_4,lm. In cases where the starting frequency cannot be reliably identified (e.g., due to lack of resolution), the frequency is estimated from the minimum frequency of the 22 mode as |m| f_min, (2,2)/2.[This fallback approximation is not always appropriate for strongly precessing systems. However, for strongly precessing systems, the relevant starting frequency can be easily identified.]In Section <ref> we will demonstrate the reliability of this procedure to extract h(t) from ψ_4. §.§ Framework for directly comparing simulations to observations I: Single simulationsIn this section, we briefly review the methods introduced in <cit.> and <cit.> to infer compact binary parameters from GW data.All analyses of the data begin with the likelihood of the data given noise, which always has the form (up to normalization)ln L(λ ;θ )=-1/2∑_k⟨ h_k(λ ,θ )-d_k |h_k(λ ,θ )-d_k⟩ _k-⟨ d_k|d_k⟩ _k,where h_k are the predicted response of the k^th detector due to a source with parameters (λ, θ) and d_k are the detector data in each instrument k; λ denotes the combination of redshifted mass M_z and the numerical relativity simulation parameters needed to uniquely specify the binary's dynamics; θ represents the seven extrinsic parameters (4 spacetime coordinates for the coalescence event and 3 Euler angles for the binary's orientation relative to the Earth); and ⟨ a|b⟩_k≡∫_-∞^∞2dfã(f)^*b̃(f)/S_h,k(|f|) is an inner product implied by the k^th detector's noise power spectrum S_h,k(f).In all calculations, we adopt the fiducial O1 noise power spectra associated with data near GW150914 <cit.>. In practice we adopt a low-frequency cutoff f_ min so all inner products are modified to⟨ a|b⟩_k≡ 2 ∫_|f|>f_ mindfã(f)^*b̃(f)/S_h,k(|f|).The joint posterior probability of λ ,θ follows from Bayes' theorem:p_ post(λ ,θ)= L(λ ,θ)p(θ)p(λ)/∫ dλ dθ L(λ ,θ)p(λ)p(θ),where p(θ) and p(λ) are priors on the (independent) variables θ ,λ. For each λ, we evaluate the marginalized likelihoodL_ marg≡∫ L(λ ,θ )p(θ )dθvia direct Monte Carlo integration, where p(θ) is uniform in 4-volume and source orientation. To evaluate the likelihood in regions of high importance, we use an adaptive Monte Carlo as described in <cit.>.We will henceforth refer to the algorithm to “integrate over extrinsic parameters” as ILE.The marginalized likelihood is a way to quantify the similarity of the data and template.If we integrate out all the parameters except total mass, we get a curve that looks like Figure <ref>. Havingin this form is the most useful for our purposes, and plots involvingwill be as a function of total mass. §.§ Framework for directly comparing simulations to observations II: Multidimensional fits and posterior distributionThe posterior distribution for intrinsic parameters, in terms of the marginalized likelihood and assumed prior p(λ) on intrinsic parameters like mass and spin, isp_ post= L_marg(λ )p(λ)/∫ dλ L_ marg(λ ) p(λ ).As wedemonstrate by concrete examples in this work, using a sufficiently dense grid of intrinsic parameters, Eq. (<ref>) indicates that we can reconstruct the full posterior parameter distribution via interpolation or other local approximations. The reconstruction only needs to be accurate near the peak. If the marginalized likelihood L_ marg can be approximated by a d-dimensional Gaussian, with (estimated) maximum valueL_ max,then we anticipate only configurations λ with ln L_ max/ L_ marg(λ)>χ^2_d,ϵ/2contribute to the posterior distribution at the 1-ϵ creditable interval, where χ^2_d,ϵ is the inverse-χ^2 distribution. [The practical significance of this threshold will be more apparent in Section <ref>, which implicitly illustrates it using one dimension.] Since the mass of the system can be trivially rescaled to any value, each NR simulation is represented by particular values for the seven intrinsic parameters (mass ratio and the three components of the spin vectors) and is represented by a one-parameter family of points in the 8-dimensional parameter space of all possible values of λ. Given our NR archive, we evaluate the natural log of the marginalized likelihood as a function of the redshifted mass ln L_ marg(M_z). As in <cit.>, our first-stage result is thisfunction,explored almost continuously in mass and discretely as our fixed simulations permit. This information alone is sufficient to estimate what parameters are consistent with the data: for example, using a cutoff such asEq. (<ref>), we identify the masses that are most consistent for each simulation. As demonstrated first in <cit.> and explored more systematically here, this likelihood is smooth and broad extending over many NR simulations' parameters. As a result, even though our function exploration is a restricted to a discrete grid of NR simulation values, we can interpolate between simulations to reconstruct the entire likelihood and hence entire posterior. We can do this because of the simplicity of the signal, which for the most massive binaries involves only a few cycles. More broadly, our method works because many NR simulations produce very similar radiation, up to an overall mass scale; as a result, as has been described previously in other contexts <cit.>, surprisingly few simulations have been needed toexplore the model space (e.g., for nonprecessing binaries). Finally, as we demonstrate repeatedly below by example, is often well approximated by a simple low-order series, typically just a quadratic. Moreover, for the short GW150914-like signals here, many nonprecessing simulations fit both observations and even precessing simulations fairly well.As a result, we employ a quadratic approximation tonear the peak under the restrictive approximation that all angular momenta are parallel using information from only nonprecessing binaries. Using this fit, we can estimatefor all masses and aligned spins and therefore estimate the full posterior distribution. Section IV B in <cit.> gives the results of this method based on the LIGO data containing GW150914. In this work, we apply this method to a larger set of examples.§ DIAGNOSTICSMany steps in our procedure to compare NR simulations to GW observationscan introduce systematic error into our inferred posterior distribution.Sources of error includethe numerical simulations' resolution; waveform extraction; finite duration; Monte Carlo integration error; the finite, discrete, and sparsely spaced simulation grid; and our fit to said grid. In the following sections, we describe tools to characterize the magnitude and effect of these systematic errors. First and foremost, we introduce the broadly-used match, a complex-valued inner product which arises naturally in data analysis and parameter inference applications. Following many previous studies <cit.>, we review how systematic error shows up as a mismatch and parameter bias. Second, we describe an analogy to the match which uses our full multimodal infrastructure and is more directly connected to our final posterior distribution: the marginalized likelihood versus mass (M), or equivalently (one-dimensional) posterior distribution implied by assuming the data must be drawn from a specific simulation up to overall unknown mass and orientation. Due to systematic error, the inferred one-dimensional distribution (or match versus mass) may change, both globally and through any concrete confidence interval (CI) derived from it.To appropriately quantify the magnitude of these effects, we introduce two measures to compare similar distributions. On the one hand, any change in the 90% CI provides a simple and easily-explained measure of how much an error changes our conclusions. On the one hand, the KL divergence (D_ KL) gives a simple, well-studied, theoretically appropriate, and numerical measure of the difference between two neighboring distributions. In this section we describe these diagnostics and illustrate them using concrete and extreme examples to illustrate how a significant error propagates into our interpretation.§.§ Inner products between waveforms: the mismatch The match is a well-used and data-analysis-driven tool to compare two candidate GW signals in an idealized setting. Unlike most discussions of the match, which derive them from the response of a single idealized instrument, we follow<cit.> and work with the response of an idealized two-detector instrument, with both co-located identical interferometers oriented at 45^o relative to one another, and the source located directly overhead this network.[Equivalently, we work in the limit of many identical detectors, such that the network has equal sensitivity to both polarizations for all source propagation directions.] As is well-known, the match arises naturally in the likelihood of a candidate signal, given known and noise-free data – or, in the notation of this work, fromEq. (<ref>) restricted to this idealized network, setting d to h_0=h(t,λ_0) and h(λ, θ)=h:ln L =-1/2{⟨ h-h_0|h-h_0⟩-⟨ h_0|h_0⟩}=-1/2{⟨ h|h⟩-2 ⟨ h_0|h⟩},whereis the real part.Again ⟨ a|b⟩ is the complex overlap (inner product) between two waveforms for a single detector as shown in Eq. (<ref>); the GW strain h=h_+-i h_× contains two polarizations, and is assumed to propagate from directly overhead the network; the likelihood reflects the response of both detectors' antenna response and noise. Eq. (<ref>) is slightly different than the the likelihood obtained in Eq. (17) of <cit.> by an overall constant. What we use, described in <cit.>, is the likelihood ratio (divided by the likelihood of zero signal). If we add this constant back into the equation, we recover Eq. (17) from <cit.>:ln L_ single=-1/2{⟨ h_0|h_0⟩+⟨ h|h⟩ -2⟨ h_0|h⟩}.This single-detector likelihood dependson the parameters λ,θ of h and λ_o,θ_0 of h_0. For the purposes of our discussion, we will include “systematic error” parameters that enhance or change the model space in λ (e.g., changes in simulation resolution). The parameters which maximize the likelihood identify the configuration of parameters that make h most similar to h_0. For a fixed emission direction from the source, three key parameters in θ dominate how h can be changed to maximize the likelihood: the event time t_ event; the source luminosity distance D_L; andthe polarization angle ψ, characterizing rotations of the source (or detector) about the line of sight connecting the source and instrument.In terms of these parameters, h = e^-2iψD_ L,ref/D_ Lh_ ref(t -t_ event|λ,θ_ rest)where h_ ref is the value of h at D_ L=D_ L,ref,t_ event=0, and ψ=0 and θ_ rest denotes the four remaining extrinsic parameters besides these three.As noted in <cit.>, a change of the polarization angle ψ corresponds to a rotation of the argument of the complex strain function, h(ψ)=e^-2iψh(ψ=0). As a result, maximizing the likelihood versus ψ corresponds to choosing a phase angle so hh_0 is purely real: max_ψ⟨ h_0|h⟩=|⟨ h_0|h⟩|.Similarlymaximizing the likelihood versus distance, the likelihood becomesmax_ψ,D_Lln L_ single=-ρ^2(1-P_*).where in this expressionρ^2=⟨ h_0|h_0⟩=⟨ h|h⟩ and the function P is P_*(h_0,h)≡ max_ψ|⟨ h_0|h⟩|/√(⟨ h_0|h_0⟩⟨ h|h⟩), This partially-maximized likelihood depends strongly on the event time.If we furthermore maximize over event time, we find the final and important relationshipsln L_ single,max =max_ψ,D_L,t_ eventln L_ single=-ρ^2(1-P), P(h_0,h)≡max_ψ,t_ event|⟨ h_0|h⟩|/√(⟨ h_0|h_0⟩⟨ h|h⟩).In the rest of this paper, we will use the mismatch ℳ between two signals:ℳ(h_0,h)=1-P(h_0,h).Because of its form – an inner product – the mismatch identifies differences between the two candidate signals; substituting this expression into the maximized ideal-detector likelihood [Eq.(<ref>)] yields:ln L_ single,max=-ρ^2ℳ.As the above relationships make apparent, a candidate signal h which has a significant mismatch cannot be scaled to resemble h_0 and therefore must be unlikely.This relationship has been used to motivate simple criteria to characterize when two signals h,h_0 are indistinguishable (or, conversely, distinguishable); working to order of magnitude [cf. Eq. (<ref>)], two signals are indistinguishable if <cit.>ℳ≤1/ρ^2.In this work, we apply the match criteria to assess when two simulations of the same or similar parameters (or the same simulation at a different mass) can be distinguished from a reference configuration. As a concrete example, discussed at greater length in Section <ref>, the top-right panel in Figure <ref> shows two plots of mismatchversus total mass.In the black curve,we calculate the match of two identical waveforms from the RIT-1a simulation: one set at a fixed total mass M=70 M_⊙ while the other changes over a given mass range.At the true total mass, the mismatch goes to zero.For comparison, the red curve in that figure shows the mismatch between another simulation h and a fixed RIT-1a (h_0), versus total mass for h.As illustrated in the top-left panel of Figure <ref>, the two simulations are not identical; hence,the mismatch in the top-right panelbetween h and h_0 never reaches zero.Moreover, due to differences in the source h_o and template family h, the location of the minimum mismatch and hence best fit occurs at a different, offset total mass, close to 50 M_⊙. As the reader will see in subsequent sections, we can also calculate the mismatch as a function of particular properties of NR simulations to see how much error is introduced, see Section <ref>.§.§ Marginalized likelihood versus massAnother simple diagnostic is the result (M) for a single simulation on some reference data (e.g., the simulation itself, or a signal with comparable physical origin). This function enters naturally into our full parameter estimation calculation; therefore, it allows us to test all of the quantities that influence our principal result directly including NR resolution, extraction radius, etc. as described below. For simplicity, as computed for the purposes of this test, this function depends on part (only l≤2 modes) of the NR radiation and the data.Figure <ref> shows a null example run with RIT-1a, a GW150914-like simulation, as a source compared against itself.As previous work from both real LIGO and synthetic data has suggested, (M) can be well-approximated by a locally quadratic fit (see Section <ref> for a more in-depth discussion of this example). §.§ Probability Density Function/KL Divergence To quantitatively assess whether two given versions of ln L(M)are demonstrably different, we employ an observationally-motivated diagnostic to prioritize agreement in regions with significant posterior support.Motivated by the applications we perform when comparing results of this kind, we translate ln L(M) into a probability distribution (i.e., assuming all other parameters are fixed):p_c(M)= 1/∫ dM e^e^.In practice, this distribution is always extremely well approximated by a gaussian, so we can further simplify by characterizing any 1d distribution by its mean M_* and variance 1/Γ_MM = σ_*^2.Using this ansatz, we can therefore define a quantity to assess the difference between any pair of results for ln L(M).In this work, we usethe KL divergence between these two approximately-normal distributions: D_KL(p_*|p)=∫ dx p(x) ln p(x)/p_*(x)=lnσ/σ_*-1/2+(x̅-x̅_*)^2+σ_*^2/2σ^2. We also will plot the derived PDF p_c(M) and evaluate the implied 1D 90% CI derived from it.The implications of a significant disagreement for this diagnostic – already illustrated via high mismatch in Figure <ref> – can be clearly seen in the 1D posterior distributions derived from the fit of (M) as shown in Figure <ref> and Figure <ref>.Loosely followingthe work in <cit.> for estimating parameter errors due to mismatch, we expect the parameter error will be a significant fraction of the statistical error.Using the notation above and approximating P ≃ 1 - 1/2Γ̅_xxδ x^2 for some nominal perturbed parameter x, we estimate the statistical error to be σ_x,stat≃ 1/ρ√(Γ_xx).Conversely, balancing mismatch and parameter biases,similar changes in likelihood occur whenδ x ≃1/Γ̅_xx^1/2 M^1/2;however, much more detailed calculations is presented in <cit.>. The above relationship illustrates how a high mismatch causes a deviation in the (M) curve as well as its corresponding posterior distribution.Figure <ref> show a comparison between two waveforms from RIT-1a and RIT-2 (red curve). With significantly different parameters (see Table <ref>), the mismatch is significantly high. This causes a radical shift in the (M) result as well as its corresponding PDF compared to to it's true value.This example will be described in greater detail in Section <ref>.§.§ Example 0: Null test/Impact of Monte Carlo Error To illustrate the use of these diagnostics, we first apply them to the special case where the data contains the response due to a known source.In this case, by construction, the match will be unity when using the same parameters.Following a similar procedureto that we would apply if we didn't know the source mass, we can also plot the mismatch h_A(M)h_A(M_*)/||h_A(M)||||h_A(M_*)||.Referring to the notation in Eq. (<ref>), we assign the RIT-1a waveform to h_0=h_RIT-1a(source) and again the RIT-1a waveform to h=h_RIT-1a (template).This plot can be seen in any of the following examples as the black curve (top-right panels from Figure <ref> and Figure <ref>).It has a peak value of unity (not plotted) and rapidly falls as one moves away from the mass corresponding to the peak match value.The left panel of Figure <ref> shows the log likelihoodprovided by ILE as a function of mass.From here we fit a local quadratic to theclose to the peak.Using the fit, we generate five random samples and use them for subsequent calculations (i.e. 1D distributions).We derived a 1D distribution using Eq. (<ref>).First and foremost, these figures illustrate the relationships between the three diagnostics.As suggested by Eq. (<ref>), the match and log likelihoodare nearly proportional up to an overall constant.Second, as required by Eq. (<ref>), the one-dimensional posterior is proportional to L_ marg.This visual illustration corroborates our earlier claim implicit in the left panel of Figure<ref>:only the part ofwithin a few of its the peak value contributes in any way to the posterior distribution and to any conclusions drawn from it (e.g., the 90% CI).Each evaluation of the Monte Carlo integral has limited accuracy, as indicated in Figure <ref>.By taking advantage of many evaluations of this integral, we dramatically reduce the overall error in the fit. To estimate the impact of this uncertainty, we use standard frequentist polynomial fitting techniques <cit.> to estimate the best fit parameters and their uncertainties (i.e., of a quadratic approximation tonear the peak): if =∑_αλ_α F_α(M_z) and γ_kk = 1/σ_k^2 is an inverse covariance matrix characterizing our measurement errors, then the best-fit estimate forand its variance is _ ,est = F (F^Tγ F)^-1γyΣ(x) = F_α(x)[(F^Tγ F)^-1]_αβ F_β(x) where y is an array representing theestimates at the data points and F is a matrix representing the values of the basis functions on the data points: F_α(x_k).The left panel of Figure<ref> shows the 90% CI derived from this fit, assuming gaussian errors. To translate these uncertainties into changes in the one-dimensional posterior distribution p_c, wegenerate random draws from the corresponding approximately multinomial distribution for fit parameters; and thereby generate random samples and hence one-dimensional distributions for p_c(M) consistent with different realizations of the Monte Carlo errors.The right panel of Figure <ref> shows five random samples from the fit in the left panel.This figure demonstrates this level of Monte Carlo error, by design, has negligible impact on the posterior distribution.To quantify the impact of Monte Carlo error on the posterior, we calculate the KL Divergence from Eq. (<ref>).In all cases, the KL divergence was small, of order10^-4, see Table <ref> for more details on D_KL and the 90% CI. In Section <ref>, we further verify this conclusion by repeating our analysis many times. §.§ Example 1: Two NR simulations with different parameters/Illustrating how sensitively parameters can be measured In this example we compare two NR simulations with significantly different parameters to demonstrate how our diagnostics handle waveforms of extreme contrast. The two NR simulations used are RIT-1a and RIT-2.As shown in Table <ref>, these simulations are both aligned spin with different magnitudes with q=1.22 and q=2.0 respectively.To illustrate the extreme differences between the radiation from these two systems, the top-left panel of Figure <ref> shows the two simulations' r h(t).Our three diagnostics equally reveal the substantial differences between these two signals.To be concrete, since these diagnostics treat data and models asymmetrically, we operate on synthetic data containing RIT-1a with inclination =π/4 in these applications. First, the top-right panel of Figure <ref> shows the results of our mismatch calculations.The black curve is the same null test mismatch calculation as in the top-right panel of Figure <ref>: it has a narrow minimum (of zero) at the true binary mass (70 M_⊙).For the red curve, we calculate the mismatch while holding RIT-2 at a fixed mass and changing the mass of RIT-1a.Using the notation in Eq. (<ref>), we assign the RIT-2 waveform to h_0=h_ RIT-2(fixed mass at M=70 M_⊙) and the RIT-1a waveform to h=h_ RIT-1a(changing mass).In this case, the match does not reach unity, differing by a few percent, whilethe peak value occurs at significantly offset parameters (here, in total mass). Second, the bottom-left panel of Figure <ref>shows the results for (M), using these two NR simulations to look at the same stretch of synthetic data including our local quadratic fit to them.Third, the bottom-right panel of Figure <ref> shows the implied one-dimensional posterior distribution derived from our fits. There is a clear shift in total mass with the null test again peaking around 70 M_⊙ and this example's peak around 50 M_⊙. There are also orders of magnitude difference between theof the two cases.These diagnostics show something that could be seen just by looking at the waveforms; however, we now have some idea on how major differences propagate through our diagnostics and how the error in each diagnostic relate to each other.For completeness, we also include the D_KL and CI for these two waveforms in Table <ref>. The D_KL as well as the CI are both considerably offset, as expected given the two significantly different simulations involved.Finally, the parameter shift seen above is roughly consistent in magnitude with what we would expect for such an extreme mismatch error, given the SNR and match: we expect using Eq. (<ref>)δ M ≃σ_Mρ M^1/2≃ 5 σ_M ≃ 5 M_⊙ (using M = 6× 10^-2,ρ=20 and σ_M = 1.1 M_⊙), or a shift in best fit of several standard deviations and many solar masses. While noticeably smaller than our actual best-fit shift, our result from Eq. (<ref>) provides a valuable sense of the order-of-magnitude biases incurred by specific level of mismatch in general.Moreover, this example is a concrete illustration of the critical need to have M≤ 1/ρ^2 to insure that any systematic parameter biases are small and under control.§.§ Example 2: Differentphysics: SEOB vs NR/Illustrating the value of numerical relativitySeveral studies have previously demonstrated the critical need for numerical relativity, since even the best models do not yet capture all available physics <cit.>.For example, these models generally omit higher-order modes, whose omission will impact inferences about the source <cit.>.To illustrate the value of NR in the context of this work, we compare parameter estimation with NR and with an analytic model.In this particular example, we use NR simulation RIT-1a including the l≤2 modes (see Table <ref>) evaluated along an inclination ι=π/4. Using this line of sight and our fiducial mass (M=70 M_⊙), higher harmonics play a nontrivial role. For our analytical model, we use an Effective-One-Body model with spin (SEOBNRv2), described in <cit.>, which was one of the models used in the parameter estimation of GW150914 <cit.> and which was recently compared to this simulation <cit.>. The top-left panel of Figure <ref> shows the time-domain strains from the NR simulation and SEOBNRv2 with the same parameters.To better quantify the small but visually apparent difference in the two waveforms, we use the diagnostics described earlier on these two waveforms.One way to characterize the differences in these waveforms is the mismatch [Eq. (<ref>)]. In the top-right panel of Figure <ref>, we calculate the mismatch by holding the SEOBNRv2 waveform at a fixed mass while changing the mass of the NR waveform shown in blue.Referring to the notation in Eq. (<ref>), we assign the SEOBNRv2 waveform to h_0=h_ SEOBNRv2 andthe RIT-1a waveform to h=h_ RIT-1a.For comparison, a mismatch calculation was done with the null test from Section <ref> (RIT-1a compared to itself) shown here in black. Two differences between the two curves are immediately apparent.First, the blue curve does not go to zero; the mismatch is a few times 10^-3, significantly in excess of the typical accuracy threshold [Eq. (<ref>), evaluated at ρ=25].Second, the minimum occurs at offset parameters. The best-fit offset and mismatch are qualitatively consistent with the naive estimate presented earlier:a high mismatch yields a high change in total mass [see Eq. (<ref>)].This simple calculation illustrates how mismatch could propagate directly into significant biases in parameter estimation. Another and more observationally relevant way to characterize the differences between these two waveforms is by carrying out a full ILEbased parameter estimation calculation.We carry out four comparisons:the null test (a NR source compared to same NR template (black));the SEOBNRv2 source compared to a SEOBNRv2 template (red); the NR source compared to a SEOBNRv2 template (cyan);and an SEOBNRv2 source compared to a NR template (blue).The bottom panels of Figure <ref> shows both the underlying (M) results; our quadratic approximations to the data; and our implied one-dimensional posterior distributions [Eq. (<ref>)].All ILE calculations were carried out with f_ min=30Hz.All four likelihoodsand posterior distributions p_c are manifestly different, with generally different peak locations and widths.Table <ref> quantifies the differences betweenthe possible four configurations, usingD_KL and 90% CI. The D_KL was always calculated by comparing one of them to the NR/NR case. These systematic differences exist even without higher modes, whose neglect will only exacerbate the biases seen here.Keeping in mind the two figures adopt a comparable color scheme,the shift in peak value and location betweenthe black and blue curves seen in the bottom panels of Figure <ref> can be traced back to the top-right of Figure <ref>: to a first approximation, systematic errors identified by the mismatch (M) show up in the marginalized likelihood (). Again, based on calculations using Eq. (<ref>), we expect the change in mass location of order unity holding all other things equal, comparable to the observed offset.In many ways, one-dimensional biases shown in the bottom-right panel understate the differences between these signals: that comparison explicitly omits the peak value of , which occurs not only at a different location but also with a different value for all four cases.As we would expect, the NR/NR case has the highestwith a peak near the true total mass 70M_⊙. The NR/SEOB case can also produce a peak near 70M_⊙; however, theis orders of magnitude lower, which translates to a lower likelihood that this was in fact the correct template.When performing a full multidimensional fit, template-dependent biases in the peak value ofcan also impact our conclusions.To summarize, we have shown that using SEOBNRv2 in place of a more precise solution of Einstein's equations introduces non-negligible systematic errors, of a magnitude comparable to the statistical error for plausible sources, and that it can impact astrophysical conclusions. §.§ Example 3: Signal duration and cutoff frequency/Illustrating the impact of simulation duration with SEOB Numerical relativity simulations have finite duration.Until hybrids <cit.> are ubiquitously available, these finite duration cutoffs willimpair the utility of direct comparison between data and multimodal NR simulations. To assess this impact of finite simulation duration, we adopta contrived but easily-controlled approach, using an analytic model where we can freely adjust signal duration.While our specific numerical conclusions depend on the noise power spectrum adopted, as it sets the required low-frequency cutoff, the general principles remain true for advanced instruments.In this example, we plot for a fiducial SEOBNRv2 source versus itself using different choices for the low-frequency cutoff(and, equivalently, different initial orbital frequencies for the binary).The left panel of Figure <ref> showsversus M.In this figure, thecurves for f_ min= 10Hz and 20Hz (brown and green) are significantly narrower and higher compared to thecurves for f_ min=30 Hz or 40Hz(red and magenta).As described in <cit.>, even though very little signal power is associated with very low frequencies for this combination of detector and source, a significant amount of information about the total mass is available there with all other parameters of the system perfectly known. These differences are immediately apparent in our one-dimensional diagnostics(M) and p_c(M), which are both narrower and more informative when more information is included (i.e., for lower f_ min).That said, our PSD does not provide access to arbitrarily low frequencies, and the lowest two frequencies have nearly identical posterior distributions, as measured by KL divergence, see Table <ref>.This investigation strongly suggests our analysis couldbe sharper with longer simulations or hybrids. That said, <cit.> demonstratedthis procedure will, for GW150914-like data and noise, arrive at similar results to an analysis which includes these lower frequencies. As noted in <cit.>, this virtue leverages a fortuitous degeneracy in astrophysically relevant observables: the limitations of our high-frequency analysis are mostly washed out due to strongdegeneracies between mass, mass ratio, and spin.§ VALIDATION STUDIES In this section we self-consistently assess our errors in h(t) and .Using the diagnostics described above, via targeted one-dimensional studies, we systematically assess the impact of Monte Carlo error; waveform extraction error; simulation resolution; and limited access to low frequency content.We will show via our diagnostics that the effects from these potential sources of error can be either ignored or mitigated (e.g., by a suitable choice of operating point for our analysis procedure, such as a high enough extraction radius).For each potential source of error, we usethe KL divergence D_KL [Eq. (<ref>)] to quantify small differences in one-dimensional posterior distributions p_c(M) [Eq. (<ref>)] derived from . We will relate our results to familiar mismatch-based measures of error.To be concrete, we will employ a target signal amplitude (SNR) ρ=25, similar to GW150914.For similarly-loud sources, the mismatch criteria [Eq. (<ref>)] suggests any parameters with mismatch below log_10(ℳ)=-2.8 will lead to “statistical errors” (associated with the width of the posterior) will be smaller than systematic biases. §.§ Impact of Monte Carlo errorWe have already assessedthe error from our Monte Carlo integrationin Section <ref>, directly propagating the (assumed correct) Monte Carlo integration error into our fit. To comprehensively demonstrate the impact of Monte Carlo integration error, we repeat our entire analysis reported in Figure <ref>multiple times.Figure <ref> shows our directly comparable results; Table <ref> reports quantitative measures of how these distributions change.Based on these quantities, we conclude the error introduced by our Mont Carlo is negligible.Our results are consistent with Section <ref>.§.§ Error budget for waveform extraction While gravitational waves are defined at null infinity, the finite size of typical NR computational domains implies a computational technique must identify the appropriate asymptotic radiation from the simulation<cit.>.This method generally has error, often associated with systematic neglect of near-field physics in the asymptotic expansion used to extract the wave (i.e., truncation error).Our perturbative extrapolation method shares this limitation.As a result, if we decrease the radius at which we extract the asymptotic strain, we increase the error in our approximation.In other words, the mismatch between the waveform extracted at r and some large radius generally decreases with r; the trend of match versus r provides clues into the reliability of our results. Figure <ref> shows an example of a mismatch between two estimates of the strain: one evaluated at finite, largest possible radius and one at smaller (and variable) radius.For context, we show the nominal accuracy requirement corresponding to a SNR=25 [see Eq. (<ref>)] as ablack dotted line. First and foremost, this figure shows that, at sufficiently high extraction radius, the error introduced by mismatch errors is substantially below our fiducial threshold for all choices of: cutoff frequency, waveform extraction location, and waveform extraction technique; see also <cit.>.Second, the second panel shows our perturbative extraction method is reasonably consistent with an entirely independent approach to waveform extraction. Agreement is far from perfect: our study also indicates a noticeable discrepancy between the results of our perturbative extraction technique and the SXS strain extraction method.Due to the good agreement reported elsewhere <cit.>, we suspect these residual disagreements arise from coordinate effects unique to our interpretation of SXS data; we will assess this issue at greater depth in subsequent work. Third and finally, as expected, comparisons that employ more of the NR signals are more discriminating: calculations with a smaller f_ min generally find a higher (i.e., worse) mismatch. Nonetheless, our mismatch calculations significantly improve at large extraction radius, when perturbative extrapolation is carried out well outside the near zone.To assess the observational impact of waveform extraction systematics, we evaluate (M) and p_c(M) using waveform estimates produced using different extraction radii.Specifically, we take a simulation; use its large-radius perturbative estimate as a source; and follow the procedures used in Figures <ref> and <ref> to produce (M) and p_c(M). Figure <ref> shows our results; for clarity, we include only the last three extraction radii(r=190M, 162M, 141M). The errors here are relatively small but bigger than expected from our match study; however, the error shown in the match only applies to changes in the peak value , which can be seen in the left panel. To again quantify these small differences, we use D_KL and CI, as reported in Table <ref>.As this table shows, the error introduced is insignificant as long as we pick a relative large extraction radius. This is almost always the case for the current simulations available.Some of the GT simulations require us to chose a lower extraction radius due to an increase in the error asthe extraction radius increases beyond a certain point, but this does not affect our overall results.§.§ Impact of simulation resolution Here we analyze errors introduced by different numerical resolutions.Higher resolutions simulations take longer to run and computationally cost more than lower resolution ones.If the effects of different resolutions are insignificant, numerical relativist will be able to run at a lower resolution while not introducing any systematic errors.Table <ref> shows a match comparison between the highest resolution RIT-1a and the two lower ones, RIT-1b and RIT-1c.The mismatches are orders of magnitudes better than our accuracy requirement (∼10^-2.8), and therefore introduce errors that are negligible.Usingas our diagnostic to compare these three simulations, we draw similar conclusions; see Figure <ref>.We again see a error so small that changes between the three curves are almost impossible to see, even far from the peak.Table <ref> quantifies these extremely small differences. In short,different resolutions have no noticeable impact on our conclusions.While this resolution study was only done for a aligned RIT simulation, similar conclusions are expected when a wider range of simulations are used. Even though in this case the mismatch and ILE studies show conclusively the minimal impact the numerical resolution has on the waveform, we generate 1D distributions from the fits for completeness.It is not surprising to see in the right panel of Figure <ref> the posteriors from the three fits match almost exactly.To quantify this similarity, we calculate D_KL as well as the CI for the corresponding PDFs.Based on the D_KL, these distributions are clearly identical and using different resolutions does not effect the waveform in any significant way.This resolution study was only done for an aligned RIT simulation;while extraction radius studies have been performed for SXS for other extraction procedures <cit.>, a similar resolution investigation needs to be done for SXS simulations for this extraction method. We hypothesize that this effect will also be minimal.§.§ Impact of low frequency content and simulation durationAs demonstrated by Example 3 in Section <ref> above, the available frequency content provided by each simulation and used to the interpret the data can significantly impact our interpretation of results.In this section, we perform a more systematic analysis of simulation duration and frequency content, again using the semi-analytic SEOBNRv2 model as a concrete waveform available at all necessary durations.Before we begin, we first carefully distinguish between two unrelated “minimum frequencies” that naturally show up in our analysis.It is easy to get confused between the low frequency cutoff (in this work called f_ min) and simulation duration (or initial orbital frequency Mω_0).The simulation duration is the true duration of the simulation, which is a property of the binary and can be drastically different over many NR simulations.The low frequency cutoff is an artificial cut to the signal that allows us to normalize the signal duration of all our waveforms.As a result, with a lower f_ min, more of the NR simulation enters into our analysis. The top panels of Figure <ref> shows the result of compare a RIT-4 source with a duration of 5.0 Hz to itself with changing f_ min.As f_ min increases, a smaller portion of the simulation waveform is being used to analyze the data.When f_ min is high, we end up cutting off more of the waveform.This results in a sharp decline in since one is now comparing less of the waveform to itself.In this panel it is clear that f_ min∼10-20Hz seems to not significantly affect ; however, the curve changes drastically when f_ min=30-40Hz.For completeness Table <ref> shows the corresponding D_KL and CI for different f_ min, again showing the similarities between the f_ min=10,20 Hz frequencies and the differences of the higher frequencies.Hybrid NR waveforms will nullify this source of error by allowing us to compare more of the waveform while at the same time allowing us to standardize durations.To investigate the shift in mass seen in Figure <ref> further, we compare a SEOBNRv2 source to a SEOBNRv2 template with the same duration/f_ min (i.e. the source has a duration of 10 Hz therefore the template has a f_ min=10 Hz).This was done to investigate the shift in total mass seen in Figure <ref> for a SEOBNRv2 source with a fixed duration compared to a SEOBNRv2 template with different low frequency cutoffs.As the bottom panels of Figure <ref> now show, this shift was a product of comparing a source and templates with different signal lengths.When we now set the same duration for the source and f_ min for the template, the ILE results and their corresponding PDFs peak around the same mass point.We still see a widening of the curves with increasing f_ min; this corresponds to a wider and shorter PDF.We calculate D_KL and CI for this case as well, see Table <ref>.These values shows that f_ min=10,20Hz are relatively similar while the higher frequencies are significantly different. § ADVANTAGES TO USING NUMERICAL RELATIVITY§.§ Impact of higher order modes Needs to be rewritten, to reflect recent prior work! Figure needs to be improvedAs shown by Example 1 in Section <ref> and by others, the semi-analytic models currently used for parameter estimation fails to capture all the information contained in the NR waveform <cit.>.Building off their insight on the power of individual modes, we picked synthetic source to focus on in this section.By construction, this source is an example of where two of the more developed waveform models (IMRPhenomPv2 and SEOBNRv4ROM) fail.Since every NR simulation is different, their dependence on higher order modes varies.The top panels of Figure <ref> show the individual SNR (normalized by the dominant (2,2) mode) of RIT-1a.The dash represents our accuracy requirement above which a mode begins to impact our results significantly.If you take the case for a system ∼70 M_⊙, higher modes' impact become significant; however, this dash line assumes a SNR set by us. If we pick a SNR closer the SNR of GW150914 (ρ∼20), the dash moves upward to -2.6. This pushes the point at which higher modes start to impact our results at higher masses. Therefore in the case of the real event, the l=2 modes were almost sufficient to recover the parameters of the source, and our results more or less agree with <cit.>.If we instead had a more exotic source, say with a higher q, more modes would be required to accurately recover the parameters.The bottom panels of Figure <ref> shows the SNR of individual modes of RIT-2 (q=2.0) normalized by the dominant (2,2) mode.If we assume the same total mass and a ρ=25, multiple modes contribute to the source.In this particular case, lℓ 3 is still not sufficient; therefore, l≤ 4 modes would be needed to extract all the information from the waveform.With the use of our method, we can now build off of previous work <cit.> and use parameter estimation with higher modes to extract all the important information in all relevant modes. §.§ Comparison ofobtained with SEOB and with NRNeeds to be rewritten or improved The agreement between NR and other semi-analytic models is heavy influenced by the particular system's dependence of higher order modes.Even for a system that, based on the top panels of Figure <ref>, does not have much contribution for higher order modes, the differences between SEOB and NR are different enough to significantly impact our final results, see Section <ref>.A study with a SEOB waveform based on a NR simulation that included more modes, RIT-2, would see a even bigger discrepancy.Using NR instead of SEOB allows for better recovery of the parameters by including higher modes and allows for the circumvention of approximations introduced in analytic models.§RECONSTRUCTING PROPERTIES OF SYNTHETIC DATA I: ZERO, ALIGNED, AND PRECESSING SPIN This section is dedicated to end-to-end demonstrations of this parameter estimation technique. Unless otherwise specified, we adopt a total binary mass of M=70 M_⊙ and use the fiducial early-O1 PSD <cit.> to qualitatively reproduce the characteristic features of data analysis for GW150914. Without loss of generality and consistent with common practice, we adopt a “zero noise" realization (i.e., the data used for each instrument is equal to its expected response to our synthetic source).Table <ref> is a list of simulations we have used as sources in our end-to-end runs; these include zero, aligned, and precessing systems all at different inclinations. Here we start with a end-to-end demonstration with zero spin from SXS. §.§ Zero Spin: A fiducial example demonstrating the method's validityWe first illustrate the simplest possible and most-well-studied scenario: a compact binary with zero spin and equal mass, as represented here by SXS-1.To enable comparison with other cases where higher-order modes will be more significant, we adopt inclinations =0,0.5,0.785,1.0,1.5,2.35.For the purposes of illustration, we present our end-to-end plots using an inclination =0.The left panel of Figure <ref> shows χ_ eff vs 1/q; the points represent the maximum log likelihoodof all the different ILE runs across parameter space.The green contour is the 90% CI derived using the quadratic fit tofor nonprecessing systems only.The colored points represent points that fall in<127 region with the red points representing higherand violet represent lower .The gray points represent points that fall between =130 and =127.The black points represent points that fall in >130.These intervals were determined using the inverse χ^2 distribution [see Eq. (<ref>)] adopting d=4 (two masses with aligned spin) for the black points and d=8 (two masses with precessing spins).This CI is consistent with the point distribution >130 (i.e. black points), which represents the points closest to the maximum.The right panel of Figure <ref> shows the χ_ eff vs M with the same green contour and black point distribution.As with the left panel, the green contour is consistent with the black point distribution.Both plots recover the true parameters (indicated by the big red dot) with regards to the confidence interval and the black point distributions.The left panel of Figure <ref> shows the χ_1z vs χ_2z where χ_1z,2z is the z component of the dimensionless spin [see Eq. (<ref>)].All the colors here represent the same as in Figure <ref>.We again see that the green contour is consistent with the black point distribution.The right panel of Figure <ref> shows the 1D posteriors for 1/q for six different inclinations. These produce distributions we expect to see; all the curves from the different inclinations lie on top of each other. This implies that higher order modes for this particular case are not expected to provide any extra information.By construction, this source needs no higher order modes to completely recover the parameters.Since all inclinations have the same distribution shape, the results here are independent of inclination at a fixed SNR. §.§ Nonprecessing binaries: unequal mass ratios and aligned spin In the previous zero spin case, the higher order modes had a minimal impact.Now we introduce an aligned spin GW150914-like simulation as the source, SXS-0233.For our total mass of M=70 M_⊙, we expect that the impact of higher order modes border on being significant.Because of this, we did 2 end-to-end runs with SXS-0233: one with l≤2 and the other with l≤ 3.The panels in Figure <ref> are the same type of plots as in the previous case; however, we have also included a contour representing the 90% CI for l≤ 3 (green dashed line).In the left panel of Figure <ref>, the posterior corresponding to l≤3 better constrains the mass ratio than that of the posterior corresponding to l≤2.In this case, including higher order modes provides more information about the mass ratio, allowing us to constrain it more tightly. The right panel of Figure <ref> is the same type of plot as the bottom panel of Figure <ref>; however, this includes the results from the l≤3 runs.Since thewas higher, the number of black and gray points slightly decreased.It is clear from these two plots that higher order modes are significant and need to be included for this source to get the best possible constrains on the parameters.The right panel in Figure <ref> shows the χ_ eff vs M; these show little difference between the l≤2 and the l≤ 3 contours.The contours agree very well with each as well as the black points' distribution in both panels of Figure <ref>.We recover the true parameters in both plots and with l≤2 and l≤3; however, we can better constrain q with higher order modes. As with the zero spin case, we plot as a function of χ_1z and χ_2z in the left panel in Figure <ref>.Here again the dashed and solid green contour represents the confidence interval for l≤2 and l≤3 respectively and are largely consistent with each other.The right panel of Figure <ref> shows the 1D distributions for 1/q for different inclination values.The difference in the curves here could be explained by higher order modes; however, more needs to be done to corroborate this hypothesis. In this particular case, higher order modes have a relatively modest impact on the posterior.The minimal impact is by design: moving away from zero spin and equal mass within the posterior of GW150914, we have explicitly selected a point in parameter space where higher-order modes have just become marginally significant. Even remaining within the posterior of GW150914, as we move towards more extreme antisymmetric spins and mass ratios, higher-order modes can play an increasingly significant role.We will address this issue further in subsequent work. §.§ Precessing binaries: unequal mass ratios and precessing spin, but short duration Since all the fits in this study have only used the nonprecessing binaries, one might come to the conclusion that this limits us to analyzing only zero spin and aligned source.We can potentially recover parameters of precessing sources if the duration of these sources are short enough; this translates to only a few cycles and therefore little to no precession before merger, see before Eq. (9) in <cit.>.Figure <ref>are the same type of plots as in Figure <ref>. Here the gray points represent points that fall between =165 and =163, and the black points represent points that fall in >165.The colored points represent the points that fall in the region <163 with the red points represent the highervalues.As with the previous cases, these intervals were determined using the inverse χ^2 distribution [see Eq. (<ref>)] adopting d=4 (two masses with aligned spin) for the black points and d=8 (two masses with precessing spins) for the gray points.As we expected, the short duration of this source allows us to recover the parameters with a fit that only uses the nonprecessing cases as shown in the left panel of Figure <ref>. Here we plot the (M) of a single null run of ILE comparing SXS-0234v2 with itself (black) andthe whole end-to-end (M) using SXS-0234v2 as the source.By construction, thefrom the null run of SXS-0234v2 is the highest (M) possible. If the maximumfrom the whole end-to-end run is close (Δln L≤1), we can recover the parameters of the simulations without fitting with the precessing systems. In this case, the Δln L=0.97.We can therefore accurately recover the parameters of this precessingsystem as evident by Figure <ref>.[When interpreting the above statement, however, it is important to note our analysis by construction uses only information f>30 Hz.If we had access to a wider range of long simulations, we could have access to information from precession cycles between 10-30Hz, even for sources of this kind and in this data.More work is needed to assess the prospects for recovery for longer, more generic sources.]We again showas a function of χ_1z and χ_2z in the left panel of Figure <ref> with all the colors and contours representing the as in Figure <ref>.The green contour are consistent with the black point distribution.We again plot the 1D distribution for 1/q for different inclinations in the right panel of Figure <ref> with all the colors corresponding to the same inclinations as in the right panel of Figure <ref>.Here we see relative consistency between the different inclinations, with a consistent trend towards extracting marginally more information as the inclination increases. We have an outlier for =1.5: a nearly edge-on line of sight.For such a line of sight, keeping in mind we tune the source distance to fix the network SNR, precession-induced modulations are amplified; this outlier could and probably does represent the impact of precession. To investigate this further, we again plot (M) of a single null run of ILE comparing SXS-0234v2 with itself (black) andthe whole end-to-end (M) using SXS-0234v2 with =1.5 as the source, see the right panel of Figure <ref>. By construction, thefrom the null run of SXS-0234v2 is the highest (M) possible.Here we find a bigger difference betweenof the null run andof the entire end-to-end run: Δln L∼1.8. We then take all the individual runs from the end-to-end runs that compared 0234v2 to itself and plot (M) for each inclination.As evident in Figure <ref>, the =1.5 curve lies well below the rest of the inclinations.More investigations are needed to be done to figure out this discrepancy; however, this could imply SXS-0234v2 has many modes that are relevant, reflecting precession-induced modulation most apparent perpendicular to J̅ the total angular momentum vector.In future work, where we attempt to recover all spin degrees of freedom for precessing sources, we will focus in particular on edge-on lines of sight like this. § CONCLUSIONS We have presented and assesseda method to directly interpret real gravitational wave data by comparison to numerical solutions of Einstein's equations.This method can employ existing harmonics and physics that has been or can be modeled.While any other method can do so as well if suitable models have been developed and calibrated, this method skips the step of translating NR results into model improvements, circumventing the effort and potential biases introduced in doing so. We also provided a detailed systematic study of the potential errors introduced in our method.We first used the overlap or mismatch to assess the difference between different simulations along fiducial lines of sight.As noted in Eq. (<ref>), we expect that is approximately proportional to the mismatch by an overall constant.We demonstrate this relationship explicitly, using NR sources and synthetic data. Once we obtained , we fitted with a simple quadratic and derived a PDF using Eq. (<ref>) with its corresponding 90% CI. Using the PDFs, we can graphically see any errors that would have been propagated through.To quantify this change, we calculated a KL Divergence between two PDFs [see Eq. (<ref>)].By using these diagnostics,we addressed and quantified systematic errors that could affect our parameter estimation results.Our validation studies systematically assessed the impact of (a)Monte Carlo error, (b) waveform extraction error, (c) simulation resolution, and (d) low frequency cutoff/signal duration via our diagnostics. * (a) Based on our results from our examples, we were confident that the error from our Monte Carlo integration would be small.To quantify the results that seem apparent by eye,we applied our diagnostics (omitting the mismatch) and found the D_KL between the PDFs (i.e. D_KL(v1,v1), D_KL(v1,v2), D_KL(v1,v3)) to be all D_KL∼10^-5. * (b)In a similar fashion, we applied our diagnostics to GW150914-like simulations from the SXS and RIT NR groups.We validated the utility of the perturbative extraction technique but noted some differences between the strain provided by SXS and perturbative extraction applied to their ψ_4 data.Based on excellent agreement between RIT (with perturbative extraction) and SXS provided strain, we expect the discrepancies relate to improper assumptions regarding SXS coordinates.More needs to be done to discover the origin of this disparity. From our match study, we determined that the impact of the error due to waveform extraction is insignificant at a largeenough extraction radius.This was validated via the D_KL between three PDFs with the highest possible extractionradii, which were all around 10^-2-10^-3.* (c)When using our mismatch study to assess the impact of resolutionerror, it was determined that the mismatch for all the different resolution was ℳ∼10^-5.Thisseemingly small difference in the waveform was then reaffirmed by the corresponding D_KL∼10^-4-10^-5.Fromour diagnostics, it was clear that the error introduced by numerical resolution was negligible.* (d) We finally used our diagnostics to the assess impact of low frequency cutoffs and signal duration.For both NR and analytic models, the available frequency content provided can significantly affect our results.After deriving our PDFs and calculating the D_KL, we found the lower f_ min (10,20 Hz) were very similar with a narrow PDF and a high peak while the higher f_ min (30,40 Hz) produced a wider PDF with a lower peak.We stress the importance of the hybridization of the NR waveforms to allow for a low f_ min to standardization NR waveforms while providing the longest waveform possible. We also provided three end-to-end examples with three different types of sources.First, we used a simple example – zero spin equal mass, where no significant higher order modes complicate our interpretation –to show our method works.Second, we examined an aligned, GW150914-like, unequal mass source.Though the leading-order quadruple radiation from such a source is nearly degenerate with an equal mass, zero spin system, this binary has asymmetries which produce higher order modes.We used our method with the l≤2 as well as the l≤ 3 modes and found we could better constrain q using higher modes.We also found significant differences between the 1D probability distributions for 1/q; this implied that higher modes were significant.Third, we used our method on a precessing but short unequal mass source.Due to its short duration of the observationally accessible signal, this comparable-mass binary has little to no time to precess in band.This allows us to recover the parameters of the binary even though we construct a fit based on the nonprecessing binaries.Even though the recovery of parameters was possible, the edge-on case for our 1D distributions were significantly different than the rest.For this line of sight, precession-induced modulations are most significant; the simplifying approximation that allowed success for the other lines of sight break down.Even though we suspect this is also due to higher order modes, more needs to be done to validate this claim.In the future, we will extend thisstrategy to recover parameters of generic precessing sources.The method presented here relies on interpolation between existing simulations of quasi-circular black hole binary mergers.For nonprecessing binaries, this three-dimensional space has been reasonably well-explored. For generic quasi-circular mergers, however, substantially more simulations may be required to fill the seven-dimensional parameter space sufficiently for this method. Fortunately,targeted followup numerical simulations of heavy binary black holes arealways possible.These simulations will be incredibly valuable to validate any inferences about binary black holemergers, from this or any other method.For this method in particular, followup simulations can be used to directlyassess our estimates, and revise them.We will outline followup strategies and iterative fitting proceduresin subsequent work.The RIT authors gratefully acknowledge the NSF for financial support from Grants: No. PHY-1505629, No. AST-1664362 No. PHY-1607520, No. ACI-1550436, No. AST-1516150, and No. ACI-1516125. Computational resources were provided by XSEDE allocation TG-PHY060027N, and by NewHorizons and BlueSky Clustersat Rochester Institute of Technology, which were supported by NSF grant No. PHY-0722703, DMS-0820923, AST-1028087, and PHY-1229173. This research was also part of the Blue Waters sustained-petascale computing NSF projects ACI-0832606, ACI-1238993, and OCI-1515969, OCI-0725070.The SXS collaboration authors gratefully acknowledge the NSF for financial support from Grants: No. PHY-1307489, No. PHY-1606522, PHY-1606654, and AST- 1333129. They alsogratefully acknowledge support for this research at CITA from NSERCof Canada, the Ontario Early Researcher Awards Program, the Canada Research Chairs Program, and the Canadian Institute for AdvancedResearch.Calculations were done on the ORCA computer cluster, supported by NSF grant PHY-1429873, the Research Corporation for Science Advancement, CSU Fullerton, the GPC supercomputer at the SciNet HPC Consortium <cit.>; SciNet is funded by: the Canada Foundation for Innovation (CFI) under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund (ORF) – Research Excellence; and the University of Toronto. Further calculations were performed on the Briarée cluster at Sherbrooke University, managed by Calcul Québec and Compute Canada and with operation funded by the Canada Foundation for Innovation (CFI), Ministére de l'Économie, de l'Innovation et des Exportations du Quebec (MEIE), RMGA and the Fonds de recherche du Québec - Nature et Technologies (FRQ-NT). The GT authors gratefully acknowledge the NSF for financial support from Grants: No. ACI-1550461 and No. PHY-1505824. Computational resources were provided by XSEDE and the Georgia Tech Cygnus Cluster.Finally, the authors are grateful for computational resources used for the parameter estimation runs provided by the Leonard E Parker Center for Gravitation, Cosmology and Astrophysics at the University of Wisconsin-Milwaukee; the Albert Einstein Institute at Hanover, Germany; and the California Institute of Technology at Pasadena, California.unsrt§ EXPLORING THE PARAMETER SPACE In this appendix, we provide additional examples of our method using numerical relativity simulations in different regions of parameter space.We demonstrate our method works reliably for extreme black hole spins (Figure <ref>) as well asin regions where fewsimulations with comparable parameters are available(Figures <ref> and <ref>). For the parameters of each source, see the following source labels (in order as they appear) in Table <ref>: RIT-5, SXS-high-antispin, SXS-χ_ eff0.4, and RIT-2. | http://arxiv.org/abs/1705.09833v1 | {
"authors": [
"Jacob Lange",
"Richard O'Shaughnessy",
"Michael Boyle",
"Juan Calderón Bustillo",
"Manuela Campanelli",
"Tony Chu",
"James A. Clark",
"Nicholas Demos",
"Heather Fong",
"James Healy",
"Daniel Hemberger",
"Ian Hinder",
"Karan Jani",
"Bhavesh Khamesra",
"Lawrence E. Kidder",
"Prayush Kumar",
"Pablo Laguna",
"Carlos O. Lousto",
"Geoffrey Lovelace",
"Serguei Ossokine",
"Harald Pfeiffer",
"Mark A. Scheel",
"Deirdre Shoemaker",
"Bela Szilagyi",
"Saul Teukolsky",
"Yosef Zlochower"
],
"categories": [
"gr-qc"
],
"primary_category": "gr-qc",
"published": "20170527154840",
"title": "A Parameter Estimation Method that Directly Compares Gravitational Wave Observations to Numerical Relativity"
} |
[email protected] Departamento de Física, Universidade Federal de Santa Catarina, 88040-900, Florianópolis, SC, Brazil [email protected] Departamento de Física, Universidade Federal de Santa Catarina, 88040-900, Florianópolis, SC, Brazil We present a Monte Carlo study of the magnetic properties of an Ising multilayer ferrimagnet. The system consists of two kinds of non-equivalent planes, one of which is site-diluted. All intralayer couplings are ferromagnetic. The different kinds of planes are stacked alternately and the interlayer couplings are antiferromagnetic. We perform the simulations using the Wolff algorithm and employ multiple histogram reweighting and finite-size scaling methods to analyze the data with special emphasis on the study of compensation phenomena. Compensation and critical temperatures of the system are obtained as functions of the Hamiltonian parameters and we present a detailed discussion about the contribution of each parameter to the presence or absence of the compensation effect. A comparison is presented between our results and those reported in the literature for the same model using the pair approximation. We also compare our results with those obtained through both the pair approximation and Monte Carlo simulations for the bilayer system. 05.10.Ln; 05.50.+q; 75.10.Hk; 75.50.GgMonte Carlo study of an anisotropic Ising multilayer with antiferromagnetic interlayer couplings N. S. Branco December 30, 2023 ================================================================================================§ INTRODUCTION The study of ferrimagnetic materials has attracted considerable attention in the last few decades, especially since a number of phenomena associated with these materials present a great potential for technological applications <cit.>. In such systems the different temperature behavior of the sublattice magnetizations may cause the appearance of compensation points, i. e., temperatures below the critical point for which the total magnetization is zero while the individual sublattices remain magnetically ordered <cit.>. Although unrelated to critical phenomena, at the compensation pointthere are physical properties, such as the magnetic coercivity of the system, that exhibit a singular behavior <cit.>. The fact that the compensation point of some ferrimagnets occurs near room temperature makes them particularly important for applications in magneto-optical drives <cit.>. Initially, compensation effects were theoretically studied in bipartite lattices with different spin magnitudes in each sublattice <cit.>. However, this is not the only possible geometry which may lead to compensation phenomena. In particular, layered ferrimagnets have been extensively studied in the recent past <cit.>. In the latter case, systems are composed of stacked planes with different magnetic properties and the realization of antiferromagnetic couplings between adjacent layers has important technological applications such as in magneto-optical recordings <cit.>, spintronics <cit.>, the giant magnetorresistance <cit.>, and the magnetocaloric effect <cit.>. In addition, the study of the magnetic properties of these systems is of great theoretical interest since it can provide insight into the crossover between the characteristic behavior of two- and three-dimensional magnets.In recent experimental works, we find examples of realization and study of such bilayer <cit.>, trilayer <cit.>, and multilayer <cit.> systems. From the theoretical stance, a bilayer system with Ising spins and no dilution has been studied via transfer matrix (TM) <cit.>, renormalization group (RG) <cit.>, mean-field approximation (MFA) <cit.>, and Monte Carlo (MC) simulations <cit.>. A similar system with both Ising and Heisenberg spins has been studied in the pair approximation (PA) both without dilution <cit.> and with dilution <cit.>. There is also a recent work considering an Ising bilayer with site dilution in an MC approach <cit.>. For the multilayer system, the PA has also been applied to the model with Ising and Heisenberg spins, both with no dilution <cit.> and with dilution <cit.>.If all layers have the same spin (e. g., spin-1/2) and we have an even number of layers, it is necessary that different layers have different number of spins in them for the existence of a compensation effect, as discussed in Refs. balcerzak2014ferrimagnetism, szalowski2014normal, diaz2016monte. It is also necessary that the layers have asymmetric intralayer exchange integrals and that the layers with stronger exchange integrals have less atoms than their weak-exchange counterparts. That is easy to achieve with site dilution. However, to the best of our knowledge, no numerical simulation methods have yet been applied to multilayer systems with site dilution.With that in mind, in this work we present a Monte Carlo study of the magnetic properties of a spin-1/2 Ising system composed of two kinds of non-equivalent planes, A and B, stacked alternately. All intralayer interactions are ferromagnetic while the interlayer interactions are antiferromagnetic. We also consider the presence of site dilution in one of the kinds of planes. The simulations are performed with the Wolff algorithm <cit.> and with the aid of a reweighting multiple histogram technique <cit.>. The model is presented in Sec. <ref>, the simulation and data analysis methods are discussed in Sec. <ref>, and the results are presented and discussed in Sec. <ref>.§ MODEL AND OBSERVABLESThe multilayer system we study consists of a simple cubic crystalline lattice such that non-equivalent monolayers (A and B) are stacked alternately (see Fig. <ref>). The A planes are composed exclusively of A-type atoms while the B planes have B-type atoms as well as non-magnetic impurities. The Hamiltonian describing our system is of the Ising type with spin 1/2 and can be written as-βℋ =∑_⟨ i∈ A,j∈ A⟩K_AAs_i s_j +∑_⟨ i∈ A,j∈ B⟩K_ABs_i s_jϵ_j +∑_⟨ i∈ B,j∈ B⟩K_BBs_i s_jϵ_iϵ_j,where the sums run over nearest neighbors, β≡ (k_BT)^-1, T is the temperature, k_B is the Boltzmann constant, the spin variables s_i assume the values ± 1, the occupation variables ϵ_i are uncorrelated quenched random variables which take on the values ϵ_i=1 with probability p (spin concentration) or ϵ_i=0 with probability 1-p (spin dilution). The couplings are K_AA>0 for an AA pair, K_BB>0 for a BB pair, and K_AB<0 for an AB pair. The corresponding exchange integrals (see Fig. <ref>) are given by J_γδ=β^-1K_γδ, where γ = A, B and δ = A, B. For the purpose of the numerical analysis to be discussed in Sec. <ref>, we define some observables to be measured in our simulation. Namely the dimensionless extensive energy is given by E≡ℋ/J_BB, and the magnetizations of A-type atoms and B-type atoms are, respectivelym_A=1/N_A∑_i∈ As_i, m_B=1/N_B∑_j∈ Bs_jϵ_j,where N_A=L^3/2 is the total numbers of A-type atoms in the system and N_B=pL^3/2 is the number of B-type atoms. The total magnetization of the system ism_ =1/2(m_A+pm_B). We also define the magnetic susceptibilitiesχ_γ =N_γ K(⟨ m_γ^2⟩-⟨ |m_γ|⟩^2),where ⟨⋯⟩ is the thermal average for a single disorder configuration whereas the over-line denotes the subsequent average over disorder configurations, K=β J_BB is the inverse dimensionless temperature, and γ=A, B,. The total number of atoms in the system is N_=N_A+N_B. § MONTE CARLO METHODS §.§ Simulational detailsWe employed the Wolf single-cluster algorithm <cit.> for the MC analysis of Hamiltonian (<ref>) on cubic lattices of size L^3 with periodic boundary conditions. We performed simulations for linear sizes L from 10 to 100 and for a range of values of the Hamiltonian parameters: 0.0<p≤ 1.0, 0.0<J_AA/J_BB≤ 1.0, and -1.0≤ J_AB/J_BB<0.0. All random numbers were generated using the Mersenne Twister pseudo-random number generator <cit.>.For each set of values chosen for the parameters above, we performed simulations in a range of temperatures close to either the critical point or the compensation point. To determine T_comp we typically divided the range in 5 to 10 equally spaced temperatures whereas for T_c we used from 8 to 17 equally spaced temperatures. In a single simulation for the largest systems considered, i. e., L=100, we performed up to 2× 10^5 steps (Wolff single-cluster updates) and discarded up to 7× 10^4 steps to account for equilibration. The total number of steps after equilibration was always at least 2000 times the relevant integrated autocorrelation time <cit.>.To analyze the data generated in the simulations for a particular range of temperatures, we use the multiple-histogram method <cit.> to compute the observables defined in the previous section at any temperature inside this range. The thermal error associated with those observables is estimated via the blocking method <cit.>, in which we divide the data from each simulation in blocks and repeat the multiple-histogram procedure for each block. The errors are the standard deviation of the values obtained for a given observable for different blocks.The process is repeated for N_s samples of quenched disorder to obtain the final estimate of our observable. We chose 10≤ N_s≤ 50, such that the error due to disorder was approximately the same as the thermal error obtained for each disorder configuration. Finally, we sum both thermal and disorder errors for an estimate of the total error. §.§ Determining T_c In this work we wish to determine the critical point accurately. This is accomplished by means of a finite-size scaling analysis <cit.>, in which we examine the size dependency of certain observables measured for finite systems of several sizes and extrapolate these results to the thermodynamic limit, i. e., L→∞. In this approach, the singular part of the free energy density for a system of size L, near the critical point, is given by the scaling formf̅_sing(t,h,L) ∼ L^-(2-α)/νf^0(tL^1/ν, hL^(γ+β)/ν)where t is the reduced temperature, t=(T-T_c)/T_c, T_c is the critical temperature of the infinite system, and h is the external magnetic field given in units of k_BT. The critical exponents α, β, γ and ν are the traditional ones associated, respectively, with the specific heat, magnetization, magnetic susceptibility, and correlation length. Various thermodynamic properties can be determined from Eq. (<ref>) and other observables have their corresponding scaling forms, e. g., for the magnetic susceptibility at null magnetic field we haveχ_ = L^γ/ν𝒳(x_t),where x_t≡ tL^1/ν is the temperature scaling variable.Eq. (<ref>) provides a powerful method to determine the critical point. It is clear from this scaling law that χ_ diverges at the critical point only in the thermodynamic limit, whereas for a finite system size L, χ_ does not diverge but has a maximum at a pseudo-critical temperature T_c(L), which asymptotically approaches the real T_c as L increases. The maximum occurs when.d𝒳(x_t)/dx_t|_T=T_c(L)=0,which yields the following relationT_c(L) = T_c+AL^-1/ν,where A is a constant, T_c is the critical temperature and ν is the critical exponent associated with the correlation length.The finite-size scaling method is applicable to different quantities, such as specific heat and other thermodynamic derivatives <cit.>. We expect the results obtained from the scaling behavior of these other quantities to be consistent, as we were able to verify in preliminary simulations. In this study, however, we focused only on the peak temperatures of the magnetic susceptibilities, defined in Eq. (<ref>), for these peak temperatures occurred fairly close to one another and were the sharpest peaks from all the quantities initially considered.To determine the pseudo-critical temperatures T_c(L), we perform simulations close to the peaks of the susceptibilities and use the multiple-histogram method to obtain χ_A, χ_B, and χ_ as continuous functions of T, as shown in Fig. <ref> for the total magnetic susceptibility of a system with p=0.60, J_AA/J_BB=0.80, J_AB/J_BB=-0.50, and L from 10 to 100. The location of the peak temperatures is automated using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method <cit.> for each of the N_s disorder configurations and the errors are estimated as discussed in Sec. <ref>. Finally, to obtain T_c, we perform least-square fits with Eq. (<ref>) using the estimates for T_c(L) as input. This equation has three free parameters to be adjusted in the fitting process and requires great statistical resolution in order to produce stable and reliable estimates of the parameters. Since the precise estimation of all of these parameters would lead to a drastic increase in computational work, and since for the present work we are not interested in a precise value for the exponent ν, we employ the same procedure presented in Ref. diaz2016monte, in which we set a fixed value for the exponent ν and perform fits with two free parameters, instead of three. These fits are made, for a fixed value of ν, for system sizes not smaller than L_ and the value of L_ that gives the best fit is located, i. e., the one that minimizes the reduced weighted sum of errors, χ^2/n_DOF, where n_DOF is the number of degrees of freedom. Next, we keep changing the values of ν and L_ iteratively until we locate the set of values that globally minimizes χ^2/n_DOF and use these values to determine our best estimate of T_c. This procedure effectively linearizes the fit, although it does not allow for an individual error estimate for the exponent ν.In Fig. <ref> we show examples of these fits of the pseudo-critical temperatures obtained from the maxima of the magnetic susceptibilities for p=0.50, J_AA/J_BB=0.80, and J_AB/J_BB=-0.50. In this figure we show the fits using the values of L_ and ν that minimize χ^2/n_DOF. We note that this method gives a very small statistical error for T_c, even negligible in some cases, but it is important to point out that this error is underestimated when compared to the actual error, obtained through a true non-linear fit. In order to obtain a more realistic error bar, we follow the criterion used in Ref. diaz2016monte, in which the values obtained from fits that give χ^2/n_DOF up to 20% larger than the minimum are considered in the statistical analysis.It is also clear from Fig. <ref> that the values of ν obtained through this process are imprecise and unreliable, however, it is worth stressing that it is not our goal to obtain a precise description of the critical behavior for the model. Therefore, the values of 1/ν presented in Fig. <ref> serve only as an “effective exponent” used to achieve a good estimate of T_c.§.§ Determining T_compThe compensation point is the temperature T_comp where =0 while , ≠ 0, as seen in Figs. <ref> and <ref>. In order to estimate that temperature we perform simulations for a range of temperatures around T_comp and obtain thevalues as a continuous function of T using the multiple-histogram method, as show in Fig. <ref> for the total magnetization of a system with p=0.80, J_AA/J_BB=0.65, J_AB/J_BB=-0.80, and several system sizes L from 20 to 100. The root of ⟨ m_(T)⟩ is then located using Brent's method <cit.>. The process is repeated for N_s configurations and the final value and associated error of T_comp are determined as discussed in Sec. <ref>. To obtain a final estimate of T_comp, it is necessary to combine the estimates for different system sizes. In Fig. <ref> we see that the different T_comp(L) are fairly close to one another. However, it is clear from Fig. <ref> that the smaller lattices provide somewhat inconsistent results. Fig. <ref> shows the size dependence of the compensation temperature estimates obtained from the same data in Fig. <ref>. We can see that, as L increases, the compensation temperature approaches a fixed value. As the compensation effect is not a critical phenomenon, we have no a priori reason to expect a particular expression for the dependence of T_comp on L. Based on the form of the curve, however, we propose a power law behavior similar to the one obeyed by the critical temperature:T_comp(L) = a+bL^-ε,where a, b, and ε are parameters to be determined in the fitting process. Also similar to the critical temperature, we lack the statistical resolution to determine the three parameters independently; thus, to obtain T_comp, we resort to the same fitting method described in Sec. <ref> to determine T_c.An alternative procedure is employed: we also fit our data toT_comp(L) = a = ,for L>L_, which corresponds to averaging the different compensation temperatures considering only the values of L after the T_comp(L) curve has approximately converged. The value of L_ is also determined by minimizing the χ^2/n_DOF of the fit. The latter method corresponds to the same one used in Ref. diaz2016monte. Table <ref> shows the results of the fits presented in Fig. <ref> with Eqs. (<ref>) and (<ref>) for different values of L_. We see that values of a obtained from each method are all consistent, irrespective of ε or L_. At first glance, we could be tempted to choose Eq. (<ref>) with L_=40 as the best fit, based on the value of χ^2/n_DOF, and use it to obtain the final estimate of T_comp. Nonetheless, we note that the fit with Eq. (<ref>) cannot be used to determine T_comp for any case from Tab. <ref> where L_≥ 40, given that ε<0 contradicts our assumption that T_comp(L) approaches T_comp asymptotically and, in these cases, the limit used to determine the compensation point, i. e., T_comp=lim_L→∞T_comp(L), does not exist. The fact that these particular values of a are so close to the other T_comp estimates is a consequence of the small values obtained for parameter b. In fact, in these cases, the term bL^-ε is less than 10^-4, at least for L≤ 100. This means that the dependence with L of Eq. (<ref>) becomes irrelevant to the fitting process, i. e., these results become equivalent to a fit using Eq. (<ref>).On the other hand, if we impose ε>0 we obtain the best fits for ε very close to zero (ε≈ 10^-6) for L_≥ 40, which is consistent with a fit using Eq. (<ref>). Yet, the fit with such a small value of ε produces imprecise estimates for a and b, with errors that reach up to three orders of magnitude higher than the parameters themselves.These results are consistent with the fact that the compensation phenomenon is not in any way related to criticality. Hence, the observables near the compensation point are not supposed to exhibit a power law scaling behavior. Consequently, the method using Eq. (<ref>) turns out to be more adequate to determine T_comp, besides being simpler and more robust. To estimate the final error bars we follow the same procedure used in Ref. diaz2016monte, i. e., we combine the error obtained in the fitting process with both the errors obtained for N_s samples and for a single sample via the blocking method. § RESULTS AND DISCUSSION The system studied in this work has three parameters in its Hamiltonian, namely the concentration of magnetic sites p, and the ratios J_AA/J_BB and J_AB/J_BB, which represent, respectively, the asymmetry between type-A and type-B intraplanar couplings and the relative strength of interplanar coupling in relation to the type-B intraplanar coupling. All these parameters play important roles in determining the behavior of the system. Our goal is to outline the contribution of each parameter to the presence or absence of the compensation phenomenon. To that end we map out the regions of the parameter space for which the system has a compensation point, as seen in Figs. <ref> and <ref>, and the regions for which the compensation effect does not take place, as seen in Figs. <ref> and <ref>.We start our analysis by fixing the values of J_AA/J_BB and J_AB/J_BB and looking at the dependence of both the critical and compensation temperatures with p,as seen in Fig. <ref>, where we plot T_c and T_comp as functions of p for J_AB/J_BB=-1.00 and for both J_AA/J_BB=0.01 and J_AA/J_BB=0.50. The solid lines are either cubic spline interpolations or linear extrapolations just to guide the eye. The vertical dashed lines mark the characteristic concentration p^∗ wherethe critical temperature and compensation temperature curves meet and below which there is no compensation. The behavior depicted in Fig. <ref> for the multilayer (3d) system is qualitatively the same as displayed by the bilayer (2d) system for both pair approximation <cit.> and Monte Carlo <cit.>. This can be seen by comparing our Fig. <ref> to Fig. 2 in reference balcerzak2014ferrimagnetism and Fig. 7 in reference diaz2016monte, since all where made using the same values for the Hamiltonian parameters. As expected, the critical temperatures obtained for the 3d system are consistently higher than those reported for the 2d one using the same approximation <cit.>. In all cases, though, we see that p^∗ is higher for J_AA/J_BB=0.50 than it is for J_AA/J_BB=0.01, which indicates that p^∗ should increase as the strength of the A intraplanar coupling increases. This is expected, since, for high J_AA/J_BB, the magnetization of planes B will never be the same (in absolute value) as the magnetization of planes A for small p. We are able to confirm this trend, for both cases of a weak and a strong interplanar coupling,in Fig. <ref>, where we show p^∗ as a function of the ratio J_AA/J_BB for J_AB/J_BB=-0.01and J_AB/J_BB=-1.00. In both cases we see that p^∗ increases monotonically as J_AA/J_BB increases. In Fig. <ref> we have the behavior of p^∗ as a function of the ratio J_AB/J_BB for several values of J_AA/J_BB. Curiously, p^∗ is not monotonic as is the case for the 2d system <cit.>. Figure 9 in Ref. diaz2016monte shows the same p^∗× J_AB/J_BB diagram for the bilayer system for the particular case of J_AA/J_BB=0.50 and, comparing the latter figure to the 3d results we see that the values of p^∗ for the 3d system are consistently lower than those for the 2d one. However, this difference gets smaller as |J_AB/J_BB| gets weaker (p^∗ for the multilayer is ≈ 8% lower than value for the bilayer for J_AB/J_BB=-1.00 and it is ≈ 3% lower for J_AB/J_BB=-0.01), which is consistent with the fact that as J_AB/J_BB→ 0 the behaviors of both multilayer and bilayer systems cross over to the 2d behavior of non-interacting planes A and B. We also notice that the values of p^∗ are less dependent on the values J_AB/J_BB for 3d than for 2d. For instance, for J_AA/J_BB=0.50, the percentile increase in p^∗ as |J_AB/J_BB| increases from 0.01 to 1.00 is ≈ 13% for the bilayer, whereas for the multilayer this increase is only ≈ 7.6%. Even if we consider the maximum percentile increase (which for 2d remains ≈ 13% for its monotonic behavior), we have, for the multilayer, at most an increase of ≈ 10% from the minimum at about J_AB/J_BB=-0.20 to the maximum at J_AB/J_BB=-1.00.Next, for fixed values of p and J_AB/J_BB, we can analyze the influence of the ratio J_AA/J_BB in the behavior of our multilayer.In Fig. <ref> we plot T_c and T_comp as functions of J_AA/J_BB for p=0.70 (Fig. <ref>) and p=0.90 (Fig. <ref>), and fixed J_AB/J_BB=-0.50 and J_AB/J_BB=-1.00 for both cases. The vertical dashed lines mark the value of J_AA/J_BB above which there is no compensation for each case.In order tocontrast the behaviors of 3d and 2d systems, we can compare Fig. <ref> with Fig. 10 in Ref. diaz2016monte, as well as Fig. <ref> with Fig. 11 in the same reference, since in both pairs of figures we have the same quantities calculated for the same values of fixed parameters. For instance, for (p, J_AB/J_BB)=(0.70, -1.00), the value of J_AA/J_BB where T_c=T_comp is ≈ 77% higher for the 3d system, whereas, for (p, J_AB/J_BB)=(0.90, -0.50), this difference drops to ≈ 10%. Still, in both cases the region of the diagram for which the system has a compensation point is larger for the multilayer. Also, in both cases we have the critical temperatures for the 3d system consistently higher than those obtained with MC simulations for the bilayer, as expected. For (p, J_AB/J_BB)=(0.70, -1.00), the critical temperature for the 3d system is almost twice as high as the 2d value at J_AA/J_BB=0.0, while at J_AA/J_BB=1.0, the 3d value is only 37% higher. For (p, J_AB/J_BB)=(0.90, -0.50), the 3d system T_c is 26% higher than the 2d one at J_AA/J_BB=0.0 and 25% higher at J_AA/J_BB=1.0. Still consistent with the 3d to 2d crossover, the difference between the 3d and 2d critical temperatures is bigger for a stronger interplanar coupling. However, this difference does not get smaller as J_AA/J_BB gets weaker, which agrees with the fact that there is no crossover from 3d to 2d as J_AA/J_BB→ 0. As a matter of fact, we are able to set J_AA/J_BB=0 and the system is still a multilayer as long as p≠ 0 and J_AB/J_BB≠ 0.We can also compare our multilayer MC results with the bilayer PA behavior reported in Ref. balcerzak2014ferrimagnetism. This is made by contrasting Fig. <ref> with Fig. 6 in the latter reference, as well as Fig. <ref> with Fig. 7 in the same reference. After correcting for the difference in temperature scale (due to the PA using s_i± 1/2 and MC using s_i± 1), as discussed in Ref. diaz2016monte, for (p, J_AB/J_BB)=(0.70, -1.00) we have the critical temperature for the 3d system in MC ≈ 90% higher than for the 2d system in the PA at J_AA/J_BB=0.0 and 12% higher at J_AA/J_BB=1.0. For (p, J_AB/J_BB)=(0.90, -0.50) the critical temperature for the 3d system in MC is ≈ 1.3% lower than for the 2d system in the PA at J_AA/J_BB=0.0 and 4.7% higher at J_AA/J_BB=1.0. Again we see that the agreement is better for weaker interplanar couplings as in this case both systems are closer to a 2d behavior. It is also worth mentioning that the PA results for the bilayer are much closer to the MC results for the multilayer than to the MC results for the bilayer. This is consistent with the fact that mean-field-like approximations tend to work better in higher dimensions. In Fig. <ref> we have T_c and T_comp as functions of J_AB/J_BB for (p, J_AA/J_BB)=(0.60, 0.25), (0.60, 0.30) (<ref>) and (p, J_AA/J_BB)=(0.70, 0.46), (0.70, 0.50) (<ref>). The vertical dashed lines mark the value of J_AB/J_BB below which there is no compensation for each case. Unfortunately no direct comparison can be made between this figure and the bilayer MC results, since the analogous figure for the bilayer (Fig. 12 in Ref. diaz2016monte) does not have the same parameters as ours. The reason we did not use the same parameter for our figure <ref> as those used for Fig. 12 in Ref. diaz2016monte is that both sets of parameters in the latter figure ((p, J_AA/J_BB)=(0.7, 0.3) and (0.9, 0.8)) are in a ferrimagnetic phase where there is always compensation for -1.0≤ J_AB/J_BB<0.0. Qualitatively, though, it is safe to say that the T_c and T_comp curves do not get further apart as J_AB/J_BB→ 0 in the multilayer system as it happens in the bilayer, for both MC <cit.> and PA <cit.>. In fact, for fixed p and J_AA/J_BB we seem to have T_c and T_comp always very close to one another, and the difference T_c-T_comp dos not increase monotonically as J_AB/J_BB increases, having a maximum value somewhere between the point where both curves meet and J_AB/J_BB=0.0. Finally, as it follows from the analyzes presented above, we can divide the parameter space of our Hamiltonian in two distinct regions of interest. One is a ferrimagnetic phase for which there is no compensation at any temperature and the second is a ferrimagnetic phase where there is a compensation point at a certain temperature T_comp. We present this resultsin Fig. <ref>, where we plot the phase diagrams for concentrations p=0.5, 0.6, 0.7, 0.8, and 0.9. For each concentration, the line marks the separation between a ferrimagnetic phase with compensation (to the left) and a ferrimagnetic phase without compensation (to the right). These diagrams show that the compensation phenomenon is favored by greater values of p, as it is clear that as p approaches 1, as long as p≠ 1, the area occupied by the ferrimagnetic phase with compensation greatly increases. It is also clear that the compensation phenomenon is only present if there is intraplanar coupling asymmetry and, as p decreases, it is necessary to increase the asymmetry for the phenomenon to occur. We also notice that, as p increases, the line separating the phases becomes more vertical, i. e., the presence or absence of compensation becomes less sensitive to the value of J_AB/J_BB. Nonetheless, on all cases presented it is possible to intersect each curve with an actual vertical line in two points, although this is easier to see for smaller values p. Therefore, there are sets of values for the parameters for which we can be in a phase without compensation, decrease |J_AB/J_BB| to get to the phase with compensation, but if we keep decreasing |J_AB/J_BB| the system goes back to the phase without compensation. This is in agreement with the discussion presented above for Fig. <ref> and it is not seen on the behavior of the bilayer for either AP <cit.> or MC <cit.>.It is possible to compare Fig. <ref> with the phase diagram for the same system and same parameters obtained with the pair approximation, presented in Fig. 2(a) of Ref. szalowski2014normal. We first notice that the area occupied by the ferrimagnetic phase with compensation (to the left of the curve) is consistently greater for MC than for the AP. The lines also become more vertical as p increases in the MC approach and, although the reentrant behavior discussed in the last paragraph is also present in the PA, it is only visible at p=0.5 in Fig. 2(a) of Ref. szalowski2014normal, while it is clearly present in MC for all values of p presented in Fig. <ref>. The PA results also indicate that there is no compensation whatsoever for p=0.6 and J_AB/J_BB below ≈ -0.86 and for p=0.5 and J_AB/J_BB below ≈ -0.36, whereas the MC results show we always have a phase with compensation for -1.0≤ J_AB/J_BB<0.0 and the concentrations considered in Fig. <ref>.§ CONCLUSION In this work we have applied Monte Carlo simulations to study the magnetic behavior of an Ising multilayer composed of alternated non-equivalent planes of two types, A and B. Both A and B intralayer couplings are ferromagnetic while the interlayer couplings are antiferromagnetic.Additionally, only the B layers are site-diluted. The simulations were performed using the Wolff algorithm and the data were analyzed with multiple histogram reweighting and finite-size scaling methods. Our main goal is to obtain the conditions of existence of a compensation point for the system, i. e., a temperature T_comp where the total magnetization is zero below the critical point T_c.We studied simple cubic lattices with linear sizes L≤ 100 to allow for a precise evaluation of both T_c and T_comp for a wide range of values of each of the Hamiltonian parameters. The results for the multilayer are compared with those for the bilayer reported in both Monte Carlo <cit.> and pair approximation <cit.> approaches. We see that the compensation phenomenon in the multilayer is favored by small but non-null dilutions and by large intralayer coupling asymmetry, as it is also the case for the bilayer. Similarly, the effect is favored by a weak interlayer coupling, although the sensitivity to this parameter is more pronounced for the 2d system than it is for the multilayer. In addition, we notice that the behavior of the multilayer gets closer to the bilayer as the interlayer coupling gets weaker. This agrees with the crossover that happens at that limit, where the multilayer becomes a set of non-interacting two-dimensional systems.A summary of the results is then depicted in a convenient way on J_AB× J_AA diagrams for several values of site concentration. 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"authors": [
"Ian Jordy Lopez Diaz",
"Nilton da Silva Branco"
],
"categories": [
"cond-mat.stat-mech"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170525213052",
"title": "Monte Carlo study of an anisotropic Ising multilayer with antiferromagnetic interlayer couplings"
} |
Measurement of Energy Spectrum of Ultra-High Energy Cosmic Rays Yoshiki Tsunesada December 30, 2023 =============================================================== Abstract We construct the nonequilibrium steady state (NESS) density operator of the spin-1/2 XXZ chain with arbitrarynon-diagonal boundary magnetic fields coupled to boundary dissipators. Thecorresponding Markovian boundary dissipation is found with which the NESS density operator is expressed in terms of the product of the Lax operators by relating the dissipation parameters to the boundary parameters of the spin chain.The NESS density operator can be expressed in terms of a non-Hermitian transfer operator (NHTO) which forms a commuting family of quasilocal charges. The optimization of the Mazur bound for the high temperature Drude weight is discussed by using the quasilocal charges and the conventional local charges constructed in the context ofthrough the Bethe ansatz.§ INTRODUCTIONThe Heisenberg spin chain with anisotropy, the so-called XXZ chain, is one of the most widely studied quantum systems. Its integrability allows us to diagonalize the Hamiltonian <cit.>, which leads to the derivation of exact physical quantities such as correlation functions <cit.>. The integrability comes from the decomposability of many-body scatterings into a sequence of two-body scatterings. The decomposability is guaranteed by the Yang-Baxter equation satisfied by the scattering matrices. However, it is still a challenging problem to unveil the model's nonequilibrium properties. Peculiar nonequilibrium behaviors of integrable systems result from the existence of sufficiently many conserved quantitiesto determineso that they are in one-to-one correspondence with the degrees of freedom. For instance, it was predicted that observables after relaxation are described by the generalized Gibbs ensemble (GGE) <cit.> by maximizing the entropy, instead of the grand canonical ensemble. Subsequently, it has been demonstrated that integrable systems exhibit generalized thermalization <cit.>.Another interesting question is, whether an integrable system showsexhibits ballistic transport at finite or high temperatures. The optimized lower bound on the ballistic transport coefficient – the so-called Drude weight – has been introduced <cit.> by using a commuting family of local charges, although this bound generically vanishes when the system possesses the ℤ_2-symmetry with respect to which the transporting current is odd.A new and fruitful approach to the above questions came recently with exact solutions of boundary driven diffusiveopen quantum systems <cit.>. The idea to derive theexplicit nonequilibrium steady state (NESS) density operator lies in the matrix product ansatz (MPA), which was originally introduced for constructing the ground state of the s=1 spin chain <cit.> and widely used to solve classical boundary driven diffusive many-body systems in one-dimension <cit.>.The matrix product states serving as the NESS density operator for the quantum boundary drivendiffusive system is given by the non-Hermitian transfer operator (NHTO) constructed from the product of the Lax operators but with the complex spin representation for the auxiliary space <cit.>. Due to the Yang-Baxter equation, the NHTO satisfies the divergence condition <cit.>, which implies bulk cancellation in the steady-state Lindblad master equation with the remainder terms localized at the boundaries.These terms are in turn compensated by the boundary dissipation, which turns the NHTO into the NESS density operator.Amazingly, the NESS density operator constructed in this way can serve, in the limit of small dissipation, as a novel quasilocal conserved quantity, which can be used to evaluate the lower bound for the high temperature Drude weight on the corresponding non-dissipative quantum system <cit.>. Indeed, the NESS satisfies both commutativity and quasilocality conditions <cit.>.What made the long-standing problem, i.e. to find the NESS of the quantum boundary-driven diffusive many-body system, solved is the complex spin representation of the auxiliary space, which has not been considered before. This extension turned out to be important also in the context of integrable systems, since the NHTO provides a new commuting family of conserved quantities which contain parts that are orthogonal to the known local conserved quantities. We aim in this paper to investigate how boundary magnetic fields imposed on the spin chain affect its nonequilibrium behavior. We derive the NESS density operator of the boundary-drivendiffusive quantum spin chain with arbitrary boundary magnetic fields. Interestingly, there always exists the corresponding boundary dissipation for which the NESS density operator is given in terms of the NHTO by properly choosing the dissipation rates as functions of boundary fields.We have also constructed the quasilocal charges for the corresponding non-dissipative spin chain. We showed that, even under the existence of arbitrary boundary magnetic fields, the NHTO forms a commuting family by keeping quasilocality. These two properties of the NHTO allow us to evaluate the lower bound for the high temperature Drude weight. We found optimization of the optimized Mazur bound due to the ℤ_2-symmetry breaking in the spin chain with arbitrarynon-diagonal boundary magnetic fields, which leads to a finite contribution of the conventional local charges to the lower bound. This paper is organized as follows. In the next section, we give the basics of the open spin-1/2 XXZ chain and the boundary dissipator of the Lindblad type. We derive the NESS density operator in Section 3. The quasilocal charges are constructed in Section 4. The lower bound for the high temperature Drude weight is also evaluated. The last section is devoted to the concluding remarks.§ THE OPEN SPIN-1/2 XXZ CHAIN WITH BOUNDARY DISSIPATION §.§ The spin-1/2 XXZ chain with non-diagonal boundaries Let us consider the spin-1/2 XXZ chain with arbitrary boundary magnetic fields: H = ∑_x=1^n-1_2^x-1⊗ h ⊗_2^n-x-1 + h_ B,L⊗_2^n-1 + _2^n-1⊗ h_ B,R,where _2^x represents the x-fold tensor product of the 2× 2 identity matrix. The Hamiltonian density for the bulk part h is expressed by the Pauli matrices σ^α (α∈{±, z}) as h = 2 σ^+ ⊗σ^- + 2 σ^- ⊗σ^+ + σ^z ⊗σ^z cosη,while the boundary Hamiltonian density is expressed as h_ B,L = 1/2σ^z sinηξ_ L + σ^+ κ_ L e^θ_ Lsinη/sinξ_ L + σ^- κ_ L e^-θ_ Lsinη/sinξ_ L, h_ B,R = 1/2σ^z sinηξ_ R + σ^+ κ_ R e^θ_ Rsinη/sinξ_ R + σ^- κ_ R e^-θ_ Rsinη/sinξ_ R,containing six free parameters ξ_ L,R, κ_ L,R, and θ_ L,R which uniquely parametrise arbitrary boundary magnetic fields. The model (<ref>) is known to be integrable in the sense that its transfer matrix forms a commuting family of infinitely many local operators. The local charges are obtained by expanding the logarithm of the transfer matrix around the permutation point. The leading term gives the momentum operator, while the next-to-leading term gives the Hamiltonian: H = d/dφ(sinη/2sinξ_ L K_1(φ,ξ_ L) + ∑_x=1^n-1 2Ř_x,x+1(φ) )|_φ=η/2 + _0 K_0(φ+η,ξ_ R)h_n,0/ K_0(φ+η,ξ_ R)|_φ=η/2.The Ř-matrix and the K-matrix satisfy the so-called RLL relation and the reflection relation, respectively: Ř_1,2(φ_1 - φ_2) _1(φ_1) _2(φ_2) = _1(φ_2) _2(φ_1) Ř_1,2(φ_1 - φ_2), Ř_2,1(φ_1-φ_2) K_2(φ_1) Ř_1,2(φ_1+φ_2) K_2(φ_2) = K_2(φ_2) Ř_2,1(φ_1+φ_2) K_2(φ_1) Ř_1,2(φ_1-φ_2),whose solutions are expressed in termsby means of the Pauli matrices σ^α (α∈ +,-,z):Ř(φ) = sinφ/2 (h + cosη) - 1 + cosφ/2sinη + 1 - cosφ/2σ^z ⊗σ^z sinη, K(φ;ξ,κ,θ) = sinξcosφ + σ^z cosξsinφ + σ^+ κ e^θsin(2φ) + σ^- κ e^-θsin(2φ).On the other hand, the Lax operator L consists of the physical space, which has the same representation as that of the Ř-matrix, and the auxiliary space, which admits any representation including the complex spin representation: (φ,s) =[ sin(φ + η_a^z) sinη_a^-; sinη_a^+ sin(φ - η_a^z) ] = ∑_α∈{+,-,0,z}_a^α(φ,s) ⊗σ_p^α,where _a^0(φ,s) = sinφcos(η_a^z), _a^z(φ,s) = cosφsin(η_a^z), _a^±(φ,s) = (sinη) _a^∓.We take the complex spin representations in the way introduced in Ref. <cit.>: _a^z = ∑_k=0^∞ (s-k) |k ⟩⟨ k|, _a^+ = ∑_k=0^∞sin(k+1)η/sinη |k ⟩⟨ k+1|, _a^- = ∑_k=0^∞sin(2s-k)η/sinη |k+1 ⟩⟨ k|by which the U_q(sl_2) algebraic relations are satisfied: [^z, ^±] = ±^±,[^+, ^-] = [2^z]_q. §.§ Boundary dissipation of the Lindblad typeWithin the theory of open quantum systems <cit.>, incoherent Markovian quantum dissipation is completely described by a set of Lindblad operators {L_μ∈ End(ℋ_p^⊗ n), μ=1,2,…}. Such a system's many-body density operator ρ(t) obeys the time evolution described by the Lindblad-Gorini-Kossakowski-Sudarshan master equation <cit.>: d/dtρ(t) = ℒ̂ρ(t) := -i[H, ρ(t)] + ∑_μ( 2L_μρ(t) L_μ^† - {L_μ^† L_μ, ρ(t)}). Throughout this paper, we consider the ultra-local Lindblad operators L_μ = ℓ_μ⊗_2^n-1 or L_μ = _2^n-1⊗ℓ_μ, ℓ_μ∈ End (ℋ_p),especially of the following forms associated with three different dissipation rates ε, ε', ε”∈ℝ^+ and two additional dissipaiton parameters α,α'∈ℂ:L_1 = √(ε) (σ_1^+ + ασ_1^0),L_2 = √(ε) (σ_n^- + α'σ_n^0),L_3 = √(ε')σ_1^z,L_4 = √(ε”)σ_n^z.One clearly obtainsNotice that the dissipators L_1 and L_3 are coupled to the left boundary of the spin chain, while L_2 and L_4 to the right boundary. We let α and α' free for the moment and determine their relations to the boundary parameters later. The Liouvillian ℒ̂ is then written as ℒ̂ = -i [H, ρ(t)] + ε𝒟̂_σ_1^+ + ασ_1^0 + ε' 𝒟̂_σ_1^z + εD̂_σ_n^- + α'σ_n^0 + ε”𝒟̂_σ_n^z,where the dissipator map is defined by 𝒟̂_L (ρ) = 2L ρ L^† - {L^† L, ρ}.From the master equation (<ref>), the density operator at time t is expressed as ρ(t) = exp(t ℒ̂) ρ(0) leading to, if the limit exists, the NESS density operator ρ_∞ = lim_t →∞exp(t ℒ̂) ρ(0). Since the NESS is invariant under the time development, the NESS density operator ρ_∞ gives the fixed point of the propagator: ℒ̂ρ_∞ = 0.§ NONEQUILIBRIUM STEADY STATE §.§ Construction of the NESS density operatorLet us first write the NESS density operator in terms of the product of non-Hermitian amplitude operators: ρ_∞ = R_∞/ R_∞,R_∞ = Ω_n Ω_n^†.Contrary to the trivial open boundary case where a particular factorization occurs in terms of Ω_n <cit.>, some modification is required for the arbitrary boundary caseunder the presence of boundary magnetic fields. The fixed point condition (<ref>) for the NESS density operator ρ_∞ under the choice of the Liouvillian (<ref>) leads to the following condition on Ω_n and Ω_n^†:iε^-1 [H, Ω_n Ω_n^†]= 𝒟̂_σ_1^+ + ασ_1^0 (Ω_n Ω_n^†) + ε'/ε𝒟̂_σ_1^z (Ω_n Ω_n^†) + 𝒟̂_σ_n^- + α'σ_n^0 (Ω_n Ω_n^†) + ε”/ε𝒟̂_σ_n^z (Ω_n Ω_n^†).In order to deal with the product of the amplitude operators, it is useful to introduce the double Lax operator acting over a tensor product of a pair of complex spin representations <cit.>.We use the Lax operator with the complex spin representation for the auxiliary space. Note that the complex spin operators have the highest weight representations given in (<ref>). Besides, we introduce another complex spin operators of the transposed lowest weight representations: t^z = ∑_k=0^∞ (t-k) |k ⟩⟨ k|, t^+ = ∑_k=0^∞sin(k+1)η/sinη |k+1 ⟩⟨ k|, t^- = ∑_k=0^∞sin(2t-k)η/sinη |k ⟩⟨ k+1|.The double Lax operators is then defined in the product representation 𝒱_s^T ⊗𝒱_t: 𝕃_x(φ,θ,s,t) = _a,x^T(φ,s) _b,x(θ,t),where 𝒱_s^T is the transposed lowest weight representation, while 𝒱_t is the highest weight representation. The double Lax operator obeys the Yang-Baxter-like equation: Ř_1,2(δ_1-δ_2) 𝕃_1(φ+δ_1, θ-δ_1, s, t) 𝕃_2(φ+δ_2, θ-δ_2, s, t) = 𝕃_1(φ+δ_2, θ-δ_2, s, t) 𝕃_2(φ+δ_1, θ-δ_1, s, t) Ř_1,2(δ_1-δ_2),from which we obtain the divergencecommutativity condition for the bulk part: [H_ bulk, 𝕃_1 ⋯𝕃_n] = 2sinη ( ∂𝕃_1 𝕃_2 ⋯𝕃_n - 𝕃_1 ⋯𝕃_n-1∂𝕃_n),where ∂𝕃_x(φ, θ, s, t)= ∂_δ( _a,x^T(φ+δ, s) _b,x(θ-δ, t) )_δ=0= ∂_φ_a,x^T(φ,s) _b,x(θ,t) - _a,x^T(φ,s) ∂_θ_b,x(θ,t).The bulk divergencecommutativity condition implies that the dissipation should be localized at the boundaries of the spin chain.Using the bulk divergencecommutativity condition (<ref>) and the boundary Hamiltonian density (<ref>), we obtain the divergencecommutativity condition for the full spin chain: [H, 𝕃^⊗ n] = [H_ bulk, 𝕃^⊗ n] + [H_ L,B, 𝕃^⊗ n] + [H_ R,B, 𝕃^⊗ n] = 2(sinη) ∂𝕃⊗𝕃^⊗ n-1 - 2(sinη) 𝕃^⊗ n-1⊗∂𝕃+ [ 1/2σ^z sinηξ_ L + σ^+ κ_ L e^θ_ Lsinη/sinξ_ L + σ^- κ_ L e^-θ_ Lsinη/sinξ_ L, 𝕃] ⊗𝕃^⊗ n-1+ 𝕃^⊗ n-1⊗[ 1/2σ^z sinηξ_ R + σ^+ κ_ R e^θ_ Rsinη/sinξ_ R + σ^- κ_ R e^-θ_ Rsinη/sinξ_ R, 𝕃]. We assume that the NESS density operator is given by the product of the Lax operators, similarly as for the trivial open boundary case <cit.>: Ω_n = 1/sin^n(φ + sη)_a⟨ 0| _1^T _2^T …_n^T |0 ⟩_a,where |0 ⟩ is the highest weight vector. Note that we have used the partially transposed Lax operator𝕃_x ∈𝒱_s^T ⊗𝒱_s̅: ^T(φ,s) = ∑_α∈{+,-,0,z}^α(φ,s) ⊗ (σ^α)^T.From the definition (<ref>), the product Ω_n Ω_n^† admits the expression in terms of the double Lax operator:Ω_n Ω_n^† = 1/sin^n(φ + sη) sin^n(φ + s̅η)_a⟨ 0| _b⟨ 0| 𝕃_1 ⋯𝕃_n |0 ⟩_a |0 ⟩_b.By applying the double highest weight vector |0 ⟩_a |0 ⟩_b to the fixed point condition (<ref>), we obtain the left and right boundary conditions: _a⟨ 0| _b⟨ 0| ( -iε^-1 2(sinη) ∂𝕃 - iε^-1[ 1/2σ^z sinηξ_ L + σ^+ κ_ L e^θ_ Lsinη/sinξ_ L + σ^- κ_ L e^-θ_ Lsinη/sinξ_ L, 𝕃] + D̂_σ_1^+ + ασ_1^0(𝕃) + ε'/εD̂_σ_1^z(𝕃) ) = 0, ( iε^-1 2(sinη) ∂𝕃 - iε^-1[ 1/2σ^z sinηξ_ R + σ^+ κ_ R e^θ_ Rsinη/sinξ_ R + σ^- κ_ R e^-θ_ Rsinη/sinξ_ R, 𝕃] + D̂_σ_n^- + α'σ_n^0(𝕃) + ε”/εD̂_σ_n^z(𝕃) ) |0 ⟩_a |0 ⟩_b = 0.Note that the dissipation operators act on an 2-by-2 matrix as D̂_σ^+ + ασ^0[ a b; c d ]= [2d + α(b+c)-b - α(a-d);-c - α(a-d) -2d - α(b+c) ], D̂_σ^- + ασ^0[ a b; c d ]= [ -2a - α(b+c)-b + α(a-d);-c + α(a-d)2a + α(b+c) ], D̂_σ^z[ a b; c d ]= [ 0 -4b; -4c 0 ]. The nontrivial solution to (<ref>) exists for the following case. We first need the spectral parameter restricted by φ = π/2.This condition coincides with the trivial open boundary case <cit.>. The dissipation rates must be related to the anisotropy and the boundary parameters of the diagonal parts: ε = -2i sinηtan(sη), ε' = -i/4sinηξ_ L, ε” = -i/4sinηξ_ R.This result includes the trivial open boundary case for ξ_ L,R = π/2 <cit.>. The parameters α and α' are determined by the boundary parameters of the non-diagonal parts: α = iε^-1sinη/sinξ_ Lκ_ L e^θ_ L = -iε^-1sinη/sinξ_ Lκ_ L e^-θ_ L, α' = -iε^-1sinη/sinξ_ Rκ_ R e^θ_ R = iε^-1sinη/sinξ_ Rκ_ R e^-θ_ R,which require θ_ L = θ_ R = iπ/2. As a result, the boundary Hamiltonian density is allowed to take only the following form: h_ B,L = 1/2σ^z sinηξ_ L - 2 σ^y κ_ Lsinη/sinξ_ L, h_ B,R = 1/2σ^z sinηξ_ R - 2 σ^y κ_ Rsinη/sinξ_ R. §.§ Rotation in the xy-planeOne may wonder why the solution to the boundary conditions requires the non-diagonal parts of the left and right boundary Hamiltonian to consist of only σ^y-terms, although the Hamiltonian itself possesses the symmetry with respect to the rotation in the xy-plane.Let us consider the transformation performed by the matrix defined by U := [e^iϕ/2 0; 0 e^-iϕ/2 ].The transformation makes σ^z invariant U σ^z U^-1 = σ^z but rotates the other components of the Pauli matrices: U σ^x U^-1 = σ^x cosϕ + σ^y sinϕ, U σ^y U^-1 = σ^x sinϕ - σ^y cosϕ,and, subsequently, U σ^+ U^-1 = e^iϕσ^+,U σ^- U^-1 = e^-iϕσ^-.Therefore, it is easily obtainedeasy to show that the bulk part of the Hamiltonian is invariant under the transformation U^⊗ n H_ bulk (U^⊗ n)^-1 = H_ bulk. Also, the σ^z-terms of the left and the right boundary Hamiltonian are invariant under the transformation. Thus, U^⊗ n rotates the Hamiltonian in the xy-plane by letting the non-diagonal boundary terms contain both σ^x- and σ^y-terms. Let us denote the transformed operator by X := U X U^-1. The boundary Hamiltonian density is then deformed as h_ B,L→h_ B,L = 1/2σ^z sinηξ_ L - 2 (σ^x sinϕ - σ^y cosϕ) κ_ Lsinη/sinξ_ L, h_ B,R→h_ B,R = 1/2σ^z sinηξ_ R - 2 (σ^x sinϕ - σ^y cosϕ) κ_ Rsinη/sinξ_ R.The transformation angle ϕ is related to the non-diagonal boundary parameters as iθ_ L,R = ϕ + π/2,which implies that the only condition for obtaining the NESS density operator is θ_ L = θ_ R.Indeed, the fixed point condition (<ref>) holds for the transformed Hamiltonian H and double Lax operator 𝕃 if we deform the Lindblad operators as D̂_σ^+ + ασ^0→D̂_σ^+ + ασ^0 = D̂_e^iϕσ^+ + ασ^0, D̂_σ^- + ασ^0→D̂_σ^- + ασ^0 = D̂_e^-iϕσ^- + ασ^0, D̂_σ^z→D̂_σ^z = D̂_σ^z.§ QUASILOCAL CHARGES §.§ Optimized Mazur bound on theDrude weightThe ballistic transport is characterized by the finite Drude weight D. The Drude weight is defined as the coefficient for the diverging part of d.c. conductivity in the context of linear-response transport. Since the Drude weight has the temperature dependence, its expansion at high temperature leads to D = β D_∞ + 𝒪(β^2),D_∞ = lim_t →∞lim_n →∞1/2tn∫_0^t dt' (J(t'), J),where J is the extensive current J := ∑_x=0^n-1_2^x⊗ j ⊗_2^n-x, j is the local current, and (A,B) is the Hilbert-Schmidt inner product defined by (A,B) :=(A^†B)/_2^n. By using a set of extensive local conserved operators {Q_r; r=1,…,n}, the Mazur bound <cit.> allows rigorous estimation of the high-temperature Drude weight from below <cit.>: D_∞≥lim_n →∞1/2n∑_r,r' (J,Q_r) (K^-1)_r,r' (Q_r',J) := D_Q,K_r,r' := (Q_r,Q_r').However, in the system with the ℤ_2-symmetry such as the spin-reversal symmetry, with respect to which the conserved charges Q_r are even and the current J is odd,the lower bound D_Q always becomes zero, which tells nothing about the ballistic behavior. The situation drastically changes by the introduction of quasilocal charges <cit.>. The quasilocal charges constructed by differentiating the NESS density operator for the corresponding boundary drivendiffusive spin chain with respect to the representation parameter s consists of both even and odd parity parts, which makes the lower bound finite. The optimized Mazur bound <cit.> was obtained as D_∞≥1/2 Re∫_𝒟_m d^2φ Z_J(φ) f(φ):= D_Z,which is bounded by the conventional Drude weight D_Z ≥ D_Q. Here we used the notation Z_J(φ) := lim_n →∞ (J,Z(φ))/n, and Z(φ) is the family of quasilocal charges which can be generated from NESS density operator of the boundary driven chain in the limit of weak driving. The function f(φ) is obtained as the solution of a complex Fredholm equation of the first kind: ∫_𝒟_m d^2φ' lim_n →∞1/n (Z(φ),Z(φ')) f(φ') = Z_J(φ), φ∈𝒟_m ⊂ℂ. §.§ Construction of quasilocal chargesThus, what we need for evaluating the lower bound of the high temperature Drude weight is to find a commuting family of quasilocal charges. Analogously to the periodic and trivial open boundary cases <cit.>, let us introduce the NHTO defined by the product of the Lax operators: W_n(φ,s) = ⟨ 0| (φ,s)^⊗_pn |0 ⟩.Note that the commutativity holds for the NHTO with any pair of spectral and representation parameters in spite of the existence of arbitrarynon-diagonal boundary magnetic fields: W_n(φ,s) W_n(θ,t)= ⟨ 0|_a ⟨ 0|_b Ř_a,b(φ-θ,s,t) ( ∏_x=1^n _a,x(φ,s) _b,x(θ,t) ) |0 ⟩_a |0 ⟩_b = ⟨ 0|_a ⟨ 0|_b ( ∏_x=1^n _a,x(φ,s) _b,x(θ,t) ) Ř_a,b(φ-θ,s,t) |0 ⟩_a |0 ⟩_b = W_n(θ,t) W_n(φ,s).The NHTO satisfies the following commutativity condition: [H, W_n(φ,s)]= -2sinη( cosφcos(sη) σ^0 ⊗ W_n-1(φ,s) - sinφsin(sη) σ^z ⊗ W_n-1(φ,s) - cosφcos(sη) W_n-1(φ,s) ⊗σ^0 + sinφsin(sη) W_n-1(φ,s) ⊗σ^z )+ sinη/sinξ_ L( -2κ_ L e^θ_ Lcosφsin(sη) σ^+ ⊗ W_n-1(φ,s) + 2κ_ L e^-θ_ Lcosφsin(sη) σ^- ⊗ W_n-1(φ,s) )- sinη/sinξ_ R( -2κ_ R e^-θ_ Rcosφsin(sη) W_n-1(φ,s) ⊗σ^- + 2κ_ R e^θ_ Rcosφsin(sη) W_n-1(φ,s) ⊗σ^+ )+ sinη/sinξ_ L( -cosξ_ L (sinη) σ^- ⊗ W_n-1^+(φ,s) + κ_ L e^θ_ L (sinη) σ^z ⊗ W_n-1^+(φ,s) )- sinη/sinξ_ R( -cosξ_ Rsin(2sη) W_n-1^-(φ,s) ⊗σ^+ + κ_ R e^-θ_ Rsin(2sη) W_n-1^-(φ,s) ⊗σ^z ).Here we introduced W_n^+(φ,s) = ⟨ 1| _1(φ,s) ⋯_n(φ,s) |0 ⟩, W_n^-(φ,s) = ⟨ 0| _1(φ,s) ⋯_n(φ,s) |1 ⟩. From now on, we show that the differentiation of the NHTO with respect to the representation parameter s forms a family of quasilocal charges. Precisely, the quasilocal charge is given by Z_n(φ) = 1/2(sinφ)^n-2ηsinη∂_s W_n(φ,s)|_s=0 - sinφcosφ/2sinη M_n^z,where M_n^z = ∑_x=0^n-1_2^x⊗σ^z ⊗_2^n-1-x.By using the time derivative of NHTO (<ref>), we have [H, Z_n(φ)]= 2sinηφ( -σ^0 ⊗ Z_n-1(φ) + Z_n-1(φ) ⊗σ^0 )+ σ^z ⊗_2^n-1 - _2^n-1⊗σ^z+ 1/2η (sinφ)^n-2sinξ_ L( -cosξ_ L (sinη) σ^- ⊗∂_s W_n-1^+(φ,s)|_s=0 + κ_ L e^θ_ L (sinη) σ^z ⊗∂_s W_n-1^+(φ,s)|_s=0)- 1/2η (sinφ)^n-2sinξ_ R( -2ηcosξ_ R W_n-1^-(φ,0) ⊗σ^+ + 2ηκ_ R e^-θ_ R W_n-1^-(φ,0) ⊗σ^z ).The third and fourth lines are evaluated through ∂_s W_n^+(φ,s)|_s=0 = ∑_k=1^n ⟨ 1| _1(φ,s) ⋯∂_s _k(φ,s) ⋯_n(φ,s) |1 ⟩|_s=0= ∑_k=1^n (sinφ)^n-k∑_α_j ∈{+,-,0,z}⟨ 1| _1^α_1(φ,s) … (∂_s _k(φ,s))^α_k |0 ⟩|_s=0σ^α_1⊗⋯⊗σ^α_k⊗_2^n-k= ∑_k=1^n 2η (sinφ)^n-k∑_α_j ∈{+,-,0,z}⟨ 1| _1^α_1(φ,0) …^α_k-1_k-1(φ,0) |1 ⟩σ^α_1⊗⋯⊗σ^α_k-1⊗σ^+ ⊗_2^n-k, W_n^-(φ,0)= ⟨ 0| _1(φ,0) ⋯_n(φ,0) |1 ⟩= ∑_α_j ∈𝒥sinφ⟨ 0| _2^α_2(φ,0) ⋯_n^α_n(φ,0) |1 ⟩σ^0 ⊗σ^α_2⊗⋯⊗σ^α_n+ ∑_α_j ∈{+,-,0,z}sinη⟨ 1| _2^α_2(φ,0) ⋯_n^α_n(φ,0) |1 ⟩σ_1^- ⊗σ_2^α_2⊗⋯⊗σ_n^α_n= ∑_k=1^n sinη (sinφ)^k-1∑_α_j ∈{+,-,0,z}⟨ 1| _k^α_k(φ,0) ⋯_n^α_n(φ,0) |1 ⟩_2^k-1⊗σ^- ⊗σ^α_k+1⊗⋯⊗σ^α_n,by using _j(φ,0) |0 ⟩ = σ_j^0 sinφ |0 ⟩, ∂_s _j(φ,s)|_s=0 |0 ⟩ = 2ησ_j^+ |1 ⟩, ⟨ 0| _j(φ,0) = σ_j^0 sinφ⟨ 0| + σ_j^- sinη⟨ 1|,and subsequently, W_n(φ,0) = (sinφ)^n _2^n,W_n^+(φ,0) = 0.Setting q_r(φ) = (sinφ)^-r+2∑_α_j ∈𝒥⟨ 1| ^α_2(φ,0) ⋯^α_r-1(φ,0) |1 ⟩σ^- ⊗σ^α_2⊗⋯⊗σ^α_r-1⊗σ^+, p^+_r(φ) = (sinφ)^-r+2∑_α_j ∈{+,-,0,z}⟨ 1| ^α_2(φ,0) ⋯^α_r-1(φ,0) |1 ⟩σ^z ⊗σ^α_2⊗⋯⊗σ^α_r-1⊗σ^+, p^-_r(φ) = (sinφ)^-r+2∑_α_j ∈{+,-,0,z}⟨ 1| ^α_2(φ,0) ⋯^α_r-1(φ,0) |1 ⟩σ^- ⊗σ^α_2⊗⋯⊗σ^α_r-1⊗σ^zleads to the simple expressions: σ^- ⊗∂_s W_n-1^+(φ,s)|_s=0 = 2η (sinφ)^n-2∑_r=2^n q_r(φ) ⊗_2^n-r, σ^z ⊗∂_s W_n-1^+(φ,s)|_s=0 = 2η (sinφ)^n-2∑_r=2^n p^+_r(φ) ⊗_2^n-r, W_n-1^-(φ,0) ⊗σ^+ = sinη (sinφ)^n-2∑_r=2^n _2^n-r⊗ q_r(φ), W_n-1^-(φ,0) ⊗σ^z = sinη (sinφ)^n-2∑_r=2^n _2^n-r⊗ p^-_r(φ).Note that the charge Z_n(φ) (<ref>) is also expressed in terms of q_r(φ): Z_n(φ) = ∑_r=2^n ∑_k=0^n-k_2^k⊗ q_r(φ) ⊗_2^n-r-k.Then the commutator (<ref>) is written by using q_r(φ) and p^±_r(φ) as [H, Z_n(φ)] = σ^z ⊗_2^n-1 - _2^n-1⊗σ^z+ (2φ - ξ_ L) sinη∑_r=2^n q_r(φ) ⊗_2^n-r- (2φ - ξ_ R) sinη∑_r=2^n _2^n-r⊗ q_r(φ)+ κ_ L e^θ_ L/sinξ_ Lsinη∑_r=2^n p^+_r(φ) ⊗_2^n-r- κ_ R e^-θ_ R/sinξ_ Rsinη∑_r=2^n _2^n-r⊗ p^-_r(φ).In order to show quasilocality of the charge Z_n(φ), it is enough to show quasilocality of q_r(φ) and p^±_r(φ). We first compute the Hilbert-Schmidt norms of these quantities in the easy-plane anisotropy regime η = π l/m for coprime l,m ∈ℤ_>0, m ≠ 0, l ≤ m.In this regime, each element of the Lax operators has the finite dimensional representation: ^0(φ) = ∑_k=0^m-1sinφcosπ lkm |k ⟩⟨ k|, ^z(φ) = -∑_k=1^m-1cosφsinπ lkm |k ⟩⟨ k|, ^+(φ) = -∑_k=1^m-2sinπ lkm |k+1 ⟩⟨ k|, ^-(φ) = ∑_k=0^m-2sinπ l(k+1)m |k ⟩⟨ k+1|.This allows the explicit calculation of the Hilbert-Schmidt norms of q_r(φ) and p^±_r(φ) as (q_r(φ), q_r(φ))= 1/2^r∑_α_j,α'_j ∈{+,-,0,z}1/(sinφ)^r-2( ⟨ 1| ^α'_2(φ) ⋯^α'_r-1(φ) )1/(sin)^r-2( ⟨ 1| ^α_2() ⋯^α_r-1() )^T ×( σ^+σ^- ⊗ (σ^α_2)^Tσ^α'_2⊗⋯⊗ (σ^α_r-1)^Tσ^α'_r-1⊗σ^-σ^+ ) = 1/4⟨ 1|T(,φ)^r-2 |1 ⟩, (p^+_r(φ), p^+_r(φ))= 1/2^r∑_α_j,α'_j ∈{+,-,0,z}1/(sinφ)^r-2( ⟨ 1| ^α'_2(φ) ⋯^α'_r-1(φ) )1/(sin)^r-2( ⟨ 1| ^α_2() ⋯^α_r-1() )^T ×( σ^zσ^z ⊗ (σ^α_2)^Tσ^α'_2⊗⋯⊗ (σ^α_r-1)^Tσ^α'_r-1⊗σ^-σ^+ ) = 1/2⟨ 1|T(,φ)^r-2 |1 ⟩, (p^-_r(φ), p^-_r(φ))= 1/2^r∑_α_j,α'_j ∈{+,-,0,z}1/(sinφ)^r-2( ⟨ 1| ^α'_2(φ) ⋯^α'_r-1(φ) )1/(sin)^r-2( ⟨ 1| ^α_2() ⋯^α_r-1() )^T ×( σ^+σ^- ⊗ (σ^α_2)^Tσ^α'_2⊗⋯⊗ (σ^α_r-1)^Tσ^α'_r-1⊗σ^zσ^z ) = 1/2⟨ 1|T(,φ)^r-2 |1 ⟩.Here we introduced T(,φ) = ∑_k=1^m-1( (cosπ lkm)^2 + φ (sinπ lkm)^2 ) |k ⟩⟨ k| + ∑_k=1^m-2|sinπ lkmsinπ l(k+1)m|/2sinsinφ (|k ⟩⟨ k+1| + |k+1 ⟩⟨ k|). Using the fact that the eigenvalues {τ_i;i=1,…,m-1} of T(,φ) satisfy 1 > |τ_1| ≥…≥ |τ_m-1|, the Hilbert-Schmidt norms of q_r(φ) and p^±_r(φ) are subjected to (q_r(φ), q_r(φ)) ≤γ e^-ξ(φ) r,(p^±_r(φ), p^±_r(φ)) ≤γ^± e^-ξ(φ) rwith the decay constant: ξ(φ) = -1/2logτ_1(φ) > 0.Therefore, Z_n(φ) is quasilocal and bounded by (Z_n(φ), Z_n(φ))= n ∑_r=2^n ( 1 - r-1/n) (q_r(φ), q_r(φ)) ≤ n γ^2 ∑_r=2^n e^-2ξ(φ) r< n γ^2/1 - e^-2ξ(φ). Besides the quasilocal charges thus constructed, the normal local charges Q_r may also contribute to the lower bound, since, unlike the periodic or trivial open boundary cases, the open XXZ chain with arbitrarynon-diagonal boundary magnetic fields does not in general have the spin-reversal symmetry and, subsequently, the conventional local charges include odd parity part as well as the even parity part. In the case where ξ_ L,R = π/2, θ_ L,R = iπ n,n ∈ℤ,the local charges consist only of even parity part and thus the normal local charges have no contribution to the lower bound. §.§ Discussion§.§.§ The open XXZ spin chain with spin-reversal symmetryWhen the spin chain possesses the spin-reversal symmetry realized by the choice of parameter values (<ref>), the optimized Mazur bound is solely evaluated through the formula (<ref>) by using the quasilocal charges constructed in the previous section (<ref>).Noting that the spin current is given by J = i ∑_j=1^n-1(σ_j^+ σ_j+1^- - σ_j^- σ_j+1^+),the overlap between the current and the quasilocal charge has a constant value Z_J(φ) = i/4. On the other hand, the inner product of the quasilocal charges is computed by using (<ref>): (Z_n(φ), Z_n(φ')) = ∑_r=2^n (n-r+1) ( (q_r(φ))^† q_r(φ) ) = n/4∑_r=2^∞⟨ 1| (φ,φ')^r-2 |1 ⟩ + 𝒪(1) = n/4⟨ 1| ( - (φ,φ'))^-1 |1 ⟩ + 𝒪(1) = -n sinφsinφ'/2 sin^2 π l/msin((n-1)(φ+φ'))/sin(m(φ+φ')) + 𝒪(1).Therefore, the complex Fredholm equation of the first kind (<ref>) is solved as f(φ) = -im/πsin^2π l/m/|sinφ|^4.Finally, the lower bound on the Drude weight is evaluated as D_Z ≥1/4sin^2π l/m/sin^2π/m( 1 - m/2πsin( 2π/m) ),which coincides with the bound for the periodic and trivial open boundary cases.§.§.§ The open XXZ spin chain without spin-reversal symmetryThe open XXZ chain with arbitrarynon-diagonal boundary magnetic fields in general does not have the spin-reversal symmetry and, consequently, its local charges contain both even and odd parity parts.By taking into account of the contributions of the local charges, the Mazur bound is expected to be further optimized. Let us consider the operator B consisting of both local and quasilocal charges: B = J - ∫_𝒟_m d^2φf(φ) Z(φ) - ∑_r=1^n α_r Q_r,where J is the time-averaged current: J := lim_T →∞1/T∫_0^T dte^iHt J e^-iHt.Note that the local charges are defined by the logarithmic derivatives of the transfer matrix: Q_r = ∂^r-1_φln V_n(φ,12) |_φ=η/2, V_n(φ,12) = (-1/sinh(φ-η/2) sinh(φ+3η/2))^n ×_a ( K(φ;ξ_ L,κ_ L,θ_ L) _1(φ,12) …_n(φ,12) K(φ+η;ξ_ R,κ_ R,θ_ R) _1(φ+η,12) …_n(φ+η,12) ),where we used the relation ^-1(φ,12) = -1/sinh(φ+η/2) sinh(φ-3η/2)(-φ+η,12).The local charges are expressed by the corresponding local densities q^(r): Q_r = ∑_x=0^n-1 (_2^x⊗ q^(r)⊗_2^n-r-x). Since the following inequality relation holds for the Hilbert-Schmidt inner product of B: 0 ≤1/2n(B,B) = D_ZQ - 1/2n∫_𝒟_m d^2φf(φ) (J,Z(φ)) - 1/2n∫_𝒟_m d^2φ f(φ) (Z(φ),J) - 1/2n∑_r=0^n α_r (J,Q_r) - 1/2n∑_r=0^n α_r (Q_r,J) + 1/2n∑_r=0^n α_r∫_𝒟_m d^2φf(φ) (Q_r,Z(φ)) + 1/2n∑_r=0^n α_r ∫_𝒟_m d^2φ f(φ) (Z(φ),Q_r) + 1/2n∫_𝒟_m d^2φ∫_𝒟_m d^2φ' f(φ) f(φ') (Z(φ),Z(φ')) + 1/2n∑_r,r'=0^n α_rα_r' (Q_r,Q_r'), the high-temperature Drude weight is bounded from below by D_ZQ ≥ F_n[f,{α_r}] := ∫_𝒟_m d^2φRe( 1/n(J,Z(φ)) f(φ) )+ ∑_r=1^n Re( 1/n (J,Q_r) α_r ) - ∑_r=1^n Re( α_r ∫_𝒟_m d^2φ 1/n(Q_r,Z(φ)) f(φ) ) - ∫_𝒟_m d^2φ∫_𝒟_m d^2φ' 1/2n(Z(φ),Z(φ')) f(φ) f(φ') - ∑_r,r'=1^n1/2n(Q_r,Q_r') α_rα_r'.The function f(φ) and the parameter α_r are determined through the variation and differentiation of F_n[f,{α_k}]: δ F_n[f] =Re∫_𝒟_m d^2φ δ f(φ)( 1/n(J,Z(φ)) - ∑_r=1^n1/n(Q_r,Z(φ)) α_r- ∫_𝒟_m d^2φ' 1/n (Z(φ),Z(φ')) f(φ') ) =0, d F_n[α_r]/d α_r =Re( 1/n(J,Q_r) - ∫_𝒟_m d^2φ 1/n(Q_r,Z(φ)) f(φ)- ∑_r'=1^n1/n (Q_r,Q_r') α_r')=0.Substituting (<ref>) into (<ref>) and taking the n →∞ limit, we find D_ZQ≥1/2∫_𝒟_m d^2φRe( Z_J(φ)f(φ) )+ 1/2∑_r=1^n Re( Q_r,Jα_r )= D_Z, Q_r,J := lim_n →∞1/n (J, Q_r).in the n →∞ limit. The first term is what was evaluated in the previous section. Although Q_r,J≠ 0 in the spin chain without spin-reversal symmetry, the second term vanishes in the thermodynamic limit, since the boundary effect on the local charges localizes at the outermost sites of the spin chain.The second inequality holds since Q_r,J≠ 0 in the spin chain without spin-reversal symmetry. IS THIS REALLY TRUE, NAMELY BREAKING OF SPIN REVERSAL SYMMETRY IS ONLY BY THE BOUNDARY TEMRS, SO ITS EFFECT WILL PROBABLY VANISH IN THE THERMODYNAMIC LIMIT?§ CONCLUSIONIn this paper, we have showed that for a certain family of the spin-1/2 XXZ chain the exact NESS density operator is derived underin the existencepresence ofarbitrary boundary magnetic fields. By properly choosing the dissipation rates as functions of boundary parameters, we found that the NESS density operator is given by the NHTO in spite of arbitrary choicea choice of diagonal and non-diagonal boundary fields.We have also derived the quasilocal charges of the corresponding spin chain without boundary dissipation. We found that the NHTO satisfies quasilocality even in the system without spin-reversal symmetry. In such a system, the conventional local charges possess the odd parity part besides the even parity part. As a consequence, we showed the existence of the further optimized Mazur bound by using the non-orthogonality of the current and the local charges. This is not due to the boundary effect on the bulk behavior but simply due to the symmetry of the Hamiltonian.that the terms coming from the odd parity part of the local charges have non-zero values, although they vanish in the thermodynamic limit.We shall mentionnote that the local charges also give non-zero contribution in the spin chain with the bulk homogeneous magnetic field, although their effect vanishes in the high temperature limit. Although we discussed the nonequilibrium behavior of the integrable spin chain with boundaries, we did not find any perception in the context of the reflection relation <cit.>. Furthermore, we did not investigate thefind any integrable NESS density operators which are not expressed in terms of the NHTO. By using the inhomogeneous rotational transformation in the xy-plane, the homogeneous version of which we used in this paper to obtain the U(1)-symmetry of the Hamiltonian, we expect that the NESS density operator is modified in such a way that includes the inhomogeneity parameters.We leave these questions as future works. THIS LAST SENTENCE I DON'T UNDERSTAND. § ACKNOWLEDGEMENTSC. M. is supported by JSPS Grant-in-Aid, No. 15K20939, Japan and JST CREST, No. JPMJCR14D2, Japan.T. P. is supported by Grants P1-0044, N1-0025 and N1-0055 of Slovenian Research Agency, and ERC Grant OMNES.abbrv | http://arxiv.org/abs/1705.09105v1 | {
"authors": [
"Chihiro Matsui",
"Tomaz Prosen"
],
"categories": [
"cond-mat.stat-mech",
"nlin.SI"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170525093048",
"title": "Construction of the steady state density matrix and quasilocal charges for the spin-1/2 XXZ chain with boundary magnetic fields"
} |
Distributionally Robust Optimisation in Congestion ControlJakub Mareček^[email protected], Robert Shorten^2, Jia Yuan Yu^3^1 IBM Research, Ireland ^2 University College Dublin, Ireland ^3 Concordia University, Canada ==================================================================================================================================================================================The effects of real-time provision of travel-timeinformation on the behaviour of drivers are considered. The model of Marecek et al.[Int. J. Control 88(10), 2015] is extended to consider uncertainty in the response of a driver to an interval provided per route. Specifically, it is suggested that one can optimiseover all distributions of a random variable associated with the driver's response with the first two moments fixed, and for each route, over the sub-intervals within the minimum and maximum in a certain number of previous realisations of the travel time per the route.§ INTRODUCTION Congestion on the roads is often due to drivers using them in a synchronized manner, “a wrong road at a wrong time”.Intuitively, the synchronisation is partly due to the reliance onthe same unequivocal information about past traffic conditions,which the drivers mistake for a reliable forecast of future traffic conditions.Perhaps, if the information about past traffic conditions were providedin a different form, the synchronisation could be reduced. This intuition led to a considerable interest inadvanced traveller information systems and models of dynamics of information provision <cit.>. In this paper, we propose and study novel means of information provision.With the increasing availability of satellite-positioning traces of individual cars,it is becoming increasingly clear that there are many approaches to aggregating the information and providing them to the public,while it remains unclear what approach is the best.Following <cit.>,we model the relationship of information provision and road use as a partially known non-linear dynamical system. In practice, our approach relies on a road network operator with up-to-date knowledge of congestion across the road network, who broadcasts travel-time information to drivers, which is chosen so as to alleviate congestion, based on an estimate of the driver's response function,e.g., up to the first two moments of some random variables involved. In terms of theory, we study non-linear dynamics, which are not perfectly known. This poses a considerable methodological challenge.We make first steps towards modelling the interactions among the road network operator and the drivers over time as a stochastic control problem and the related delay-tolerant and risk-averse means of information provision. In an earlier paper <cit.>, we have studied the communication of a scalar per route at each time, specific to each driver. In another recent paper <cit.>,we have studied the communication of two scalars (an interval) per route (or road segment) at each time,with the same information broadcast to all drivers. There, the intervals were based on the minimum and maximum travel time over the segment within a time window. In this paper, we propose an optimisation procedure, where one considers sub-intervals of the interval. Across all three papers, we show that congestion can be reduced by withholding some information, while ensuring that the information remains consistent with the true past observations. Let us consider the travel time over a route as a time series. Broadcasting the most recent travel time,an average over a time window,or any other scalar function over a time window,may lead to a suboptimal “cyclical outcome,” where drivers overwhelmingly pick the supposedly fastest route, leading to congestion therein, and another route being announced as the fastest, only to become congested in turn. On the other hand, depriving the drivers of any information leads to a suboptimal outcome,where each driver acts more or less randomly. We illustrate our findings on an intentionally simple model. § RELATED WORK Recent studies <cit.> have focussed ona dynamic discrete-time model of congestion, where a finite population of N drivers is confronted with M alternative routes at every time step.The time horizon is discretized into discrete periods t=1,2,…. At each time, each driver picks exactly one route, and is hence “atomic”. Let a_t^i denote the choice of driver i at time t and n^m_t = ∑_i 1_[a_t^i = m] be the number of drivers choosing route 1 ≤ m ≤ M at time t. Sometimes, we use n_t to denote the vector of n^m_t for 1 ≤ m ≤ M. The travel time c_m(n^m_t) of route m at time t is a function of the number n^m_t of drivers that pick m at time t,c_m: ℕ→ℝ_+. The social cost C(n_t) weights the travel times of the routes at time t with the proportions of drivers taking the routes,C(n_t) ≜∑_m = 1^Mn^m_t/N· c_m(n^m_t).Notice that in the case of two alternatives, M = 2, C(n_t) becomes a function of n^1_t only,with n^2_t beign equal to N - n^1_t: C(n_t) = n^1_t/N· c_1(n^1_t) + N - n^1_t/N· c_2(N - n^1_t).The social or system optimum at every time step t is n^* ∈min_0 ≤ n ≤ N C(n). Notice that the travel time is, in effect, a time-series,with a data point per passing driver.Often, however, one may want to aggregate the time series, for instance in order to communicate travel times succinctly. Essentially, <cit.> discuss various means of aggregating the history of travel times c_m(n^m_t') for all 1 ≤ m ≤ M and for all times t' < t in past relative to present t. Every driver i takes route a_t^ibased on the history of s_t', t' ≤ t received up to time t. In keeping with control-theoretic literature, a mapping of such a history to a route is called a policy. Ω denotes the set of all possible types of drivers and μ a probability measure over the set Ω, which describes the distribution of the population of drivers into types. We refer to <cit.> for the measure-theoretic definitions.Sending of the most recent travel time or any other single scalar value per route uniformly to all drivers is not socially optimal <cit.>. One option for addressing this issue is to vary the scalar value sent to each user.<cit.> studied a scheme,where the network operator sends a distinct s^i_t ≜ (y^m,i_t, 1 ≤ m ≤ M) ∈ℝ^M to each driver i at time t, wherey^m,i_t ≜ c_m(n^m_t-1) + w^i,m_t,and the sequence of random noise vectors { w^i,m_t : t=1,2,…} is such that for all t, w^i,m_t = 0, and w^i,m_t - w^i,m'_t is normally distributed with mean 0 and variance σ^2 for 1 ≤ m ≠ m' ≤ M. These properties of w^i,m_t assure that no driver is being disadvantaged over the long run, but the absolute value of w^i,m_t may vary across drivers i at a particular time t.Considering the introduction of such driver-specific randomisation may not be desirable, <cit.> presented a scheme that broadcasts two distict scalar values per route to all drivers, where the two distinct scalars for a particular route are the same for all the drivers at a particular time. For M routes, one has s_t ≜ (u^m_t, u^m_t, 1 ≤ m ≤ M) ∈∈ℝ^2M, where u^m_t≜ c_m(n^m_t-1) + ν^m_t - δ^m/2, u^m_t≜ c_m(n^m_t-1) + ν^m_t + δ^m/2,m ∈{A,B}, where ν_t^m are uniform random variables with support: (ν^m_t)= [-δ^m/2,δ^m/2]. Notice that <cit.> use δ and γ to denote the non-negative constants δ^A and δ^Bin the case of M=2, and hence use (δ,γ)-interval to denote such s_t.Let Ω be a finite subset of = [0,1] and assume thateach driver 1 ≤ i ≤ N is of type ω∈Ω and follows the policy π^ω:a_t^i ≜π^ω(s_t) ≜min_m=1^M ωu^m_t + (1-ω) u^m_t.in response to s_t. Observe that for ω = 0, policy π^0 models a risk-averse driver, who makes decisions based solely on u^m_t. Similarly, π^1 and π^1/2 model risk-seeking and risk-neutral drivers, respectively. Under certain assumptionsbounding the modulus of continuity of functions c_A, c_B, …, cf. <cit.>, one can show that this results in a stable behaviour of the system.Considering that any randomisation may be undesirable, <cit.> suggested broadcasting a deterministically chosen interval for each route. In one such approach, called r-extreme <cit.>, one simply broadcasts the maximum and minimum travel time within a time window ofr most recently observed travel times. In another variant, called exponential smoothing <cit.>, one broadcasts a weighted combination of the current travel-time and past travel times, alongside a weighted combination of the current variance of the travel times and the previously sent information about the variance.Under some additional assumptions, one can analyse the resulting stochastic (delay) difference equations: Using results developed in the theory of iterated random functions <cit.>, <cit.> show that the r-extreme schema yields ergodic behaviour when the distributionof types of drivers changes over time in a memory-less fashion. <cit.> extended the result to populations, whose evolution is governed by a Markov chain, which allows, e.g., for different distributions at different times of the day, such as at night, during the morning and afternoon peaks, and all other times. In Table <ref>, we present an overview of these schemata.We should like to stress that the above is not a comprehensive overview of related work. We refer to <cit.> for pioneering studies in the field as well as to <cit.> for extensive, book-lengthoverviews of further related work.§ DISTRIBUTIONALLY ROBUST OPTIMISATION In this paper, we suggested broadcasting a deterministically chosen interval for each route,where the deterministic choice is based on optimisation over subintervals of theinterval given by the minimum and maximum over a time window of a finite, fixed length r.For 1 < r < t, we define s_t = (u_t^1, u_t^1, u_t^2, u_t^2, …, u_t^M, u_t^M, ) to be r-supported, whenever min_j=t-r,…,t{ c_m(n^m_j) }≤u_t^m < u_t^m ≤max_j=t-r,…,t{ c_m(n^m_j) }.Notice that r-extreme s_t is a special case of r-supported s_t. To study the effects of broadcasting r-supported s_t, we need to formalise themodel of the population. Clearly, one can start with:[Full Information] Let us assume that Ω is a finite set. Further, let us assume the number of drivers of type ω at time t+1 is N μ_t+1(ω) and that N μ_t+1(ω)is known to the network operator at time t. Assumption <ref> is very restrictive.Instead, we may want to assume that μ_t are independently identically distributed (i.i.d.) samples of a random variable.[One could go further still and assume time-varying distributions of μ_t, or more general structures, yet. We refer to <cit.> for an example, but note that such assumptions do not allow for the efficient application of methods of computational optimisation, in general. In this paper, we hence consider the i.i.d. assumption.]In the tradition of robust optimisation <cit.>, one could assume that a support of the random variable is known and optimise social cost over all possible distributions of the random variable with the given support. That approach, however, tends to produce overly conservative solutions, when it produces any feasible solutions at all. In the tradition of distributionally robust optimisation (DRO) <cit.>, one could assume that a certain number of moments of the random variable are known and optimise social cost over all possible distributions of the random variable with the given moments. We suggest to use DRO with the first two moments: [Partial Information] Let us assume that Ω is a finite set. Let us assume the number of drivers of type ω at time t+1 is N μ_t+1(ω), but that the distribution of μ_t+1 is unknown at time t, except for the first two moments of the distribution of μ_t+1, denoted E, Q:E= [ E_1; E_2; ⋮; E_|Ω|; ] = [ μ_t+1(1);⋮; μ_t+1(Ω);]Q=[ Q_11… Q_1|Ω|;⋮ ⋮; Q_|Ω|1… Q_|Ω||Ω|;] =[ [ μ_t+1(1); μ_t+1(2);⋮; μ_t+1(|Ω|);][ μ_t+1(1); μ_t+1(2);⋮; μ_t+1(|Ω|);]^T ] and let us assume E, Q are known to the network operator at time t. Notice that Assumption <ref> is much more reasonable than Assumption <ref>.The authorities can compute an unbiased estimate of the first two momentsusing readily-available statisticalestimation techniques <cit.>.In contrast, ascertaining the actual realisation of the random variable inreal time seems impossible, andestimating more than two moments of a multi-variate random variable remains a challenge, as the requisite number of samples grows exponentially with the order of the moment, which in turn makes the computations prohibitively time consuming. In short, we believe that Assumption <ref> presents a suitable trade-off between realism and practicality. Next, one needs to decide on the objective, which should be optimised.Clearly, even a finite-horizon approximation of the accumulated social cost is a challenge. Beyond that, we can show a yet stronger negative result:Under Assumption <ref>,there exist c_A, c_B, and an initial s_1 broadcast, such that it is undecidable whether iterates s_t ∈^n, n ≥ 2 induced by policies π^ω responding to intervals broadcast converge to a points_T ∈^n from s_1, such that s_t for all t > T is equal to s_T.The proof is based on the results of <cit.> that given piecewise affine function g : R^2 → R^2 and an initial point x_0 ∈ R^2,it is undecidable whether iterated application g … g(x_0) reaches a fixed point, eventually, and the fact we make no assumptions about the functions c_m. Although Proposition <ref> does not rule out weak convergence guarantees in the measure-theoretic sense under Assumption <ref>, for instance, some assumptions concerning the functions c_m do simplify the matters considerably.To formulate such an assumption, observe that the function g corresponds to a composition of the social cost (<ref>) and the policy (<ref>). In particular:u^m_t+1= min{u^m_t, c_m(n^m_t)}, u^m_t+1 = max{u^m_t, c_m(n^m_t)},wherein one applies c_m to values of n^m_t: n^m_t= ∑_i 1_(a^i_t = A_m)= ∑_i ∑_ω∈Ω 1_(a^i_t = A_m | driveriis of type ω )μ_t(ω) = N ∑_ω∈Ω 1_⋀_s ≠ r ( ωu^m_t + (1-ω) u^m_t < ωu^s_t + (1-ω) u^s_t )μ_t(ω),whereby one obtains u^m_t+1, u^m_t+1 as a function of u^m_t, u^m_t.We refer to the proof of Theorem 1 in <cit.> for a detailed discussion of this signal-to-signal mapping and properties of μ_t. One may hence obtain a signal-to-signal mapping g of more desirable properties by restricting oneself to a particular class of c_m, and hence to a particular class of social costs (<ref>). In particular, we restrict ourselves to:For any functions c_m convex on [0, 1],there exist solvers for the minimisation of the unconstrained social cost C (cf. Eq. <ref>), with guaranteed convergence to a stationary point. Using a wealth of results <cit.> on the optimisation of DC (“difference of convex”) functions, we can show: Under Assumption <ref>,a stationary point of:min_ s_t = (u^m_t, u^m_t, 1 ≤ m ≤ M) ∈ P(S_t, Ω) C(n_t) can be computed up to any fixed precision in finite time, where P ⊆^2M is the set of r-supported signals (<ref>)and functions c_m, 1 ≤ m ≤ M are convex on [0, 1]. Let us introduce an auxiliary indicator variable and a non-negative continuous variable:x_t,m^ω = 1_[π^ω(s_t) = m] = 1if ω selects actionmat timet0otherwisey_t,i,j^ω =- g^ω_i,jifg^ω_i,j < 00otherwisey_t,i,j^ω =g^ω_i,jifg^ω_i,j≥ 00otherwisewhere g^ω_i,j≜ωu_t^i + (1-ω) u_t^i - ωu_t^j - (1-ω) u_t^j.See that n_t^m = ∑_ω∈Ω x_t,m^ω n_t(ω).Sometimes, we use x_t to denote a matrix of x_t,m^ω for all ω∈Ω, 1 ≤ m ≤ M.It is easy to show there exist a lifted polytope P' such that:min_ s_t∈^2M∑_m=1^M n^m_t/N· c_m(n^m_t),s ∈ P(S_t,Ω) = min_ s_t∈^2M, y_t, y_t ∈^M(M-1)|Ω| x_t ∈{0, 1 }^M|Ω|∑_m=1^M n^m_t/N· c_m(n^m_t) (s_t, x_t, y_t, y_t ) ∈ P'(S_t,Ω),The definition of the polytope P' depends on the policies defined by Ω and the history of signals S_t. Specifically:ωu^i_t + (1-ω) u^i_t - ωu^j_t - (1-ω) u^j_t≤ y_t,i,j^ω ∀ i, j, t, ω ωu^j_t + (1-ω) u^j_t - ωu^i_t - (1-ω) u^i_t≤ y_t,i,j^ω ∀ i, j, t, ω - Z (1 - x_t,m^ω) ≤ y_t,m,i^ω≤ Z (1 - x_t,m^ω) ∀ t, m, ω- Z x_t,m^ω≤ y_t,m,i^ω≤ Z x_t,m^ω ∀ t, m, ωu_t^m ≥ u_t^m ≥min_j=t-r,…,t{ c_m(n^m_j) } ∀ m, r, t u_t^m ≤max_j=t-r,…,t{ c_m(n^m_j) } ∀ m, r, t ∑_m = 1^M x_t,m^ω =1∀ t, ω y_ω, y_ω≥0∀ωwhere the max, min operators are applied tothe revealed realisations of the random variable n^m_j, and hence yield constants, rather than bi-level structures. Further, Z is a sufficiently large constant,max_m = 1,2,…,M, m'∈{1,2, …, M }∖{m} u^m_t,u^m_t,u^m'_t,u^m'_t{ |ωu^m_t + (1-ω) u^m_t - ωu^m'_t - (1-ω) u^m'_t| }≤max_x { C(x) }. The integer component can be solved by branching, whereby the Lagrangian gives us an unconstrained relaxation of the original problem.Hence, by Proposition <ref>, the stationary point can be computed up to any precision in finite time. Under Assumption <ref>,let us consider functions c_m, 1 ≤ m ≤ M convex on [0, 1] and min_ (u^m_t, u^m_t, 1 ≤ m ≤ M) ∈ P(S_t, Ω)sup_D∼ (E, Q)C( _D n_t )where D∼ (E, Q) in the inner optimisation problem suggests optimisation over the infinitely many distribution functions of Ω with the first two moments of Assumption <ref>, and P ∈^2Mis the set of r-supported signals (<ref>). A stationary point of the distributionally robust optimisation problem (<ref>)can be computed up to any fixed precision in finite time. Notice that we can reformulate the problem (<ref>) as an integer semidefinite program by the introduction of a new decision variable W in dimension |Ω| × |Ω|, vector w_ω∈^|Ω|, and scalar q_ω, in addition to the variables introduced in the proof of Proposition <ref>: min_ (u^m_t, u^m_t, 1 ≤ m ≤ M) ∈ P(S_t, Ω)sup_D ∼ (E, Q)C( _D n_t ) =min_ s_t∈^2Mx_t ∈{0, 1 }^M|Ω| sup_ D ∼ (E, Q) ∑_m=1^M ( ∑_ω∈Ω( x_t,m^ω_D μ_t(ω)) · c_m ( ∑_ω∈Ω x_t,m^ω N _D μ_t(ω)) )s.t._D μ_t = E _D [ μ_t μ_t^T ] = Qs ∈ P'(S_t, Ω) = min_ s_t∈^2M,y_t, y_t ∈^M(M-1)|Ω|x_t ∈{0, 1 }^M|Ω| W_ω∈^|Ω| × |Ω|w_ω∈^|Ω|, q_ω∈ ∑_m=1^M ∑_ω∈Ω( x_t,m^ω e_ω w_ω)/N· c_m ( ∑_ω∈Ω( x_t,m^ω e_ω w_ω) )s.t.∑_ω∈Ω[ W_ω w_ω; w_ω^T q_ω ] = [ Q E; E^T 1 ][ W_ω w_ω; w_ω^T q_ω ]≽ 0 ∀ω∈Ω (s_t, x_t, y_t, y_t ) ∈ P'(S_t,Ω),where e_ω are vectors with only the ω^th entry of 1 and others 0.The first equality follows from the definition of C (<ref>). The second equality follows from the work of Bertsimas et al. <cit.> on minimax problems, and specifically from Theorem 2.1 therein. Although Theorem 2.1 does not consider integer variables explicitly, it is easy to see that for each of the 2^M|Ω| possible integer values of x_t, the equality holds, and hence it holds generically.See also the lucid treatment of Mishra et al. <cit.>.Computationally, one can apply branching to the integer variables x_t, as in the proof of Proposition <ref>, which leaves one with a semidefinite program with a non-convex objective. There, one can formulate the augmented Lagrangian, which is non-convex, but well-studied <cit.>. For instance, it can be reformulated to a “difference of convex” form and Proposition <ref> can be applied. Let us multiply C(·) by N to study the 2 M terms one by one.We want to show that the rest is a sum of a convex and concave terms. Let us see that for i = 1, 2, …, M - 1, we have the term n_t^i c_i(n_t^i), which is convex in n_t^i, considering that for convex and non-decreasing g and convex f, we know g(f(x)) is convex. For i = M, we have the terms c_i( 1-∑_i = 1^M-1 n_t^i) and a M - 1 terms from- (∑_i = 1^M-1 n_t^i) c_i( 1-∑_i = 1^M-1 n_t^i). Considering that convexity is preserved by affine substitutions of the argument, the former term is convex for the affine subtraction and convex c_i. Considering the additive inverse of aconvex function is a concave function, we see - n_t^i n_t^M(·) is concave. The proposition follows from the following Proposition <ref>.Alternatively, one may consider polynomial functions c_m, where the minimum of the social cost C can be computed up to any fixed precision in finite time by solving a number of instances of semidefiniteprogramming (SDP).§ A COMPUTATIONAL ILLUSTRATION For optimisation problems such as (<ref>) and (<ref>), there are solvers based on sequential convex programming with known rates of convergence <cit.>. In our computational experiements, we have extended a sequential convex programming solver of Stingl et al. <cit.>, which handles polynomial semidefinite programming of (<ref>), to handle mixed-integer polynomial semidefinite programming.Specifically, Stingl et al. replace nonlinear objective functions by block-separable convex models, following the approach of Ben-Tal and Zhibulevsky <cit.> and Kočvara and Stingl <cit.>.In our experiments, we have considered the same set-up as in <cit.>, where M=2 andtwo Bureau of Public Roads (BPR) functions are used for thecosts, as presented in Figure <ref>. The population is given byΩ = { 0, 0.5, 1, Uniform(0, 1), Uniform(0, 1) },the initial signal is s_1 = (0.5, 1, 0.6, 0.9), and κ = 0.15, μ_t(ω) ∼Uniform(1/5 - κ, 1/5 + κ)∀ω∈Ω, t > 1,and N=30. These settings have been chosen both for the simplicity of reproduction as well as to allow for comparisonwith plots presented in <cit.>.Figure <ref> illustrates the cost {C(n_t)} over time 1 ≤ t ≤ 20,for three lengths r of the look-back, r = 1 (top), r = 3 (middle), r = 5 (bottom), with error bars at one standard deviation capturing the variability over the sample paths. It seems clear that the r-supported scheme (in dark blue, Eq. <ref>) is only marginally worse than the full-information optimum (in red, Eq. <ref>), which is “pre-scient” and hence impossible to operate in the real-world. Also, it seems clear that for low values of r, there is not enough data to estimate the second moments, and hence the use of the first moment (in green) behaves similarly to the use of the first two moments (in dark blue). Both compared to the use of the first moment and to the previously proposed r-extreme scheme (in light blue), the r-supported scheme yields costs with less prominent extremes, even after averaging over the sample paths. Further, Figure <ref> illustrates the the process {C(n_t)} averaged over 1 ≤ t ≤ 20 for varying r, again with error bars at one standard deviation. It shows that employing r-supported scheme (in dark blue, Eq. <ref>)allows for a reduction of the social cost,when compared to r-extreme singalling (in light blue), across a range of the length r of the look-back interval. Again, it seems clear that the r-supported scheme (in dark blue) is only marginally worse than the full-information optimum (in red, Eq. <ref>). Finally, we note that the forThe stationary point (<ref>) can be computed up to precision 10^-6 in about 15 seconds on a basic laptop with Intel i5-2520M,although the run-time does increase with the number of routes.This is much more than the run-time of the previously proposed r-extreme scheme. An efficient implementation of the r-supported scheme remains a major challenge for future work. § CONCLUSIONS In conclusion, there are multiple ways of introducing “uncertainty” into the behaviour of the road user in terms of the route choice.Previously, the addition of zero-mean noise with a positive variance σ <cit.>, broadcasting intervals such as (δ,γ) intervals <cit.> and r-extreme intervals (minima and maxima over a time window of size r) <cit.>, and intervals based on exponential smoothing <cit.>,have been shown result in the distribution of drivers over the road network converging over time, under a variety of assumptions about the evolution of the population over time. This paper studied the optimisation of the social cost over sub-intervals within the minima and maxima over a time window of size r, under a variety of assumptions.This paper is among the first applications of distributionally robust optimisation (DRO)in transportation research. while other recent work considered its use in stochastic traffic assignment <cit.>, where it presents a tractable alternative to multinomial probit <cit.>, and in traffic-light setting <cit.>. We envision there will be a wide variety of further studies,once the power of DRO is fully appreciated in the community. This work opens a number of questions in cognitive science, multi-agent systems,artificial intelligence, and urban economics. How do humans react to intervals, actually? How to invest in transportation infrastructure, knowing that information provision can be co-designed to suit the infrastructure? Future technical work may include the study of variants of the proposed scheme, such asbroadcasting s_t such that _j=t-r,…,t{ c_m(n^m_j) }≥u_t^m≥min_j=t-r,…,t{ c_m(n^m_j) }, _j=t-r,…,t{ c_m(n^m_j) }≤u_t^m≤max_j=t-r,…,t{ c_m(n^m_j) },wheredenotes the average. One could also employ risk measures such as value at risk (VaR) and conditional value at risk (CVaR) for a given coefficient α and distribution function L with support { c_m(n_t^m) }_t,as suggested inTable <ref>. Further studies of (weak) convergence properties <cit.>, including the rates of convergence,and further developments of the population dynamics <cit.> would also be most interesting. Beyond transportation, one could plausibly employ similar techniques in related resource-sharing problems (e.g., ad keyword auctions,dynamic pricing in power systems, announcements of emergency evacuation routes) in order to improve the variants of social costs therein.§ ACKNOWLEDGEMENTThis research received funding from the European Union Horizon 2020 Programme (Horizon2020/2014-2020) under grant agreement number 688380. Robert Shorten has been funded by Science Foundation Ireland under grant number 11/PI/1177.is-plain | http://arxiv.org/abs/1705.09152v1 | {
"authors": [
"Jakub Marecek",
"Robert Shorten",
"Jia Yuan Yu"
],
"categories": [
"math.OC"
],
"primary_category": "math.OC",
"published": "20170525124956",
"title": "Distributionally Robust Optimisation in Congestion Control"
} |
Institute of Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, ChinaInstitute of Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, ChinaInstitute of Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, ChinaInstitute of Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, ChinaInstitute of Particle and Nuclear Physics, Henan Normal University, Xinxiang 453007, China The B_c^∗ → B_u,d,sV, B_u,d,sP decays are investigated with the QCD factorization approach, where V and P denote the ground SU(3) vector and pseudoscalar mesons, respectively. The B_c^∗ → B_u,d,s transition form factors are calculated with the Wirbel-Stech-Bauer model. It is found that branching ratios for the color-favored and Cabibbo-favored B_c^∗ → B_sρ, B_sπ decays can reach up to O(10^-7), which might be measurable in the future LHC experiments.12.15.Ji 12.39.St 13.25.Hw 14.40.Nd Possibility of searching for B_c^∗ → B_u,d,sV, B_u,d,sP decays Gongru Lu December 30, 2023 ====================================================================== § INTRODUCTION The vector B_c^∗ meson, a spin-triplet ground state, consists of two heavy quarks with different flavor numbers B = C = ±1, i.e., b̅c for B_c^∗+ meson and bc̅ for B_c^∗- meson. With nonzero bottom and charm numbers, the bottom and charm quarks of the B_c^∗ meson cannot annihilate into gluons and photons via the strong and electromagnetic interactions, respectively, unlike the decay modes of the unflavored J/ψ(1S) and Υ(1S) mesons. The B_c^∗ meson serves as a unique object in studying the heavy quark dynamics, which is inaccessible through both charmonium and bottomonium.The B_c^∗ meson lies below the B_qD_q (q = u, d, s) meson pair threshold. And the mass splitting m_B_c^∗ - m_B_c ≈ 50 MeV <cit.> is less than the pion mass. Hence, the B_c^∗ meson decays via the strong interaction are strictly forbidden. The electromagnetic transition process, B_c^∗ → B_cγ, dominates the B_c^∗ meson decays, but suffers seriously from a compact phase space suppression, which results in a lifetime of τ_B_c^∗ ∼ O(10^-17s) <cit.>. Besides, the B_c^∗ meson decays via the weak interaction, although with very small decay rates, are allowable within the standard model.The B_c^∗ meson has a relatively large mass. In addition, both constituent quarks b and c of the B_c^∗ meson can decay individually. Therefore, the B_c^∗ meson has rich weak decay channels. The B_c^∗ meson weak decays, similar to the pseudoscalar B_c meson weak decays <cit.>, can be divided into three classes: (1) the c quark decay with the spectator b quark, (2) the b quark decay with the c quark as a spectator, and (3) the b and c quarks annihilation into a virtual W^± boson. This property makes the B_c^∗ meson another potentially fruitful laboratory for studying the weak decay mechanism of heavy flavor hadrons.The study of B_c^∗ weak decays might be interesting, but has not really started yet. One of the major reasons is the extraordinary difficulty of producing the B_c^∗ meson. The production cross section for the B_c^∗ meson in hadronic collisions via the dominant process of g + g → B_c^∗ + b + c̅ <cit.> is at least at the order of α_s^4. The nature of QCD's asymptotic freedom implies a much small possibility of creating two heavy quark pairs (bb̅ and cc̅) from the vacuum at the ultrahigh energy. Fortunately, the high luminosities of the running LHC and the future Super proton proton Collider (SppC, which is still under discussion today) will promisingly improve this situation. It is expected that a huge amount of the B_c^∗ data samples would be accumulated, and offer a valuable opportunity to investigate the B_c^∗ weak decays.As is well known, there exist some hierarchical structures among the Cabibbo-Kobayashi-Maskawa (CKM) matrix elements. The CKM coupling strength for the bottom quark weak decay is proportional to |V_cb| ∼ O(λ^2) or |V_ub| ∼ O(λ^3), while the CKM coupling strength for the charm quark weak decay is proportional to |V_cs| ∼ O(1) or |V_cd| ∼ O(λ), with the Wolfenstein parameter λ ≈ 0.2 <cit.>. The B_q (q = u, d, s) weak decays are induced dominantly by the bottom quark decay with the phenomenological spectator scheme. The B_c^∗ → B_qV, B_qP decays are actually inducedby the charm quark weak decay, where V and P denote respectively the lightest 9-pelts SU(3) vector and pseudoscalar mesons. With respect to the B_q weak decays, the B_c^∗ → B_qV, B_qP decays are favored by the CKM matrix elements. In this paper, we will study the B_c^∗ → B_u,d,sV, B_u,d,sP weak decays with the QCD factorization (QCDF) approach <cit.>, in order to provide an available reference for the future experimental investigation. There is a more than 2.0 σ discrepancy between the value for CKM matrix element |V_cs| obtained from semileptonic D decays and that from leptonic D_s decays[1] <cit.>. The B_c^(∗) → B_sV, B_sP decays, together with semileptonic D decays and leptonic D_s decays, will provide |V_cs| with more stringent constraints. [1]The value for CKM matrix element |V_cs| is |V_cs| = 0.953±0.008±0.024 from semileptonic D decays, and |V_cs| = 1.008±0.021 from leptonic D_s decays <cit.>. In addition, some of the B_c weak decays, for example, the B_c → B_sπ decay <cit.>, have been measured now. One possible background might come from the B_c^∗ decays, due to a slightly larger production cross section σ(B_c^∗) than σ(B_c) in hadronic collisions <cit.>, and the nearly equal mass m_B_c^∗ ≃ m_B_c <cit.>. Hence, the study of the B_c^∗ → B_u,d,sV, B_u,d,sP decays will be helpful to the experimental analysis on the B_c → B_u,d,sV, B_u,d,sP decays.This paper is organized as follows. The theoretical framework and decay amplitudes will be presented in Section <ref>. Section <ref> is the numerical results and discussion. The last section is a summary. § THEORETICAL FRAMEWORK §.§ The effective Hamiltonian Using the operator product expansion and the renormalization group (RG) method, the low-energy effective weak Hamiltonian describing the B_c^∗ → B_u,d,sV, B_u,d,sP decays has the following general structure <cit.>, H_ eff= G_F/√(2)∑_q,q^'=s,d V_cq^∗ V_uq^'{C_1(μ)Q_1(μ) + C_2(μ)Q_1(μ) }+h.c., where the Fermi coupling constant G_F ≃ 1.166×10^-5GeV^-2 <cit.>; V_cq^∗ V_uq^' is a product of the CKM matrix elements. Using the Wolfenstein parameterization, there are <cit.>V_cs^∗V_ud = 1 - λ^2 - 1/2 A^2 λ^4+ 1/2 A^2 λ^6 { 1 -ρ^2-η^2 -2 (ρ-i η) } +O(λ^8), V_cs^∗V_us = λ - λ^3/2 - λ^5/8 - λ^7/16 - 1/2 A^2 λ^5 + 1/2 A^2 λ^7 {1/2 - ρ^2-η^2 -2 (ρ-i η) } +O(λ^8), V_cd^∗V_ud = -V_cs^∗V_us-A^2 λ^5 (ρ-i η) +O(λ^8), V_cd^∗V_us = - λ^2+ 1/2 A^2 λ^6 {1-2 (ρ-i η) } +O(λ^8), where the values for these Wolfenstein parameters A, λ, ρ and η are given in Table <ref>.The renormalization scale μ separates the physical contributions into two parts. The hard contributions above the scale μ are summarized into the Wilson coefficients C_i(μ). With the RG equation for C_i(μ), the Wilson coefficients at an appropriate scale μ_c ∼ O(m_c) for the charm quark decay are given by <cit.>C⃗(μ_c)=U_4(μ_c,m_b) U_5(m_b,m_W) C⃗(m_W) ,where m_W, m_b and m_c are the mass of the W boson, b quark and c quark, respectively. Here U_f(m_2,m_1) denotes the RG evolution matrix for f active flavors. The initial values for the Wilson coefficients C⃗(m_W) at scale μ_W = m_W to a desired order in α_s can be calculated with perturbation theory. The expressions for the RG evolution matrix U_f(m_2,m_1) and Wilson coefficients C⃗(m_W), including both leading order (LO) and next-to-leading order (NLO) corrections, have been presented in Ref.<cit.>. The contributions below the scale μ are included in the hadronic matrix elements (HME) where the local four-quark operators Q_i are sandwiched between the initial and final states. The expressions for the four-quark operators in question areQ_1= [ q̅_α γ_μ (1-γ_5) c_α] [ u̅_β γ^μ (1-γ_5) q^'_β], Q_2= [ q̅_α γ_μ (1-γ_5) c_β] [ u̅_β γ^μ (1-γ_5) q^'_α], where the subscripts α and β are color indices. It should be pointed out that (1) because the contributions from the penguin operators and annihilation topologies are proportional to the CKM factor V_cb^∗V_ub ∼ O(λ^5) and therefore negligible in the actual calculation of branching ratio <cit.>, only the contributions of tree operators are considered here. (2) The participation of the strong interaction, especially, the nonperturbative QCD effects, makes the theoretical treatment of HME very complicated. The main problem at this stage is how to effectively factorize HME into hard and soft parts, and how to evaluate HME properly.§.§ Hadronic matrix elements Hadronic matrix elements might be the most intricate part in the calculation of heavy flavor weak decay, due to the entanglement of perturbative and nonperturbative contributions. Phenomenologically, one has to turn to some approximation and assumption, which bring uncertainties and model dependence to theoretical predictions. A simple approximation is the naive factorization ansatz (NF) according to Bjorken's color transparency argument, which says that the colorless energetic hadron has flown away from the weak interaction point during the formation time of the emission hadron <cit.>. With the NF approach, HME is parameterized as a product of decay constants and hadron transition form factors <cit.>. A major flaw of the NF approach is the disappearance of scale dependence and strong phases from HME, which results directly in a scale-sensitive nonphysical prediction and none of CP violation for nonleptonic meson weak decays. In order to overcome these shortcomings of the NF approach, nonfactorizable contributions to HME should be carefully considered, as commonly recognized. Some QCD-inspired models, such as, the QCDF approach <cit.>, the soft and collinear effective theory <cit.>, the perturbative QCD approach <cit.>, and so on, have been developed recently, based on the Lepage-Brodsky treatment on exclusive processes <cit.> and some power counting rules in the expansion in α_s and Λ_ QCD/m_Q, where α_s is the strong coupling, Λ_ QCD is the QCD characteristic scale, and m_Q is the mass of a heavy quark. In these QCD-inspired models, HME is generally written as a convolution integral of hadron's distribution amplitudes (DAs) and hard rescattering kernels. A virtue of the QCDF approach is that the NF's result can be reproduced, if both the nonfactorizable contributions and the power suppressed contributions are neglected <cit.>.For the B_c^∗ → B_qV, B_qP decays (q = u, d, s), the spectator quark is a heavy quark — the bottom quark. It is generally assumed that the bottom quark in both the B_c^∗ and B_q mesons is nearly on shell, and that the gluon exchanged between the heavy spectator quark and other quarks is soft. The virtuality of emission gluon from the spectator quark is of order Λ_ QCD^2. The contributions of spectator scattering are power suppressed relative to the leading order contributions <cit.>. In addition, it is supposed that the recoiled B_q meson should move slowly in the rest frame of the B_c^∗ meson. There should be a large overlap between the B_c^∗ and B_q mesons. The recoiled B_q meson cannot be clearly factorized from the B_c^∗B_q system due to the soft and nonperturbative contributions.The B_c^∗B_q system should be parameterized by some physical from factors. Hence, with the QCDF approach, up to leading power corrections of order Λ_ QCD/m_Q, hadronic matrix elements have the following structure <cit.>, ⟨B_qM|Q_i|B_c^∗⟩=f_M∑_j F^B_c^∗→B_q_j∫dxH_ij(x) ϕ(x)=f_M∑_j F^B_c^∗→B_q_j { 1+α_s r_j+⋯}, where f_M is the decay constant for the light final M (≡ V and P) meson; F^B_c^∗→B_q_j is a transition form factor; H_ij(x) is a hard rescattering kernel; ϕ(x) is a DA of parton momentum fraction x.For the light pseudoscalar P and longitudinally polarized vector V mesons, the leading twist DAs are expanded in terms of the Gegenbauer polynomials <cit.> ϕ_P(x) = 6 x x̅ { 1+ ∑_n=1a_n^PC_n^3/2(x-x̅) },ϕ_V(x) =6 x x̅ { 1+ ∑_n=1a_n^VC_n^3/2(x-x̅) }, where x̅ = 1 - x; a_n^P,V is a nonperturbative parameter, also called the Gegenbauer moment. The expressions for the Gegenbauer polynomials C_n^3/2(z) are C_1^3/2(z) = 3 z,C_2^3/2(z) = 3/2 (5 z^2-1), ⋯§.§ Decay amplitudes The typical Feynman diagrams for the B_c^∗ → B_sπ decay within the QCDF framework are shown in Fig.<ref>, where no hard gluons are exchanged between the spectator quark and other partons. There is no gluon exchange in factorizable topology of Fig.<ref>(a), so the emitted hadron matrix element is entirely separated from that of the B_c^∗B_s system. In this approximation, the hard rescattering kernel H_ij = 1 and the integral in Eq.(<ref>) reduces to the normalization condition for distribution amplitude. According to the QCDF power counting rules, the leading order contributions come from the factorizable topology of Fig.<ref>(a), and recover the NF's results at the order of α_s^0. For the radiative correction diagrams in Fig.<ref>(b-e), hard gluons are exchanged between the emission meson and the B_c^∗B_s system. The hard rescattering kernel H_ij and x-integral in Eq.(<ref>) are nontrivial. It has already been shown <cit.> that although both collinear and soft divergences exist for each of diagrams in Fig.<ref>(b-e), infrared divergences cancel after summing up the vertex corrections. The strong phases could then come from HME. The renormalization scale μ dependence of HME is recuperated from the nonfactorizable contributions, which will reduce partly the μ-dependence of Wilson coefficients.After a straightforward calculation using the QCDF master formula Eq.(<ref>), the amplitudes for the B_c^∗ → B_qM decays (q = u, d, s) are written asA(B_c^∗→B_qM)= ⟨B_qM| H_ eff|B_c^∗⟩= G_F/√(2)V_cq^∗ V_uq^' a_i ⟨M|j^μ|0⟩ ⟨B_q| j_μ|B_c^∗⟩. With the naive dimensional regularization scheme, the effective coefficientsare <cit.>: a_1 =C_1^ NLO+1/N_c C_2^ NLO + α_s/4π C_F/N_cC_2^ LOV,a_2 =C_2^ NLO+1/N_c C_1^ NLO + α_s/4π C_F/N_cC_1^ LOV,V = 6 log( m_c^2/μ^2) -18 - ( 1/2+i 3 π) +( 11/2-i 3 π) a_1^M - 21/20 a_2^M +⋯, where N_c = 3 and C_F = 4/3; C_1,2^ NLO,LO are Wilson coefficients containing NLO or LO contributions; a_i^M is a Gegenbauer moment. For the transversely polarized vector meson, the vertex factor V = 0 beyond the leading twist DAs. For convenience, the numerical values for a_1,2 of the B_c^∗ → B_qπ decay are listed in Table <ref>.There are some comments on the coefficients a_1,2. (1) The first two terms on the right hand side of Eq.(<ref>) and Eq.(<ref>) correspond to the leading order contributions. The third terms correspond to nonfactorizable contributions. The NF scenario follows when one neglects the nonfactorizable contributions, i.e., V = 0. (2) Nonfactorizable vertex corrections to HME are of order α_s. They include the dependence on the renormalization scale. It is shown <cit.> that with the RG equations for the Wilson coefficients at leading order logarithm approximation, one can obtainμd/dμa_1,2=0 . In principle, the residual scale dependence could be compensated by higher order corrections to HME. (3) Compared with the LO contributions, nonfactorizable contributions are generally suppressedby α_s and the factor 1/N_c (see Eq.(<ref>) and Eq.(<ref>)). Because the LO contributions of a_2 are color-suppressed, vertex corrections multiplied by the large Wilson coefficient C_1^ LO could be sizable to branching rates of the a_2-dominated heavy flavor decays. The coefficients a_1,2 contain strong phases via the imaginary parts of vertex corrections. Correspondingly, strong scattering phase of a_1 (a_2) is small (large). This argument is also confirmed by the numerical results for a_1,2 in Table <ref>. (4) With the QCDF approach, nonfactorizable radiative corrections to HME occur first at order α_s as well as the leading strong phases at order α_s. In addition, it should be pointed out that nonfactorizable power corrections beyond leading order are neglected here. For the charm quark decay, power Λ_ QCD/m_c is comparable to α_s. The strong phases due to soft (hard) interactions are of order Λ_ QCD/m_c (α_s). One should not expect these phases to have great precision, as stated in Ref.<cit.>. (5) With the QCDF approach, the values for a_1,2 are close to those for the charm quark decay <cit.>, |a_1,2| ≈ |C_1,2|, and basically consistent with those of the large-N_c approach <cit.>. The hadronic matrix elements of diquark current operators are defined as <cit.>: ⟨V(ϵ,p)|q̅_1 γ^μ (1-γ_5) q_2|0⟩=f_V m_V ϵ^∗μ,⟨P(p)|q̅_1 γ^μ (1-γ_5) q_2|0⟩=-i f_P p^μ, ⟨B_q(p_2)|q̅ γ_μ (1-γ_5) c|B_c^∗(p_1,ϵ)⟩=-ε_μναβ ϵ^νq^α(p_1+p_2)^β V(q^2)/m_B_c^∗+m_B_q-i 2 m_B_c^∗ ϵ·q/q^2q_μA_0(q^2) -i ϵ_μ( m_B_c^∗+m_B_q )A_1(q^2)-i ϵ·q/m_B_c^∗+m_B_q ( p_1 + p_2 )_μA_2(q^2)+i 2 m_B_c^∗ ϵ·q/q^2 q_μA_3(q^2) , where f_V and f_P are the decay constants of vector V and pseudoscalar P mesons, respectively; q = p_1 - p_2; ϵ is the polarization vector of vector mesons; V(q^2) and A_0,1,2,3(q^2) are the B_c^∗ → B_q transition form factors. To eliminate singularities at the pole of q^2 = 0, a relation, A_0(0) = A_3(0), is required, with A_3(q^2) given by <cit.>: 2 m_B_c^∗ A_3(q^2) =(m_B_c^∗+m_B_q) A_1(q^2)+(m_B_c^∗-m_B_q) A_2(q^2) .In the bottom conservation transition B_c^∗ → B_q, both the initial and final mesons contain a heavy bottom quark. After a sudden kick, the B_q meson would move slowly, even remain nearly intact, with respect to the B_c^∗ meson. Therefore, the zero-recoil configuration (q^2 = 0) would be a good approximation. Simultaneously, the emission meson would take up most of the energy available and fly rapidly away from the interaction point. This fact not only reproduces the NF scenario (Fig.<ref>(a)) but also requires the exchanged gluon in vertex corrections (Fig.<ref>(b-e)) to be hard. Due to the large virtuality of gluon exchanged between the emitted light meson and the B_c^∗B_q system, perturbative calculation of nonfactorizable vertex corrections with the QCDF approach should be applicable and reliable.With the form factors given above, the decay amplitudes are expressed as A(B_c^∗→B_qV)=-i G_F/√(2)m_Vf_V V_cq^∗ V_uq^' { a_1 δ_B_q,B_d,s + a_2 δ_B_q,B_u} { (ϵ_B_c^∗·ϵ_V^∗) ( m_B_c^∗+m_B_q )A_1+(ϵ_B_c^∗·p_V) (p_B_c^∗·ϵ_V^∗) 2 A_2/m_B_c^∗+m_B_q +i ϵ_μναβ ϵ_B_c^∗^μ ϵ_V^∗ν p_B_c^∗^α p_V^β 2 V/m_B_c^∗+m_B_q},A(B_c^∗→B_qP)= √(2)G_Fm_B_c^∗(ϵ_B_c^∗·p_B_q)f_P A_0V_cq^∗ V_uq^' { a_1 δ_B_q,B_d,s + a_2 δ_B_q,B_u}.The B_c^∗ → B_qV decay amplitude is a sum of S-, P-, D-wave amplitudes <cit.>, i.e., A(B_c^∗→B_qV)= a (ϵ_B_c^∗·ϵ_V^∗) + b/m_B_c^∗ m_V(ϵ_B_c^∗·p_V) (p_B_c^∗·ϵ_V^∗) + i c/m_B_c^∗ m_Vϵ_μναβ ϵ_B_c^∗^μ ϵ_V^∗ν p_B_c^∗^αp_V^β, with a, b, c, the S-, D- and P-wave amplitudes respectively, in the notation of <cit.>, a=F( m_B_c^∗+m_B_q )A_1,b=F 2 m_B_c^∗ m_V/m_B_c^∗+m_B_q A_2,c=F 2 m_B_c^∗ m_V/m_B_c^∗+m_B_q V ,F=-i G_F/√(2)m_Vf_V V_cq^∗ V_uq^' { a_1 δ_B_q,B_d,s + a_2 δ_B_q,B_u}. From the above expressions, one can find that the P- and D-wave amplitudes are suppressed by a factor of 2 m_B_c^∗ m_V/(m_B_c^∗+m_B_q)^2 relative to the S-wave amplitude. The relations among the helicity amplitudes and the S-, P-, D-wave amplitudes are <cit.>H_± = a ± c √(y^2-1), H_0 = -a y-b (y^2-1), y =p_B_c^∗·p_V/ m_B_c^∗ m_V =m_B_c^∗^2-m_B_q^2+m_V^2/ 2 m_B_c^∗ m_V, p_ cm = √( [m_B_c^∗^2-(m_B_q+m_V)^2] [m_B_c^∗^2-(m_B_q-m_V)^2] )/ 2 m_B_c^∗, p_ cm^2 = m_V^2 (y^2-1), where p_ cm is the common momentum of final states in the rest frame of the B_c^∗ meson.We assume that the vector mesons are ideally mixed in the singlet-octet basis, i.e., ϕ = ss̅ and ω = (uu̅+dd̅)/√(2). As for the pseudoscalar η and η^' mesons, they are usually written as a linear superposition of states in either flavor basis or the singlet-octet basis. Here, we adopt the quark flavor basis description proposed in Ref. <cit.>, i.e.,([ η; η^' ]) =([cosϕ -sinϕ;sinϕcosϕ ])([ η_q; η_s ]),where η_q = (uu̅+dd̅)/√(2) and η_s = ss̅; the mixing angle ϕ ≈ (39.3±1.0)^∘ <cit.>. Due to the symmetric flavor configurations of both η_q and η_s states, we assume that DAs for η_q and η_s states are similar to DAs for pion. It should be pointed out that the contributions from possible cc̅ and gluonium compositions are not considered in our calculation for the moment, because (1) the final states with B_q meson and cc̅ or gluonium states lie above the B_c^∗ meson mass; (2) the fraction of gluonium components in η and η^' is rather tiny <cit.>. Thus, the amplitudes for the B_c^∗ → B_uη, B_uη^' decays are written as A(B_c^∗→B_uη) = cosϕA(B_c^∗→B_uη_q) -sinϕA(B_c^∗→B_uη_s) , A(B_c^∗→B_uη^') = sinϕA(B_c^∗→B_uη_q) +cosϕA(B_c^∗→B_uη_s) .§.§ Form factors The hadron transition form factors are the basic input parameters for decay amplitudes [see Eq.(<ref>) and Eq.(<ref>)]. It is assumed <cit.> that form factors come mainly from soft contributions, and form factors are generally regarded as nonperturbative parameters in the QCDF master formula of Eq.(<ref>). Fortunately, form factors are universal. Form factors determined by other means or extracted from data can be employed here to make predictions. Phenomenologically, form factors are written as overlap integrals of wave functions.Here, we will employ the Wirbel-Stech-Bauer model <cit.> for evaluating the form factors. With a factorization of spin and spatial motion, wave function is written asϕ^(j,j_z)(k⃗_⊥,x)= ϕ(k⃗_⊥,x) |s,s_z;s_1,s_2⟩, where k⃗_⊥ and x are the transverse momentum and longitudinal momentum fraction, respectively; j (s) is the total angular momentum (spin); j_z (s_z) is the magnetic quantum number; s_1 and s_2 are spins of valence quarks. j = s = 1 for the ground vector B_c^∗ meson, and 0 for the ground pseudoscalar B_u,d,s meson. The spatial wave function of a relativistic scalar harmonic oscillator potential is given by <cit.>ϕ(k⃗_⊥,x)=N_m √(x (1-x)) exp{-k⃗_⊥^2/2 ω^2} exp{-m^2/2 ω^2 (x-1/2 -m_q_1^2-m_q_2^2/2 m^2)^2}, where parameter ω determines the average transverse momentum of partons, i.e., ⟨k⃗_⊥^2⟩ = ω^2; m is the mass of the concerned meson; m_q_1 (m_q_2) is the constituent mass of the decaying (spectator) quark carrying a gluon cloud; N_m is a normalization factor determined by∫d^2k_⊥∫_0^1dx |ϕ(k⃗_⊥,x)|^2=1.The form factors at zero momentum transfer are given by <cit.>A_0(0)=A_3(0)= ∫d^2k_⊥∫_0^1dx ϕ_B_c^∗^(1,0)(k⃗_⊥,x) σ_z^(1) ϕ_B(k⃗_⊥,x), J= √(2) ∫d^2k_⊥∫_0^1dx/x ϕ_B_c^∗^(1,-1)(k⃗_⊥,x)i σ_y^(1) ϕ_B(k⃗_⊥,x), V(0)= m_c-m_q/ m_B_c^∗-m_B_q J, A_1(0)= m_c+m_q/ m_B_c^∗+m_B_q J, where σ_z,y^(1) are Pauli matrixes acting on the spin indices of the decaying quark q_1.It has been shown <cit.> that the form factors are sensitive to the choice of parameter ω. And it is argued <cit.> that parameter ω is not expected to be largely different for various mesons due to the flavor independence of the QCD interactions. Thus the same ω might be applied to all mesons with the same spectator quark. The motion of the spectator (bottom) quark is nearly nonrelativistic in the B_c^∗ → B_q transition. Thus, nonrelativistic QCD (NRQCD) effective theory <cit.> could be used to deal with both B_c^∗ and B_q mesons. According to the NRQCD power counting rules, the average transverse momentum is the order of ω ≈ m α_s. In order to see the parameter ω effects on the form factors, we explore two scenarios. One is the same parameter ω for both the B_c^∗ and B_q mesons, and the other is ω = m α_s, i.e., ω ≈ 1.24 GeV for the B_c^∗ meson, 1.10 GeV for the B_s meson, and 1.09 GeV for the B_u,d mesons. The numerical results for form factors are shown in Table <ref>.There are some comments on the form factors. (1) From the expressions in Eq.(<ref>) and Eq.(<ref>), it is seen that due to the factor m_c-m_q/ m_B_c^∗-m_B_q ≈ 1 and m_c+m_q/ m_B_c^∗+m_B_q ≪ 1, one can obtain a relation, A_1(0) < V(0). (2) Compared with the integrand in Eq.(<ref>), there is a factor 1/x for the integrand in Eq.(<ref>) with longitudinal momentum fraction 0 < x < 1. Thus, it is expected to have in general A_0,3(0) < V(0). (3) With the relation of form factors in Eq.(<ref>), A_2(0) is significantly enhanced by a factor of 2 m_B_c^∗/ m_B_c^∗-m_B_q (or m_B_c^∗+m_B_q/ m_B_c^∗-m_B_q) relative to A_3(0) (or A_1(0)). These relations are comprehensively verified by the numerical results for form factors in Table <ref>.In addition, from the numbers in Table <ref>, it is seen that (1) the form factors increase as parameter ω increases, due to the fact that the overlap between wave functions of B_c^∗ and B_q mesons increases as parameter ω increases, as shown in Fig.<ref>. (2) The flavor symmetry breaking effects on form factors are small, but the isospin symmetry is basically held. (3) The values for A_2(0) (V(0)) are about ten (five) times as large as those for A_1(0), as explained above. The large values forA_2 and V would enhance the contributions from the D- and P-wave amplitudes (see Eq.(<ref>) and Eq.(<ref>)).§ NUMERICAL RESULTS AND DISCUSSION In the rest frame of the B_c^∗ meson, branching ratios are defined asBr(B_c^∗→BV)= 1/24π p_ cm/m_B_c^∗^2Γ_B_c^∗ {|H_+|^2+|H_0|^2+|H_-|^2},Br(B_c^∗→BP)= 1/24π p_ cm/m_B_c^∗^2Γ_B_c^∗ | A(B_c^∗→BP)|^2, where Γ_B_c^∗ is the full width of the B_c^∗ meson.Because the electromagnetic radiation process B_c^∗ → B_cγ dominates the B_c^∗ meson decay, to a good approximation, Γ_B_c^∗ ≃ Γ(B_c^∗→B_cγ). However, there is still no experimental information about the partial width Γ(B_c^∗→B_cγ) now, because the photon from the B_c^∗ → B_cγ process is too soft to be easily identified. The information on Γ(B_c^∗→B_cγ) comes mainly from theoretical estimation on the magnetic dipole (M1) transition, i.e., <cit.>Γ(B_c^∗→B_cγ) =4/3 α_ emk_γ^3 μ^2_h, where α_ em is the fine-structure constant of electromagnetic interaction; k_γ is the photon momentum in the rest frame of initial state; μ_h is the M1 moment of B_c^∗ meson. There are plenty of theoretical predictions on Γ(B_c^∗→B_cγ), for example, the numbers in Tables 3 and 6 in Ref.<cit.>. However, these estimations still suffer from large uncertainties due to our lack of a precise value for μ_h. To give a quantitative evaluation, Γ_B_c^∗ = 50 eV will be fixed in our calculation for the moment. The value of 50 eV seems reasonable since it is close to the value given by the potential model (PM) which produces good agreement with experiment for the measured J/ψ → η_cγ decay rate. The value for the charm quark magnetic moment μ_c obtained from the charmoniumM1 decay width can now be used to predict the B_c^∗ → B_cγ decay width, with a very small b quark magnetic moment μ_b = -0.06 μ_N givenin Ref.<cit.>.The numerical values for other input parameters are listed in Table <ref>. Unless otherwise stated, their central values will be fixed as the default inputs. Our numerical results are presented in Table <ref>. The following are some comments.(1) According to the relative sizes of coefficients a_1,2 and CKM factors, the B_c^∗ → B_qV, B_qP decays could be classified into six cases (see Table <ref>). There is a clear hierarchical relation among branching ratios, i.e., Br(case 1-I) ∼ O(10^-7), Br(case 1-II) ∼ O(10^-8), Br(case 1-III) ∼ O(10^-9), and Br(case 2-I) ∼ O(10^-8), Br(case 2-II) ∼ O(10^-9), Br(case 2-III) ∼ O(10^-10).(2) Branching ratios for the B_c^∗ → B_qV decays are generally larger than those for the B_c^∗ → B_qP decays with the same final B_q meson, where V and P have the same quark components. There are two reasons for this. One is the decay constant relation f_V > f_P, and the other is three partial wave contributions to the B_c^∗ → B_qV decays rather than only the P-wave contributions to the B_c^∗ → B_qP decays.It should be pointed out that although the P- and D-wave amplitudes for the B_c^∗ → B_qV decays are enhanced by large values for the form factors V and A_2, they are suppressed by a factor of 2 m_B_c^∗ m_V/(m_B_c^∗+m_B_q)^2 relative to the S-wave amplitude, as discussed above. In addition, the P- and D-wave contributions to helicity amplitudes H_± in Eq.(<ref>) and H_0 in Eq.(<ref>) are future suppressed respectively by factors of √(y^2-1) and (y^2-1)/y relative to the S-wave contribution. Take the B_c^∗ → B_sρ decay for example, 2 m_B_c^∗ m_ρ/(m_B_c^∗+m_B_s)^2 ≈ 7%, √(y^2-1) ≈ 0.7 and (y^2-1)/y ≈ 0.4, resulting in the polarization fractions f_0 ≈ 60%, f_+ ≈ 30% and f_- ≈ 10% with f_0,+,- ≡ |H_0,+,-|^2/|H_0|^2+|H_+|^2+|H_-|^2. (3) The branching ratios for the B_c^∗ → B_sρ, B_sπ decays can reach up to O(10^-7). With the estimated production cross section of the B_c^∗ meson ∼ 30 nb at LHC<cit.>, it is expected to have more than 10^10 B_c^∗ mesons per ab^-1 dataat LHC, corresponding to more than 10^3 events of the B_c^∗ → B_sρ, B_sπ decays. Therefore, even with the identification efficiency, the B_c^∗ → B_sρ, B_sπ decays might be measurable in the future.(4) Branching ratios for the B_c^∗ → B_qV, B_qP decays are several orders of magnitude smaller, especially for the a_1 dominant decays, than those for the B_c → B_qP, B_qV decays <cit.>. This fact might imply that possible background from the B_c^∗ → BV, BP decays could be safely neglected for an analysis of the B_c → B_qP, B_qV decays, but not vice versa, i.e., one of main pollution for the B_c^∗ → B_qV, B_qP decays would likely come from the B_c decays.(5) It is seen clearly that the numbers in Table <ref> are very sensitive to the choice of the parameter ω. In addition, with a different value for Γ_B_c^∗, branching ratios in Table <ref> should be multiplied by a factor of 50eV/Γ_B_c^∗. Of course, many factors, such as the choice of scale μ, higher order corrections to HME, q^2-dependence of form factors, final state interactions, etc., are not carefully considered in detail here, but have effects on the estimation and deserve more dedicated study in the future. § SUMMARY With the running and upgrading of the LHC, there are certainly huge amounts of the B_c^∗ mesons. This would provide us with a possibility of searching for the B_c^∗ weak decays in the future. In this paper, the B_c^∗ → B_qV, B_qP decays (q = u, d and s), induced by the charm quark weak decay, are studied phenomenologically with the QCDF approach. The form factors for the B_c^∗ → B transitions are calculated using the Wirbel-Stech-Bauer model. The nonfactorizable contributions from the vertex radiative corrections are considered at the order of α_s. 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"authors": [
"Junfeng Sun",
"Yueling Yang",
"Na Wang",
"Qin Chang",
"Gongru Lu"
],
"categories": [
"hep-ph",
"hep-ex"
],
"primary_category": "hep-ph",
"published": "20170526083211",
"title": "Possibility of searching for $B_{c}^{\\ast}$ ${\\to}$ $B_{u,d,s}V$, $B_{u,d,s}P$ decays"
} |
firstpage–lastpage The HDUV Survey: A Revised Assessment of the Relationship between UV Slope and Dust Attenuation for High-Redshift Galaxies Naveen A. Reddy1,10, Pascal A. Oesch2,3, Rychard J. Bouwens4, Mireia Montes3, Garth D. Illingworth5, Charles C. Steidel6, Pieter G. van Dokkum3, Hakim Atek3, Marcella C. Carollo7, Anna Cibinel8, Brad Holden5, Ivo Labbé4, Dan Magee5, Laura Morselli9, Erica J. Nelson9, & Steve Wilkins8 ================================================================================================================================================================================================================================================================================================= Supermassive primordial stars forming in atomically-cooled halos at z ∼15-20 are currently thought to be the progenitors of the earliest quasars in the Universe. In this picture, the star evolves under accretion rates of 0.1 – 1 until the general relativistic instability triggers its collapse to a black hole at masses of ∼10^5 . However, the ability of the accretion flow to sustain such high rates depends crucially on the photospheric properties of the accreting star, because its ionising radiation could reduce or even halt accretion. Here we present new models of supermassive Population III protostars accreting at rates 0.001 – 10 , computed with thestellar evolution code including general relativistic corrections to the internal structure. We use the polytropic stability criterion to estimate the mass at which the collapse occurs, which has been shown to give a lower limit of the actual mass at collapse in recent hydrodynamic simulations. We find that at accretion rates higher than 0.001 the stars evolve as red, cool supergiants with surface temperatures below 10^4 K towards masses >10^5 , and become blue and hot, with surface temperatures above 10^5 K, only for rates ≲0.001 . Compared to previous studies, our results extend the range of masses and accretion rates at which the ionising feedback remains weak, reinforcing the case for direct collapse as the origin of the first quasars.quasars: supermassive black holes - early universe - dark ages, reionization, first stars - stars: Population III - galaxies: high-redshift– stars: massive§ INTRODUCTIONThe properties and evolution of supermassive stars (SMS), with masses ≳ 10^4 , have been studied since the early 1960s <cit.>. But the existence of such stars has only recently been suspected to be necessary to explain the formation of quasars by z ≳ 7, such as ULAS J1120+0641, a 2 × 10^9 black hole at z = 7.1 <cit.> and SDSS J010013.02+280225.8, a 1.2×10^10black hole at z=6.3 (; see ). The origin of these supermassive black holes (SMBHs) may not have been 10-500 Population III (Pop III) star BHs at z∼20-25 because they might not have achieved the rapid, sustained growth needed to exceed 10^9 by z≳7 <cit.>. Supercritical accretion by Pop III star BHs could allow them to grow to such masses at early times even with limited duty cycles <cit.>, but it is not known if such processes operate in primordial accretion discs or for the times required to produce massive seeds. The seeds of the first quasars may instead have been 10^4-10^5 BHs that formed via direct collapse.In this picture, a primordial halo forms in close proximity to nearby star-forming regions with strong Lyman-Werner (11.18 - 13.6 eV) UV and H^- photodetachment (> 0.755 eV) fluxes that sterilise the halo by effectively destroying the main coolant, molecular hydrogen H_2 (; but see also ). Due to the absence of H_2 molecules, the gas temperature rises to 10^4 K, preventing fragmentation and star formation before the halo's mass reaches a few 10^7 . At such masses, the halo gas finally becomes gravitationally unstable and begins to contract towards the centre with very high accretion rates of 0.1 – 10 , forming a 10^4-10^5 star in less than the lifetime of the star on the main sequence <cit.>. The dynamics of these flows on the smallest scales are not yet fully understood, but in the simulations performed to date a massive line-cooled disc forms that rapidly feeds the growth of a single object at its centre. Fragmentation, if it occurs, is minor and the clumps mostly spiral into the central object <cit.>. It is expected that stars in this mass range will collapse directly to BHs without exploding, with masses equal to the progenitors due to the inefficiency of radiative mass losses in metal-free stars. An observational candidate for a direct collapse black hole (DCBH) has now been found, CR7, a Ly-α emitter at z=6.6 <cit.>. Current models favor the BH interpretation of CR7 over a Pop III starburst <cit.>.A number of studies have recently examined the evolution of supermassive Pop III protostars growing by accretion at the high rates expected for atomically-cooled haloes. <cit.> followed the growth of such objects up to ∼10^5 for different constant accretion rates and found that for rates ≳0.1 the protostars remain red and cool until they reach a few 10^4 . <cit.> studied the evolution of supermassive Pop III stars in clumpy accretion scenarios and suggest that the protostar could intermittently become blue and hot at low masses but eventually evolved onto a redder, cooler track. But the codes used in these studies did not include the general relativistic (GR) corrections to hydrostatic equilibrium, and thus the runs were stopped before the stellar mass exceeds 10^5 . Indeed, above this mass, the GR effects are expected to become significant, in particular by triggering the collapse into a black hole through the GR instability <cit.>. <cit.> included the post-Newtonian corrections to their models, computing the internal structure of stars accreting at rates 0.1 – 10 towards masses of 1-8×10^5 . At these masses, the criterion of <cit.> for GR stability, based on the assumption of polytropic structures, indicates instability. However, no evolutionary tracks were displayed in this study, and the properties of the radiative feedback in their models, in particular in the highest mass-range, are not available.In a first paper <cit.>, we modeled the growth of supermassive Pop III stars at accretion rates of 0.01 - 10with thestellar evolution code. Thecode includes a self-consistent treatment of the hydrodynamics, taking into account the post-Newtonian correction, so that the collapse can be followed without the use of any external criterion. We found that the mass at collapse varied from ∼7.5×10^4 to 3.2×10^5 . However, numerical difficulties in the integration of the atmosphere in case of accretion did not allow us to study the photospheric properties of our models, and to establish the evolutionary tracks and ionising effect of the radiation field.The actual growth rates of supermassive Pop III stars may be crucially dependent on their internal structure and surface temperature. If the star is red and cool, accretion proceeds at rates set by the cosmological flows assumed in previous evolution models. But if the star becomes compact, blue and hot its ionising radiation could reduce accretion and the final mass of the star. The accretion geometry is also critical: the ram pressure of spherical inflows at the rates previously studied would almost certainly prevent radiation from the star from ionising the flows even if it is blue and hot <cit.>. Accretion through a disc, which is more likely, could result in bipolar radiation breakout that disperses the flows <cit.>. Clumpy accretion can likewise result in hot protostars that suppress their own growth at early times <cit.>.In the present work, we re-examine the evolution of supermassive primordial protostars accreting at high rates (0.001 – 10 ). We present models computed with thestellar evolution code, and describe the properties of their internal structures and evolutionary tracks. We focus on the ionising properties of the radiation field, in order to evaluate the potential of these stars to regulate their own growth. In addition, we use the polytropic criterion to study the development of the GR instability in the stellar interior, expected to trigger the collapse of the protostar into a black hole. In Sect. <ref> we describe ourmodels. In Sect. <ref> we examine their interior structure, and surface properties. In Sect. <ref> we study how these surface properties depend on the treatment of their external layers, we estimate the mass at which the GR instability triggers the collapse into a black hole, and we compare our results with those of previous studies. We summarise our conclusions in Sect. <ref>.§ NUMERICAL METHODThecode is a one-dimensional hydrostatic stellar evolution code that numerically solves the four usual equations of stellar structure (e.g. ) with the Henyey method. The energy generation rate includes both nuclear reactions and gravitational contraction, opacities are derived from the OPAL tables <cit.>, and convection is approximated by mixing-length theory. A general description of the code for the case without accretion is given in <cit.>.Accretion has recently been implemented in thecode as described in <cit.>, and we recall here only the main ideas. The accretion rate is a free parameter, fixed externally. Here we consider the following constant rates:=0.001, 0.01, 0.1, 1, 10Since the code is hydrostatic, we model only the stellar interior, without the accretion shock. Moreover, the code does not include any contribution to the luminosity from the accretion energy, and we assume that the entropy of the accreted material is the same as that of the stellar surface. This assumption corresponds to accretion onto the star through a disc, for which any entropy excess can be efficiently radiated away in the polar direction before being advected in the stellar interior (cold disc accretion, ).GR effects are expected to be important in SMS, and to account for them we apply the first order post-Newtonian Tolman-Oppenheimer-Volkoff (TOV) correction in the equation of hydrostatic equilibrium. We replace the Newtonian gravitational constant G byG_ rel=G(1+Pρ c^2+2GM_r rc^2+4π Pr^3 M_rc^2)where P is the pressure, ρ the mass density, c the speed of light and M_r the mass enclosed in a radius r. This approximation to GR is the same as that in().We emphasise that, as usual, the outer regions of the star are not included in the calculation of the stellar interior. For numerical stability, one has to neglect the production and absorption of energy in the layers with M_r/M>, whereis fixed externally. In these layers, the structure equations are integrated assuming a flat luminosity profile. Decreasingfavours numerical convergence, while increasing it improves physical consistency. In all the models described in the present work, we fix a value ofwhich is constant during the evolution. Depending on the models, we consider either =0.999 or =0.99. The consequences of this assumption are discussed in Sect. <ref>.Convective zones are determined according to the Schwarzschild criterion. For numerical stability, we do not include any overshooting. The consequences of these choices are discussed in Sect. <ref>.§ MODELS §.§ Initial setupsAccretion at high rates onto low-mass hydrostatic cores makes numerical convergence difficult. Thus for 0.1, 1 and 10 we initialise our models with a mass of M_ ini=10 , while for 0.01 and 0.001we take M_ ini=2 . The chemical composition of the initial models is homogeneous, with a hydrogen mass fraction of X=0.7516, a helium mass fraction of Y=0.2484, and a metallicity Z=1-X-Y=0. We include deuterium with a mass fraction of X_2=5×10^-5 (). The chemical composition of the accreted material is identical to that of the initial protostellar seeds. The initial structures correspond to polytropes with n≃3/2, with flat entropy profiles, so that the stars start their evolution as fully convective objects. The central temperatures are 4.1×10^5 K and 6.6×10^5 K for the 10 and 2 initial models, respectively, which is below the temperature required for D-burning (≃1-2×10^6 K). We choose the initial time-step in order to ensure that the mass accreted in the first time-step does not exceed 0.1 . As a consequence, the initial time-step depends on the accretion rate, and is given by t= 0.1 /. We take =0.999 as a fiducial value, except for 0.001 . The motivations and consequences of this choice are discussed in Sect. <ref>. For reasons of numerical stability, we do not include the GR correction in the model at =0.001 . This model never exceeds 10^4significantly, so that we expect GR effects to be negligible in this case. §.§ Evolutionary tracks and internal structuresThe evolutionary tracks on the Hertzsprung-Russel (HR) diagram are shown on Fig. <ref> for the five accretion rates. For all the models, the luminosity increases monotonically as the stellar mass grows by accretion, except in the very early evolution. The mass-luminosity relation is nearly independent of the accretion history (Fig. <ref>, lower panel). But the evolution of the effective temperature differs significantly between models at various rates. After an adjustment phase, the tracks converge towards two distinct asymptotic regimes in the HR diagram: the high- regime (≳0.01 ) gives a nearly vertical track in the red, along the Hayashi limit, while the low- regime (<0.01 ) leads to the blue, along the Zero-Age Main Sequence (ZAMS). In the high- regime, the effective temperature is locked around 5000 – 6000 K, and thus never exceeds 10^4 K before the luminosity reaches 10^10 . The star is bloated up, with a radius larger than 1000 , as a red supergiant protostar <cit.>. In the low- regime, the track evolves immediately towards the blue, approaching ≃10^5 K before the luminosity exceeds 10^6 . The increase in effective temperature is only stopped when the star reaches the ZAMS and stops contracting. Thus the location of the star on the HR diagram is confined between two limits: the Hayashi limit in the red and the ZAMS in the blue. Each of these limits corresponds to the asymptotic track of our models according to their accretion regime, low- or high-, i.e. depending if the rate is above or below a critical value ∼0.005 . Notice that the Hayashi limit reflects the physics at the stellar surface, while the ZAMS limit reflects the physics at the centre. Models at =0.001or with ≥0.1converge to their asymptotic track relatively early, before the luminosity exceeds significantly 10^6 . For the intermediate case =0.01 , the track remains longer between the two asymptotic limits, showing oscillations inin the range 10 000 – 30 000 K. Convergence towards the Hayashi limit occurs eventually when the luminosity has grown to 2×10^7 .The internal structure of these models is illustrated in Fig. <ref>. In addition, Fig. <ref> shows the evolution of their central temperatures and surface luminosities. All five models start with a fully convective structure and a central temperature T_c<. The star takes its energy from Kelvin-Helmholtz (KH) contraction, loosing entropy (L_r/ M_r=-T s/ t>0) and increasing T_c. As T_c increases, the opacity in the centre becomes low enough for a radiative core to form and grow in mass. The growth of the radiative core follows the isotherms, which reflects the fact that the transition from convection to radiation is an effect of the temperature increase, through the opacity. Only for Ṁ=10 , intermediate convective zones survive in between the radiative regions. Since these zones correspond to the Lagrangian layers of the initial seed, we expect their properties to reflect the choice of the initial structure. The reduction of opacity in the central regions produces an increase in the internal luminosity in radiative regions (L_r/ M_r>0). The entropy produced in these regions is absorbed by the cold external convective layers with high opacity (L_r/ M_r<0). As the internal temperature increases, the boundary between these two regions moves outwards in mass. When this luminosity wave <cit.> reaches the stellar surface, the radius increases abruptly, by more than one order of magnitude for ≥1 , by a factor of a few for ≤0.01 . Fig. <ref> shows the evolution of the luminosity wave for the model at 0.1 .In the while, T_c>≃1-2×10^6 K (Fig. <ref>, upper panel; see also the isotherm of 10^6 K on Fig. <ref>), and deuterium starts burning: in the radiative core for ≤0.1 , in the central convective zones for ≥1 (Fig. <ref>). Once deuterium is exhausted in the centre, the D-burning region moves outwards in mass (shell D-burning), following the isotherms (Fig. <ref>). The convective core that formed in the high- models survives the D exhaustion, but remains confined in the same Lagrangian layers that correspond to the initial seed, and thus contracts with it. Notice that neither this convective zone nor the plateau in central temperature visible on Fig. <ref> (at 30 <M<100) are due to D-burning, as one could naively believe. Test computations without deuterium confirmed that this features appear in any case. Actually, our computations show that D-burning has no significant impact on the stellar structure of the models described here.After the luminosity wave has reached the surface, all the layers of the star loose entropy (L_r/ M_r=-T s/ t>0). For low enough accretion rates (∼0.001 ), the stellar radius decreases as a consequence, despite the new mass which is continuously accreted, and the star can contract towards the ZAMS. For highhowever, the entropy losses are not efficient enough to make the stellar radius to decrease. Despite the contraction of all the layers, the new material that lands on the stellar surface makes the stellar radius to increase monotonically for the rest of the evolution.In order to illustrate the origins of this difference between the low- and high- regime, we plot on Fig. <ref> the timescales for accretion and KH contraction. The KH time gives the timescale for thermal adjustment in the stellar interior, and indicates the time it takes for the star to contract towards the ZAMS if accretion stops. We use t_ KH=GM^2/RL for M=500 , with R and L from the ZAMS of <cit.>. The accretion time is simply M/Ṁ for the same value of the mass, and gives the timescale of the evolution in the accretion phase. The timescales balance changes according to the accretion rate. For =0.001 , the KH time is shorter by one order of magnitude than the accretion time, while for ≥0.1 it is longer by one order of magnitude or more (three orders of magnitude for 10 ). For the intermediate case =0.01 , both timescales are similar. As a consequence, in the low- regime, the star has the time to contract towards the ZAMS before its mass increases significantly, while in the high- regime the mass increases too fast, and the stellar radius grows.After this point, the behaviour of the various models remains qualitatively different depending on the regime, low- or high-. In the low- regime, the internal structure is qualitatively similar to that of stars at present days <cit.>. However, due to the lower opacity, the star remains more compact and convective zones are thiner. As a consequence, shell D-burning occurs in the radiative core instead of the convective envelope. In the model at 0.001 , the swelling leads to a maximum radius of 32 (instead of 48.6in the present-day case, see model CV2 in ). Then, at M=8.5 , the star becomes fully radiative and contracts. The central temperature increases (Fig. <ref>, upper panel), triggering the -reaction. The energy produced by the -reaction remains negligible. The total luminosity is dominated by the gravitational contribution at this stage (KH contraction, L_r/ M_r≃-T s/ t). However, the reaction produces enough ^12C in the centre in order to trigger the CNO cycle, which becomes the dominant energy source for the rest of the evolution (Fig. <ref>, lower panel). As a consequence, a convective core forms at M=40 and grows in mass (Fig. <ref>, lower panel), while the central temperature is locked at T_c=1.26×10^8 K due to the thermostatic effect of H-burning (Fig. <ref>, upper panel). The radius reaches a minimum of 3.3at M=50 , and then grows continuously until the end of the computation, according to the homologous relation T_c∝ M/R and the fact that T_c≃ cst. At M=11 650 , numerical convergence becomes too difficult and we stop the computations, while H-burning is still proceeding.In the high- regime, the structure evolves in a qualitatively different way. Due to the fast mass load at the surface, the radius can not contract and the star remains large. This leads to low temperatures in the outer regions, which keep thus a high opacity and stay convective. Below this convective envelope, intermediate convective zones appear. A high accretion rate favours the formation and the development of these convective regions, because of the timescales balance: the higher the rate, the shorter the time for the star to radiate the entropy contained in the deepest regions before reaching a given mass. For =10 , in addition to the convective core described above, an intermediate convective region forms in the Lagrangian layers that were accreted during the swelling (see the iso-mass of 100 on the upper panel of Fig. <ref>). This convective zone results from the accretion of entropy when the luminosity wave crosses the surface. When the peak of the wave approaches the surface (L_r/ M_r=-T s/ t<0), the surface entropy increases suddenly. Through the assumption of cold disc accretion, the specific entropyof the material that is accreted increases then. After the passage of the peak, the surface can radiate its entropy efficiently (L_r/ M_r=-T s/ t>0), anddecreases suddenly. For the layers that are accreted at this stage, the decrease in the accreted entropy results in a negative entropy gradient (s/ M_r<0), which drives convection. Then, entropy is redistributed on a thermal timescale in the interior, but this mechanism remains inefficient at highbecause of the timescales balance between the KH and accretion times. This is why this intermediate convective zone appears only in the model at >1 . Despite the growth of the stellar radius, each Lagrangian layer contracts at this stage. As a consequence, the central temperature increases until H-burning starts (Fig. <ref>). The physics of H-burning is the same as in the low- regime: the -reaction is triggered first, and produces the ^12C that allows the CNO-cycle to operate as the dominant energy source for the rest of the evolution (Fig. <ref>, lower panel). A convective core forms (Fig. <ref>), and the central temperature is locked at T_c≃1.25-2×10^8 K by the thermostatic effect of H-burning (Fig. <ref>, upper panel). Notice that the higher the accretion rate, the higher the mass at which H-burning starts. This is due to the timescale balance: the higher the rate, the shorter is the accretion time compared to the KH time and thus the higher the mass accreted during the KH contraction towards the ZAMS. Once the convective core has formed, the evolution proceeds in a regular way. The stellar structure is made of three zones: the convective core, that grows in mass and radius along the isotherms (T≃10^8 K), the convective envelope, that covers less than 10% of the stellar mass (less than 2% during most of the evolution), and an intermediate radiative region in between. While H burns in the convective core, D burns in a thin shell of the intermediate radiative zone, following the isotherms T≃. As the stellar mass grows by accretion, the stellar radius continues to increase monotonically.Our models run until they reach different final masses, for reasons that are discussed below (Sect. <ref>). However, we notice that none of our models in the high- regime shows a decrease of the stellar radius when the stellar mass exceeds 10^4 . For ≥1 , our models reach final masses of several 10^5 , with a radius that is still growing. As a consequence, the effective temperature remains lower than 10^4 K until the stellar mass exceeds 3.5×10^5 . This is in contrast with the results of previous studies, as discussed below (Sect. <ref>).In the intermediate case 0.01 , after the swelling, the radiative star starts to contract towards the ZAMS as in the low- regime. But before it contracts significantly, in contrast to the 0.001case, the Eddington factor exceeds 50% at the surface (Fig. <ref>). As a consequence, a second swelling occurs. After several oscillations, at a mass of 600 ,exceeds 90%, the radius expands to ≃5000 and the model converges towards the high- regime. Notice that in contrast to the models with ≥0.1 , the model at 0.01evolves to the red because of the high radiation pressure (>90%), and not because of the high gas pressure related to the high entropy content which cannot be radiated away. In this intermediate case, oscillating between the two regimes, the contribution from radiation pressure is critical in determining the asymptotic behaviour. Notice also that several intermediate convective zones survive between the convective core and the convective envelope in the model at 0.01 . They reflect the accretion history of entropy, in a similar way as the intermediate convective zone of the model at =10 . However, at such low rate, thermal adjustment is efficient enough to redistribute the entropy before these zones join the convective core. §.§ Ionising feedback and Lyman-Werner fluxOur models show that the effective temperature of a star accreting at 0.001 – 0.1depends sensitively on the rate. To determine if the star can regulate accretion onto itself by radiative feedback, we compute the number of ionising photons per second by integrating the black-body spectrum above the ionising energy:= 4π R^2∫_ hν>13.6 eVF_ν hν dν= 8π^2R^2 c^2h^2∫_ hν>13.6 eV(hν)^2 e^hν/k-1 dνWe compute also the Lyman-Werner (LW) flux (11.18 - 13.6 eV):=8π^2R^2 c^2h^2∫_ hν=11.18 eV^ 13.6 eV(hν)^2 e^hν/k-1 dν The evolution ofandas a function of the stellar mass is shown on Fig. <ref> for the five accretion rates. The two asymptotic limits in the evolutionary tracks reflect in two asymptotic limits in the ionising and LW fluxes. In the low- case, the ionising photon rate exceeds 10^45 s^-1 before the stellar mass reaches 20 . This increase slows down only whenreaches the ZAMS limit. Thengrows slower, exceeding 10^50 s^-1 when the mass is M>100 . In the high- regime in contrast, after a short jump corresponding to the adjustment phase to the asymptotic behaviour, the ionising photon rate remains lower than 10^45 s^-1 until M=10^4 , and lower than 10^50 s^-1 until M≃3×10^5 . The fluxes grow slowly as the mass increases, following the Hayashi limit. For the intermediate rate 0.01 , the ionising and LW photon rates follow first the ZAMS limit. The oscillations in effective temperature hardly impact the fluxes, becauseremains in the range 10 000 – 30 000 K. But when the model converges to the Hayashi limit, at M=600 , the fluxes switch to the high- regime for the rest of the evolution.§ DISCUSSION §.§ Effect of changing the value of As mentioned in Sect. <ref>, thecode assumes that the energy generation rate is zero and the luminosity profile is flat in the external layers of the star, those above a given value of =M_r/M. In our models above, we used =0.999, except for the case =0.001where we used =0.99. Here we study the effect of changing the value of . To that aim, we present models at >0.001with =0.99, and a model at 0.001with =0.999. Decreasingreduces physical consistency, but makes numerical convergence easier. For <1 , models with =0.99 are started from the 2 initial protostellar seed, instead of the 10 one (Sect. <ref>). For =0.001 , the model with =0.999 is started from the 10seed.The evolutionary tracks are shown on Fig. <ref> for accretion rates of 0.1, 0.01 and 0.001 . Due to the use of various initial models, the early evolution differs between the = 0.999 and 0.99 cases. But at the stage where the models at 0.001 and 0.1 converge to their respective asymptotic behaviours, their evolutionary tracks become nearly independent of . The slight shift in the tracks at 0.1 reflects simply the dependence of the exact location of the Hayashi limit on the treatment of the external layers of the star, since this limit results from the opacity in the external layers. The same is true for the 1 case, not shown here. In contrast, the asymptotic track in the low- regime (model at 0.001 ) is not affected by a change in , because the location of the ZAMS is fixed by the physics in the centre of the star, and a change in the treatment of the outer layers has no impact. This justifies our choice of =0.99 for the model at 0.001 described in Sect. <ref>.Only for the intermediate case at 0.01 , the asymptotic behaviour differs significantly between = 0.999 and 0.99. After several oscillations in , when the model at = 0.999 converges to the Hayashi limit in the red, with <10^4 K, the model at = 0.99 remains in the blue, close to the low- tracks, with ≃70 000 K. Thus at this intermediate rate, the asymptotic behaviour is switched from one limit to the other by a change in . We notice also that before the convergence to the asymptotic track, the amplitude of the oscillations inis reduced by the decrease in .In order to study how this effect impacts the ionising flux, we computeaccording to Eq. (<ref>) for the same models. The result is shown on Fig. <ref>. As expected from the evolutionary tracks, for ≥0.1and =0.001 ,is nearly unaffected by a change in . Only in the intermediate case 0.01 ,differs between models at =0.999 and 0.99. The reduction of the amplitude of the -oscillations in the model at =0.99 makesto follow the ZAMS limit closer. But more importantly, as the model with =0.999 converges to the high- regime, the corresponding value ofdecreases suddenly by 8 orders of magnitude, from 10^51 to 10^43 s^-1. At the same stage, the model with =0.99 remains in the blue, withgrowing slowly (>10^50 s^-1). The 8 orders of magnitude difference is maintained as the stellar mass approaches 10^4 .This example shows that the choice ofis critical in order to determine properly the ionising properties of stars accreting at a rate between 0.001 and 0.1 . Difficulties in numerical convergence make it impossible to use >0.999. However, the models described above show that an increase inleads to larger radii and lower effective temperatures. But the Hayashi limit preventsto decrease under 5000 – 6000 K. Since our model at =0.01and =0.999 reaches the Hayashi limit at 600 , we do not expect a further increase into modify the track in the supermassive range. Actually, regarding the above examples, one can expect convergence to the high- asymptotic track to occur earlier at higher . Thus a further increase incould potentially reduce the value ofcloser to 0.001 , and bring definitely the intermediate 0.01 rate to the high- range. An accurate treatment of the external layers of the accreting star is thus required in order to determine the exact value of , but our models suggest a value closer to 0.005 than to 0.05 . §.§ Final mass at the onset of the collapseDespite the absence of hydrodynamics in our models, one can estimate the stage at which the GR instability triggers the collapse using the polytropic criterion of <cit.> <cit.>. According to this criterion, the star becomes unstable when the adiabatic index Γ is reduced under a critical value Γ_ crit. For a classical star (GM_r/rc^2=0), this critical value is simply 4/3. The first order relativistic development (GM_r/rc^2<<1) in the polytropic case givesΓ_ crit=43+K 2GM_r r c^2 ,where K is a constant that depends on the polytropic index (K≃1.12 for n=3). For a purely radiation-supported star (P=P_ rad, P_ gas=0), the adiabatic index is exactly 4/3, and the star is unstable. The first order development in terms of β=P_ gas/P, the ratio of gas pressure to total pressure, isΓ=43+β6 .Thus for stars that are dominated by radiation (β<<1), the stability criterion Γ>Γ_ crit can be expressed asβ6>K 2GM_r r c^2 . The two members of Eq. (<ref>) are compared in Fig. <ref> for our models with =10, 1 and 0.1at various stellar masses, using K=1.12. The right column shows the internal profiles of these quantities at the end of the computations, while the left one shows it at an earlier stage. The regions that are unstable according to the polytropic criterion are also shown as red areas on Fig. <ref>. In all the cases, the stellar structure consist in a convective core (coloured regions at the left-hand sides of each plot), an intermediate radiative zone (grey regions) and a convective envelope (coloured region at the right-hand sides). Since the stability criterion of Eq. (<ref>) is based on polytropic structures, the changes in the polytropic indices between the various regions of the star complicates the analysis. In each case, most of the mass of the star is contained in the radiative region, so that one could naively expect the n=3 criterion to be relevant. However, the high-entropy accreted envelope is not approximated by an n=3 polytrope in the present case. Moreover, we notice that the regions that are unstable according to the criterion are located deep in the radiative zone, close to the edge of the convective core, where the first order relativistic correction to Γ_ crit has its maximum. In the model at 10 , the criterion indicates instability in the radiative zone at M=261 000 . As evolution proceeds, the unstable zones grow, joining eventually the convective core. In the model at 1 , the criterion indicates instability at M=116 000 . The 0.1 model was stopped at M=0.7×10^5 for numerical reasons, when β/6 was approaching 1.12×2GM_r/rc^2 close to the convective core. We summarise these results in Table <ref>. The final masses are compared on Fig. <ref> with those of <cit.> and <cit.>, as discussed below (Sect. <ref>). §.§ Comparison with previous studiesThe evolution of Pop III stars under high accretion rates has been computed by several authors under various physical conditions. <cit.> and <cit.> used rates in the range 0.001 - 1to produce models towards 1000 with the assumption of spherical accretion, in which all the entropy of the accretion shock is advected in the stellar interior. This assumption is expected to produce stars that are more bloated than in the disc case, due to their high amount of internal entropy. In these models, stars evolve as red supergiant protostars along the Hayashi line if >, contract to the ZAMS if <, and oscillate inbetween these two limits if ≃. <cit.> obtained a critical value of =0.004 . In <cit.>, a model at =0.006 oscillates inwithout converging to the Hayashi limit before the end of their computation, at M=600-700 . <cit.> concluded that this expansion indicates the end of the accretion phase, since radiation pressure close to the Eddington limit is expected to reverse an accretion flow having a spherical symmetry. In contrast, at lower rates, the star contracts towards the ZAMS despite accretion. But at the high effective temperatures of the ZAMS, the ionising effect of the radiation field becomes significant, and a largeregion, with high pressure, is expected to form around the star, preventing accretion above ∼50<cit.>.All these computations were stopped at 1000due to convergence difficulties in the code used, that solves the structure equations with a shooting method. <cit.> used instead the numerical code stellar <cit.>, based on the Henyey method, like thecode, to push the computations towards higher masses. In contrast to the previous studies, entropy is accreted according to cold disc accretion, like in our models. They confirmed that at rates abovethe star evolves as a red supergiant protostar, with<10^4 K. But the value they obtain forexceeds 0.01 , in contrast to <cit.>. Indeed, after several oscillations in , their model at 0.01 remains in the blue until the end of their computation, at M≃2000 and L≃10^8 . Only at 0.1 the star converges to the Hayashi limit, already at M≃30 , and evolves then as a red supergiant protostar. Exploring the supermassive range, the supergiant models of <cit.> start to decrease in radius at M≃3×10^4 . At this point, the effective temperature grows, which could potentially lead to the formation of anregion during further evolution, not followed in their models. Their computation stops at 10^5 . Although their code does not include the GR correction in general, they computed a few test-models with it, and found the correction to be negligible below 10^5 , when they stopped their computations. The difference in the exact value ofbetween <cit.> and <cit.> is expected to come from the change in the accretion of entropy, between hot spherical and cold disc accretion. Indeed, the reduction of the entropy accreted in the disc case compared to the spherical case makes accreting stars more compact, so that a higher rate is needed in order to produce red supergiant protostars. Thus the results of <cit.> and <cit.> indicate ≃0.004 for spherical accretion and >0.01 for disc accretion.Notice that <cit.>, using analytical considerations based on timescales comparisons, obtained that stars accreting at a rate above 0.14 continue to expand in radius until the mass reaches 10^6 , in contrast with the results of <cit.>.Our models, based on cold disc accretion, explore the highest mass-range, above 10^5 , until the star collapse into a black hole. We confirm the main features obtained in the studies mentioned above: stars accreting above a critical rateevolve as red supergiants along the Hayashi limit, while underthey contract towards the ZAMS. However, two differences appear between our models and those of <cit.>. First, our results indicate <0.01 , since after several oscillations in effective temperature our model at 0.01 converges to the Hayashi limit and evolve as a red supergiant protostar when its mass exceeds 600 . This fact suggests thatis closer to the value obtained by <cit.> for spherical accretion than to the one obtained by <cit.> in the disc case. Our numerical experiment described in Sect. <ref> suggests that this discrepancy is related to the treatment of the energy equation in the external layers of the star. Indeed, neglecting the production of energy on the external 1% of the stellar mass gives an evolutionary track at 0.01 which is similar as that of <cit.>. In other words, our models show that an accurate treatment of the energy equation in the external layers can reduce the value ofby nearly one order of magnitude, from ∼0.05 to ∼0.005 , extending the range of accretion rate at which the ionising feedback remains negligible.The second difference between our models and those of <cit.> concerns the evolution in the supermassive range. In contrast to the models of <cit.>, our supergiant models never show a decrease of the stellar radius at M>10^4 . Until the highest mass (≃5×10^5 for =10 ), the radius continues to expand andremains ≃10^4 K. Thus our models confirm the analytical results of <cit.>, in contrast to <cit.>, and we do not expect the ionising flux to grow significantly at M>10^5 . This result extends the mass-range in which stars accreting at high rate do not significantly ionise their surrounding.Recently, <cit.> presented models of Pop III stars accreting at rates 0.1 – 0.3 – 1 – 10 , including the post-Newtonian correction. Their model run until final masses of 1.2 – 1.9 –3.5 – 8.0×10^5 , respectively. At these masses, the polytropic criterion of Eq. (<ref>) with n=3 indicates instability in the convective core. Actually, the criterion indicates instability in the core already before the star reaches these final masses, as it is visible on their Fig. 3. No evolutionary track of their models are provided, and we do not know if expansion stops in their model when the stellar mass exceeds 10^4 .In a previous paper <cit.>, we used thecode to follow the evolution of Pop III stars accreting at high rates until the final collapse. Since thecode includes hydrodynamics, the onset of the collapse can be followed self-consistently, without the use of the polytropic criterion considered in the present work and in <cit.>. The collapse is triggered at masses of 1.5-3.3×10^5 for =0.1-10 .The final masses are shown on Fig. <ref> as a function of the accretion rate, for the various studies. Models of <cit.>, of <cit.> and of the present work all agree with the fact that the mass at collapse remains in the same order of magnitude (∼1-8×10^5 ) over the two orders of magnitude range from 0.1 to 10 in , with a slight increase as a function of . However, strong discrepancies appear in the exact values of the mass. <cit.> obtained final masses that are larger to ourby a factor 1.5, nearly independently of the accretion rate. Thus, despite the discrepancies in , the dependence ofonis similar between <cit.> and us. In <cit.>, the slope of the curve (, ), i.e. the dependence ofon , is weaker than in the present models and those of <cit.>. In particular, for =10 , the final mass in <cit.> is only 60% of that of the present work.The validity of the polytropic criterion used in the present work can be questioned, since the stellar structures considered here are not polytropes. Indeed, <cit.> showed that with a self-consistent treatment of hydrodynamics the star remains stable well after the polytropic criterion indicates instability in the convective core. This suggests that the criterion provides only a lower limit for the final mass of SMSs.Moreover, the final masses of our runs are not necessary the masses at collapse, first because numerical instability could be responsible for the impossibility to make the code to converge at higher masses, second because the GR instability is a pulsational instability <cit.>. Marginal stability allows hydrostatic equations to be solved during that stage, and a hydrostatic code can evolve through this phase without noticing anything. As a consequence, a fully consistent treatment of hydrodynamics is necessary to capture the GR instability.A detailed treatment of the accretion of entropy and of convection itself are expected significantly effect the results. Any departure from cold disc accretion could impact the stellar structure, in particular at high rates, for which the accretion history of entropy plays a more important role than internal entropy redistribution in shaping the entropy gradient, as described in Sect. <ref>. Since the entropy gradient determines the presence of convection, a change in the accretion of entropy could impact significantly the mass at which the collapse occurs. More importantly, the treatment of convection, based on the mixing-length theory, appears as critical. In the present work, the triggering of convection is based on the Schwarzschild criterion. The use of the Ledoux criterion, that takes the chemical gradient into account as a stabilizing effect, favours purely radiative transport and reduces convective regions. Moreover, we do not include overshooting in our models. Including overshooting would increase the mass of the convective core, and thus decrease potentially the collapse masses. As a consequence, a precise estimation of the stellar mass at collapse is currently problematic, and one can interpret the various curves of Fig. <ref> as providing the envelope of the real - relation.§ CONCLUSIONIn the present work, we described new models of Pop III protostars accreting at high rates (0.001 – 10 ) towards the mass at which the GR instability is expected to trigger the collapse into a black hole. We described the evolutionary tracks and internal structures of these models, and studied the properties of their radiative feedback. We confirm the results of previous studies that stars accreting at a rate above a critical valueevolve as red supergiants, following the Hayashi limit in the red part of the HR diagram with a weak ionising feedback, while for lower rates the star contracts towards the ZAMS in the blue with a strong ionising feedback.In contrast to previous studies, our models show that Pop III protostars accreting at rates ≳0.1 continue to expand in radius in the highest mass-range (>10^5 ), until it reaches GR instability. Thus no significant increase in the ionising effect of the radiation field are expected in this mass-range as long as rapid accretion proceeds. In addition, compared to previous studies, our models reduce the value ofin the case of cold disc accretion, by nearly one order of magnitude, from ∼0.05 to ∼0.005 . Thus our results extend the range of masses and accretion rates at which the ionising feedback remains negligible.Using the polytropic criterion of <cit.> for GR instability, we estimated the mass at which the protostar collapses into a black hole. We obtained collapse masses that remain in the same order of magnitude (ranging from 0.7 to 5.5×10^5 ) for accretion rates that varies in the two orders of magnitude range 0.1 – 10 . Inside this interval, the collapse mass increases with . Our final masses are in the interval of the various values obtained in previous studies, and the dependence onis qualitatively in agreement with these studies. Discrepancies remain in the exact value of the mass at collapse. We interpret them as the result of the various treatment of entropy accretion and of convection, and in particular the use of the polytropic criterion instead of a fully consistent treatment of the hydrodynamics.§ TABLES Tables <ref> to <ref> display the age, mass, radius, luminosity, effective temperature, ionising flux and Lyman-Werner flux. The ionising and LW fluxes are computed from Eq. (<ref>) and (<ref>). The age is counted since M=0, i.e. t=M/Ṁ. The time-steps are computed so that the differences in log(L/) and log between two consecutive points does not exceed 0.1 and 0.025, respectively. § ACKNOWLEDGEMENTS LH, RSK and DJW were supported by the European Research Council under the European Community's Seventh Framework Programme (FP7/2007 - 2013) via the ERC Advanced Grant `STARLIGHT: Formation of the First Stars' (project number 339177). Part of this work was supported by the Swiss National Science Foundation. DJW was supported by STFC New Applicant Grant ST/P000509/1.mnras | http://arxiv.org/abs/1705.09301v1 | {
"authors": [
"Lionel Haemmerlé",
"Tyrone E. Woods",
"Ralf S. Klessen",
"Alexander Heger",
"Daniel J. Whalen"
],
"categories": [
"astro-ph.SR"
],
"primary_category": "astro-ph.SR",
"published": "20170525180009",
"title": "The Evolution of Supermassive Population III Stars"
} |
Study of Non-Standard Charged-Current Interactions at the MOMENT experiment Yibing Zhang December 30, 2023 =========================================================================== We present a simple result that allows us to evaluate the asymptotic order of the remainder of a partial asymptotic expansion of the quantile function h(u) as u→ 0^+ or 1^-. This is focussed on important univariate distributions when h(·) has no simple closed form, with a view to assessing asymptotic rate of decay to zero of tail dependence in the context of bivariate copulas. The Introduction motivates the study in terms of the standard Normal. The Normal, Skew-Normal and Gamma areused as initial examples. Finally, we discussapproximation to the lower quantile of the Variance-Gamma and Skew-Slash distributions.Keywords: Asymptotic expansion; asymptotic tail dependence; Quantile function; Regularly varying functions; Skew-Slash distribution; Variance-Gamma distribution. § INTRODUCTION This paper is motivated by the need for a generally applicable procedure to study the asymptotic behaviour as u→ 0^+ of F_i^-1(u), i=1,2 and C(u,u) = P(X_1≤ F_1^-1(u), X_2 ≤ F_2^-1(u)) where the F_i^-1(u)'s are the inverse of continuous and strictly increasing cdf's F_i(u), i=1,2, and a bivariate copula function respectively.A random vector X = (X_1, X_2)^⊤ with marginal inverse distribution function F_i^-1(u), i=1,2 has coefficient of lower tail dependence λ_L if the limit λ_L = lim_u→ 0^+λ_L(u) exists, whereλ_L(u)= P(X_1≤ F_1^-1(u) | X_2≤ F_2^-1(u))= P(X_1≤ F_1^-1(u), X_2≤ F_2^-1(u))/P(X_2≤ F_2^-1(u)) = C(u ,u)/u.If λ_L=0 then X is said to be asymptotically independent in the lower tail. In this situation in particular the asymptotic rate of approach to the limit 0 of λ_L(u) is an indication of the strength of asymptotic independence. The classical case is the bivariate Normal with correlation coefficient ρ as discussed in <cit.>. It was shown in <cit.> that if λ(u) = 2Φ(Φ^-1(u)√(1-ρ/1+ρ)) ∼ u^θL(u)where Φ(x), -∞<x<∞ is the cdf of the standard Normal, and L(u) is a slowly varying function as u→ 0^+, then λ_L(u) ∼u^θL(u)/θ+1.<cit.> also showed in their Theorem 3 that λ(u)∼ u^1-ρ/1+ρL(u) whereL(u)∼ 2√(1+ρ/1-ρ)(-4πlog u)^-ρ/1+ρ.The proof within their Theorem 2 depended heavily on the very specific asymptotic relation as x→ -∞ between the cdf Φ and the corresponding standard Normal pdf f.We were unable to use in that paper, for this purpose, the expression for the quantile function Φ^-1(u) of the standard Normal distribution y(u) = -√(-2log(u√(-4πlog u))),given for example by <cit.> from a truncated expansion of Φ^-1(u), since we did not know the asymptotic order of the reminder Φ^-1(u) - y(u). We were able to prove that Φ^-1(u) ∼ y(u) as u→ 0^+, but inasmuch as this asserts only that Φ^-1(u) = y(u)(1+o(1)) = -√(-2log u)(1+o(1)),we can only be sure of the dominant term of a truncated asymptotic expansion, and this was inadequate to proceed from (<ref>). However our general result below, when applied to the standard Normal, gives Φ^-1(u) = y(u) (1+O(log|log u|/(log u)^2)).Since Φ(y) ∼ - y^-1√(2π)e^-y^2/2as y→ -∞, from (<ref>)λ(u) ∼ k(u) e^-2(y(u)√(1-ρ/1+ρ)(1+O(log|log u|/(log u)^2)))^2where k(u) = 2/√(-2log u)√(2π)√(1+ρ/1-ρ); = k(u)e^-2(1-ρ/1+ρ)y^2(u)(1+O(log|log u|/(log u)^2))^2= k(u)e^-2(1-ρ/1+ρ)y^2(u)(1+O(log|log u|/(log u)^2)) =k(u) e^-2(1-ρ/1+ρ)(y^2(u)+o(1))as u→ 0^+, since y^2(u)O(log|log u|/(log u)^2) = O(log|log u|/|log u|), since y^2(u)∼ -2log u as u→ 0^+. Thus λ(u) ∼ k(u)e^-2(1+ρ/1-ρ)y^2(u)and the right hand side simplifies to the right hand side of (<ref>). In this note we present a simple result that allows us to evaluate the asymptotic order of the difference between y(u) and h(u) as u→ 0^+ or 1^-, where h(u) is the quantile function corresponding to a cumulative distribution function g(·) for important distributions where h(u) has no closed form, and y(u) is an asymptotic closed form expression. This is the general result which we discuss in Section 2. In Section 3, we will illustrate our results by considering the quantile function for the Generalised Gamma-type tail which has various commonly used distributions such as Normal, Skew-Normal, Gamma, Variance-Gamma and a Skew-Slash as special cases. Detailed extreme value structure of such distributions is important in a financial mathematics context. These individual examples will be discussed in Section 4.§ MAIN RESULT Our main result is summarised into the following Theorem. Suppose that g(x) is a strictly positive continuous and strictly increasing cumulative distribution function (cdf) on (-∞, A], A<0. Suppose further that some function y(u) → -∞ as u→ 0^+ satisfies y(g(x)) = x(1+O(ζ(x)))as x→ -∞, such that ζ(x)→0, x → -∞.Suppose finally that ζ(x) = ψ(-x) with ψ(w) = w^ρ L(w)for some constant ρ≥0 and function L(w), w>0, slowly varying at 0. If h(u), u∈ (0, g(A)] is the inverse function of g(·), then h(u) = y(u) (1+O(ζ(y(u))).We begin with the fact that g(h(u))= u⇒ y(g(h(u))) = y(u)⇒ h(u)(1+O(ζ(h(u)))) = y(u), using (<ref>)Thus h(u)/y(u)→ 1 as u→ 0^+ and henceζ(h(u))/ζ(y(u)) = ψ(-1/h(u))/ψ(-1/y(u))→ 1by the Uniform Convergence Theorem of slowly varying function of <cit.>. From (<ref>), ζ(h(u))/ζ(y(u))→ 1 ⇒ O(ζ(h(u))) = O(ζ(y(u)) as u→ 0^+.Finally, h(u)(1+O(ζ(h(u))))= y(u) ⇒h(u) = y(u)(1+O(ζ(h(u))))^-1 ⇒h(u) = y(u)(1+O(ζ(h(u))))⇒h(u) = y(u)(1+O(ζ(y(u)))).Notethat the formulation of the theorem requires, in the case ρ=0 only,that the function L(w) → 0, w → 0. The condition required in Theorem <ref> is simply that the correction term in (<ref>) is related to a regular varying function which is quite general and should apply to a wide range of distributions. The result in Theorem <ref> not only ensures h(u) and y(u) will be asymptotically equivalent, it will also stipulate how accurate y(u) will be for h(u). We express as a corollary to Theorem <ref> the corresponding result for the upper tail quantiles.Suppose that g(x) is a strictly positive continuous and strictly increasing cumulative distribution function (cdf) on [A, ∞), A>0. Suppose further that some function y(u) →∞ as u→ 1^- satisfies y(g(x)) = x(1+O(ζ(x)))as x→∞, such that ζ(x)→0, x →∞.Ifζ(x) = ψ(x) with ψ(w) = w^ρ L(w)for some constant ρ≥ 0 and function L(w), w>0, slowly varying at 0. If h(u)= g^-1(u), u∈ [g(A), 1) is the inverse function of g(·) i.e. the upper tail quantile function then h(u) = y(u) (1+O(ζ(y(u))). In the next section, we will illustrate our results by considering the quantile function corresponding to a cdf that has a Generalised Gamma-type tail behaviour. § QUANTILE FOR GENERALISED GAMMA-TYPE TAIL BEHAVIOUR §.§ Lower Tail We are interested in approximation to the quantile function for the Generalised Gamma-type tail.We consider a cdf g to have a Generalised Gamma-type (lower) tail behaviour if g can be expressed asg(x) = a|x|^b e^-c|x|^d(1+O(|x|^e)),x<0,for some constants a, c, d, e>0 and b∈.Several distributions of current interesthave such tails, and we discuss them asspecial cases in the next section.Suppose that an approximation to the quantile function h(u) = g^-1(u)tobe y(u) = -{-b/cd[log(cd/|b|(u/a)^d/b/| logcd/|b|u^d/b|)] }^d, forsmall u>0.Thiscan be obtained via the recursive method for an inverse function (see Chapter 2.4 of <cit.> for instance) on g(·). Then-y(g(x))= {-b/cd[log(cd/|b|(a|x|^b e^-c|x|^d(1+O(|x|^e))/a)^d/b/|logcd/|b|(a|x|^b e^-c|x|^d(1+O(|x|^e)))^d/b|)]}^d= {-b/cd[ log(cd/|b||x|^d e^-cd/b|x|^d(1+O(|x|^e))/|logcda^d/b/|b||x|^d e^-cd/b|x|^d(1+O(|x|^e))|)]}^d= {-b/cd[ - cd/b|x|^d+ log(cd/|b||x|^d)+log(1+O(|x|^e))-log|- cd/b|x|^d+log(cda^d/b/|b||x|^d)+ log(1+O(|x|^e))|]}^d= { -b/cd[- cd/b|x|^d+ log(cd/|b||x|^d) +O(|x|^e) -log(cd/|b||x|^d)- log(1- log(cda^d/b/|b||x|^d)+O(|x|^e)/cd/b|x|^d)]}^d= {-b/cd[-cd/b|x|^d + O(log |x|/|x|^d) + O(|x|^e)]}^d=|x|(1+O(|x|^e+d)+O(log|x|/|x|^2d))=|x|(1+ O(log|x|/|x|^2d)),if d≤ e|x|(1+O(|x|^d+e)),otherwise.As y(u)∼ - (-clog u)^d), the behaviour of the quantile function h(·) ish(u) =y(u)(1+O(log|(-clog u)^d|/(log u)^2)) = y(u)(1+O(log|log u|/(log u)^2)),if d≤ ey(u)(1+O(|log u|^e/d+1)),otherwise,by Theorem <ref>.We next expand y(u) to obtain an expansion for h(u) with an error term of appropriate lower order. After some algebra,y(u) = -(-clog u )^d[1- blog|log u|/d^2log u - blog(a^d/b/c)/d^2log u + b^2/2d^3(d-1)(log|log u|)^2/(log u)^2+b^2/d^3(d-1)log(a^d/b/c)log|log u|/(log u)^2+ b^2/d^3[(2d-2) (log(a^d/b/c))^2- log(cd/|b|)]/(log u)^2+ O((log|log u|)^3/(|log u|)^3) ].As a result, when d≤ e, h(·) becomesh(u) = y(u)(1+O(log|log u|/(log u)^2)) =-(-clog u)^d - blog|log u|/cd^2(-clog u)^1-d - blog(a^d/b/c)/cd^2(-clog u)^1-d - b^2(2d-2)(log|log u|)^2/c^2d^3(-clog u)^2-d+ O(log|log u|/|log u|^2-d); On the other hand, when d>e, h(·) becomesh(u) =y(u)(1+O(|log u|^e/d+1))= -(-clog u)^d - blog|log u|/cd^2(-clog u)^1-d- blog(a^d/b/c)/cd^2(-clog u)^1-d + O(1/|log u|^e/d+1-d). §.§ Upper Tail Similarly, a cdf g(·) is said to have a Generalised Gamma-type (upper) tail behaviour if g can be expressed as g(x) = 1- ax^b e^-cx^d(1+O(x^e)),x →∞,for some constants a, c, d, e>0 and b∈, which wouldsuggest the following approximation to the upper tailquantile function: y(u) ={-b/cd[log(cd/|b|((1-u)/a)^d/b/| logcd/|b| (1-u)^d/b|)] }^d, asu→ 1^-.Then by Corollary <ref>, the upper tail quantile function h(·) can be expressed ash(u)= y(u)(1+O(log|log (1-u)|/(log(1- u))^2)),if d≤ e;y(u)(1+O(|log (1-u)|^e/d+1)),if d<e; = (-clog(1-u))^d + blog|log(1-u)|/cd^2(-clog(1-u))^1-d + blog(a^d/b/c)/cd^2(-clog(1-u))^1-d+ b^2(2d-2)(log|log(1-u)|)^2/c^2d^3(-clog(1-u))^2-d+ O(log|log(1-u)|/|log(1-u)|^2-d),if d≤ e; (-clog(1-u))^d + blog|log(1-u)|/cd^2(-clog(1-u))^1-d+blog(a^d/b/c)/cd^2(-clog(1-u))^1-d+ O(1/|log (1-u)|^e/d+1-d),if d>e.as u → 1^-.§ APPLICATIONS In this section, we will illustrate our results withapplication tothe Normal, Skew-Normal, Gamma, Variance-Gamma and Skew-Slash distributions.§.§Standard Normal From <cit.> Chapter VII Lemma 2, the form of g(·) isg(x) = √(2π)|x|e^-x^2/2(1+O(x^2)), for x<1.If we compare this with (<ref>), we have a= √(2π), b = -1, c = 2 andd= e = 2. This means that the approximation y(·) becomes: y(u) = -{log((√(2π) u)^-2/| log(u) ^-2|)}^2 = - √(-2log(u√(4π|log u|)))and is the same as (<ref>). As in this case we have d=e, we can obtain the behaviour of the quantile function by using using (<ref>) and (<ref>) and get h(u) = y(u)(1+O(log|log u|/(log u)^2)) = - √(-2log u) + log|log u|/2√(-2log u) + log 4π/2√(-2log u) + (log|log u|)^2/8(-2log u)^3/2 + O(log|log u|/|log u|^3/2). Ifwe use only the dominating term of y(u) in (<ref>) and soput y^*(u) = -√(-2log u)then after some algebra we have y(g(x)) = x(1+O(ζ(x)))whereζ(x) = log |x|/x^2.As a result, h(u) = y^*(u)(1+ O(log|log u|/log u))by using Theorem <ref>.We can see that y^*(u) is a less accurate approximation to h(u) than y(u), and the expression for h(u) that it gives only improves on y^*(u) by giving the correct order of the difference h(u) -y^*(u).We are aware that there exists a whole field of literature on finding an efficient and accurate approximation in the numerical sense for the Normal quantile functions, such as <cit.>, <cit.> to the more recent <cit.>. <cit.> provide a substantial bibliography on this subject. But that is not our focus and so our methodology will most likely be outperformed by the more sophisticated approximation in the literature. Take the approximation to the Normal quantile in Section 2.2.1 in <cit.> for example which givesy_V(u) = c_3√(-2log u) + c_2' + c_1'√(-2log u)+c_0'/-2log u+d_1√(-2log u)+d_0for e^-37^2/2 <u<0.0465 where c_3=-1.000182518730158122,c_0'= 16.682320830719986527, c_1'= 4.120411523939115059, c_2'= 0.029814187308200211, d_0= 7.173787663925508066, d_1= 8.759693508958633869.To see how the expansions perform against the approximations, we plot the standard Normal quantile function, denoted as Φ^-1(u), in R against y_V(·), y(·) and y^*(·) and the results are shown in Fig. <ref>. From Fig. <ref>, we can see that h_V(·) gives almost identical result to the built-in Normal quantile function in R and is better than our expansion y(u). We also see that y^*(·) is an “order” worse than the other methods. §.§ Skew-Normal The quantile function of the Skew-Normal distribution is another example that is covered by our result. The distribution was first introduced by <cit.> and has developed into an extensive theory presented in a recent monograph by<cit.>. A random variable is said to have a standard Skew-Normal distribution if its density function isf(x) = 2/√(2π)e^-2x^2Φ(λ x),x∈where λ∈ R controls the skewness of the distribution. When λ=0, it reduces to the standard Normal as special case so we shall exclude the case of λ=0 from our subsequent discussion. Using Lemma 2 of<cit.> or <cit.>, we have g(x)=πλ(1+λ^2)|x|^2e^-2(1+λ^2)x^2(1+O(x^2)),λ>0; 2/√(2π)|x|e^-2x^2(1+O(x^2)),λ<0,x→ -∞.This means if we compare (<ref>) with (<ref>), we havea = πλ(1+λ^2), b = -2, c = (1+λ^2)/2 and d = e = 2 when λ>0; a = 2/√(2π), b = -1, c= 1/2, d = e = 2 when λ<0.This means that the approximation y(·) becomes:y(u) = - {2/(1+λ^2)[log((1+λ^2)/2(u πλ(1+λ^2))^-1/|log(1+λ^2)/2(u)^-1|)]}^2, λ>0;- {log((u√(2π)/2)^-2/|log(u)^-2|)}^2,λ<0; =-√( - 2/1+λ^2log(- 2πλ u log(2u/(1+λ^2)))) ,if λ>0; -√(-2log(u/2√(-4πlog u)))) ,if λ<0.As in this case we have d = e, we can obtain the behaviour of the quantile function by using (<ref>) and (<ref>) to get h(u) = y(u) (1+O(log|log u|/(log u)^2))= -√(-2/1+λ^2log u) + log| log u|/(1+λ^2)√(-2/1+λ^2log u) + log(2πλ)/(1+λ^2)√(-2/1+λ^2log u)+ (log|log u|)^2/2(1+λ^2)^2(-2/1+λ^2log u)^3/2+O(log|log u|/|log u|^3/2),λ>0; - √(-2log u ) + log|log u|/2√(-2log u) + logπ/2√(-2log u)+ (log|log u|)^2/8(-2log u)^3/2 +O(log|log u|/|log u|^3/2),λ<0. The expression (<ref>) justifies the use of expression (<ref>) above in Theorem 2 of <cit.>.§.§ Gamma Using (<ref>), one can also determine the accuracy of the gamma-like upper tail. Suppose that X∼Γ(α, β) and let its pdf be f(x) = β^α/Γ(α)x^α-1e^-β x,x>0. Then g(x) = 1- β^α-1/Γ(α) x^α-1e^-β x(1+O(x)),as x→∞. This means that if we compare (<ref>) with (<ref>), we havea = β^α-1/Γ(α), b = α-1, c = β and d = e =1. By using (<ref>) an approximation to the upper quantile function when u→ 1^- would be y(u) = {1-α/β[log(β/|α-1|(1-u/β^α-1/Γ(α))^α-1/|log(β/|α-1|(1-u)^α-1)|)]}=-βlog((1-u)Γ(α)/|log((1-u) β^α-1/|α-1|^α-1)|^α-1)As in this case we have d = e again, we have for the upper quantile functionh(u)=y(u)(1+O(log|log(1-u)|/(log(1-u))^2)) = -βlog((1-u)Γ(α)/|log((1-u) β^α-1/|α-1|^α-1)|^α-1)(1+O(log|log(1-u)|/(log(1-u))^2)) =-βlog(1-u) + α-1/βlog|log(1-u)| - βlogΓ(α)+ O(log|log(1- u)|/|log(1-u)|)as u→ 1^- by using (<ref>). InSection 4 of<cit.>, via their Theorem 1, the authors obtained h(u) = y(u)( 1 + o(1)),u → 1^-with y(u) =-βlog(1-u).§.§ Variance-Gamma and Skew-SlashFinally, wepropose an approximation to the lower tail quantile function of the Variance-Gamma (VG) and a Skew-Slash distribution, as they can have similar tail structure. A random variable X is said to have a skew VG distribution introduced in <cit.> and further studied in<cit.>, <cit.> and<cit.>, if X is defined by a Normal variance-mean mixture as X|Y ∼ N(μ+θ Y, σ^2 Y), where μ, θ∈ and σ^2>0. The distribution of Y is Γ(ν, ν) with ν>0 such that EY = 1. When μ=0 and σ =1, from <cit.>, we haveg(x)= |x|^ν-1e^-(√(2/ν+θ^2)+θ)|x|/ν^νΓ(ν)(2/ν+θ^2)^2ν(√(2/ν+θ^2)+θ)(1+O(|x|)) =a |x|^ν-1e^-(√(2/ν+θ^2)+θ)|x|(1+O(|x|))as x→ -∞, where a>0 isdefined by the above.. By comparing with (<ref>), a is defined as above, b = ν-1, c = √(2/ν+θ^2)+θ and d = e=1 which in turn would suggest the following approximation to the lower quantile function:y(u)= √(2/ν+θ^2)+θlog(u/a(√(2/ν+θ^2)+θ/|log[ u(√(2/ν+θ^2)+θ/|ν-1|)^ν-1]|)^ν-1) ∼ log u/√(2/ν+θ^2)+θ,asu→ 0^+ by using (<ref>). We can then obtain the behaviour of the quantile function ash(u) = √(2/ν+θ^2)+θlog(u/a(√(2/ν+θ^2)+θ/|log[ u(√(2/ν+θ^2)+θ/|ν-1|)^ν-1]|)^ν-1)(1+O(log|log u|/(log u)^2))= log u/√(2/ν+θ^2)+θ - (ν-1)log|log u|/√(2/ν+θ^2)+θ - (ν-1)log(a^ν/1-ν/√(2/ν+θ^2)+θ)/√(2/ν+θ^2)+θ+ O(log|log u|/|log u|)as u→ 0^+, by using (<ref>) and (<ref>). A Skew-Slash distribution on the other hand was first proposed in multivariate form in<cit.> and further studied in<cit.>. A random variable X is said to have a Skew-Slash distribution if X is defined by a Normal variance-mean mixture as X|Y ∼ N(μ+θ/Y, σ^2Y), where μ, θ∈ and σ^2>0. The distribution of Y is Beta(λ,1); that is, its pdf is: f(y) = λ y^λ - 1, 0 <y <1 ; =0 otherwise. Here we only consider the case of θ>0so that the tail behaviour is similar to that of the Variance-Gamma in (<ref>). When μ=0, σ=1 and θ>0, from <cit.>, we have g(x) = λθ^λ-1/2 |x|^-(λ+1) e^-2θ|x|(1+O(|x|)),as x→ -∞. By comparing this with (<ref>), we have a = λθ^λ-1/2, b = -(λ+1), c= 2θ, and d=e=1, which in turnsuggests the following approximation to the lower quantile function: y(u) = -(λ+1)/2θ[log(2θ/λ+1(u/λθ^λ-1/2)^-λ+1/|log2θ/λ+1u^-λ+1|)] ∼log u/2θ,as u→ 0^+ by using (<ref>). <cit.> use the fact that h(u) = y(u)(1+o(1)), u→ 0^+ with y(u) = log u/2θ. We canobtain the behaviour of the quantile function as h(u) =-(λ+1)/2θ[log(2θ/λ+1(u/λθ^λ-1/2)^-λ+1/|log2θ/λ+1u^-λ+1|)] (1+O(log|log u|/(log u)^2))= log u/2θ + (λ+1)log|log u|/2θ - log(λ(2θ)^2λ)/2θ+O(log|log u|/|log u|).as u→ 0^+ by using (<ref>) and (<ref>) once again. § ACKNOWLEDGEMENTSThe authors thank the referee for a careful reading of the original version, anda list of suggestions which have resulted in a much improved paper.[Abramowitz and Stegun(1964)]AS1964 Abramowitz, M. and Stegun, I.E.(ed.), Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables. National Bureau of Standards, Washington.[Arslan(2008)]A2008 Arslan, O. (2008) An alternative multivariate skew-slash distribution. Statistics and Probability Letters 78:2756–2761.[Azzalini(1985)]A1985Azzalini, A. (1985) A class of distributions which includes the normal ones.Scand. J. Statist.12:171–178.[Azzaliniand Capitanio(2014)]AC2014Azzalini, A. and Capitanio, A. (2014) The Skew-Normal and Related Families. IMS Monographs. Cambridge University Press, Cambridge. [Beasley and Springer(1977)]BS1977 Beasley, J. D. and Springer S. G. (1977). The percentage points of the Normal Distribution, Applied Statistics 26, 118–121.[Capitanio(2010)]C2010Capitanio, A. (2010) On the approximation of the tail probability of the scalar skew-normal distribution.Metron 68:299–308.[De Bruijn(1961)]D1961De Bruijn, N.G. (1961) Asymptotic Methods in Analysis. 2nd Ed. North-Holland Publishing Co., Amsterdam. [Embrechts, McNeil and Straumann(2011)]EMS2001 Embrechts, P., McNeil, A. and Straumann, D. (2001). Correlation and dependency in risk management: properties and pitfalls. In, Dempster, M. andMoffatt, H., eds. Risk Management: Value at Risk and Beyond. Cambridge University Press, Cambridge, pp.176-223. [Madan, Carrand Chang(1998)]MCC1998Madan, D.B., Carr, P and Chang, E.C. (1998) The variance gamma process and option pricing. European Finance 1:39–55. [Feller(1968)]F1968Feller, W. (1968) An Introduction to Probability Theory and Its Applications 1. Wiley, New York.[Fungand Seneta(2011)]FS2011Fung T. and Seneta, E. (2011) The bivariate normal copula function isregularly varying. Statistics and Probability Letters 81:1670–1676.[Fungand Seneta(2016)]FS2016Fung, T. and Seneta, E. (2016) Tail asymptotics for the bivariate skew normal. Journal of Multivariate analysis144:129–138.[Ledfordand Tawn(1997)]LT1997 Ledford, A.W.and Tawn J.A. (1997) Modelling dependence with joint tail regions. Journal of the Royal Statistical Society: Series B 59:475–499. [Ling and Peng(2015)]LP2015 Ling, C. and Peng, Z. (2015) Tail dependence for two skew slash distributions. Statistics and Its Interface 8:63–69.[Schoutens(2003)]S2003Schoutens, W.(2003) Lévy Processes in Finance: Pricing Financial Derivatives. Wiley, New York.[Seneta(2004)]S2004 Seneta, E. (2004) Fitting the variance-gamma model to financial data. Journal of Applied Probability 41A (Heyde Festschrift):177–187. [Seneta(1976)]S1976Seneta, E. (1976) Regularly Varying Functions. Lecture Notes in Mathematics 508. Springer, Berlin.[Soranzo and Epure(2014)]SE2014 Soranzo, A. and Epure, E. (2014). Very simply explicitly invertible approximations of normal cumulative and normal quantile function. Applied Mathematical Sciences 8, 4323–4341.[Tjetjepand Seneta(2006)]TS2006Tjetjep, A. andSeneta E. (2006) Skewed normal variance-mean models for asset pricing and the method of moments. International Statistical Review 74:109–126.[Voutier(2010)]V2010 Voutier, P.M. (2010). A new approximation to the normal distribution quantile function. arXiv:1002.0567 [stat.CO] | http://arxiv.org/abs/1705.09494v2 | {
"authors": [
"Thomas Fung",
"Eugene Seneta"
],
"categories": [
"math.ST",
"stat.TH",
"60E05, 41A60"
],
"primary_category": "math.ST",
"published": "20170526092257",
"title": "Quantile function expansion using regularly varying functions"
} |
1]Camilo Sanabria 1]Esteban Vargas [1]Department of Mathematics, Universidad de los Andes, Bogota, ColombiaCompetent hosts and endemicity of multi-host diseases [===================================================== In this paper we propose a method to study a general vector-hosts mathematical model in order to explain how the changes in biodiversity could influence the dynamics of vector-borne diseases. We find that under the assumption of frequency-dependent transmission, i.e. the assumption that the number of contacts are diluted by the total population of hosts, the presence of a competent host is a necessary condition for the existence of an endemic state. In addition, we obtain that in the case of an endemic disease with a unique competent and resilient host, an increase in its density amplifies the disease.§ INTRODUCTIONThe abundance of hosts of a vector-borne disease could influence the dilution or amplification of the infection. In <cit.>, the authors discusse several examples where loss of biodiversity increases disease transmission. For instance, West Nile virus is a mosquito-transmitted disease and it has been shown that there is a correlation between low bird density and amplification of the disease in humans <cit.>. One of the suggested explanations of this phenomenon is that the competent hosts persist as biodiversity is lost, meanwhile the density of the species who reduce the pathogen transmission declines.This is the case of the Lyme disease in North America, which is transmitted by the blacklegged tick Ixodes pacificus. The disease has the white-footed mouse Peromyscus leucopus as competent host, which are abundant in either low-diversity or high-diversity ecosystems. On the other hand, the opossum Didelphis virginiana, which is a suboptimal host and acts as a buffer of the disease, is poor in low-diversity forest <cit.>.Symmetrically, the dilution effect hypothesizes that increases in diversity of host species may decrease disease transmission <cit.>. The diluting effect of the individual and collective addition of suboptimal hosts is discussed in <cit.>. For example, the transmission of Schistosoma mansoni to target snail hosts Biomphalaria glabrata is diluted by the inclusion of decoy hosts. These decoy hosts are individually effective to dilute the infection. However, it is interesting to notice that their combined effects are less than additive <cit.>.The objective of this paper is to study the behavior of a vector-borne disease with multiple hosts when changes in biodiversity occur.More precisely, we present a mathematical framework that simultaneously explains why the accumulative effect of decoy hosts is less than additive and how competent and resilient host amplify the disease. To model a vector-borne disease with multiple hosts we use a dynamical system that was created based on <cit.>. We suggest a mathematical interpretation of competent and suboptimal host using the basic reproductive number of the cycle formed by the host and the vector. Furthermore, we assume that the abundances of the hosts follow a conservation law given by community constraints and with it we attempt to capture how a disturbance of the ecosystem leads to changes in the density of the hosts. We also give a mathematical interpretation of what a resilient species is using the conservation law. In this way, we are able to measure the effect on the dynamics of the disease due to different changes in the biodiversity. We show that in the case of endemic diseases these effects are determined by the effectiveness of the hosts to transmit the disease and the resistance of the hosts to biodiversity changes.In section <ref> we present the variables and the equations of the model. Section <ref> is divided in three subsections. In subsection <ref> we derive some properties of the basic reproductive number and we show how an endemic state implies the existence of a competent host. From these properties we explain why the combined effect of decoy hosts is less than additive and how biodiversity loss can entail amplification of the disease. Subsection <ref> introduces the community constraints that leads us to a definition of resilient host. In subsection <ref> we consider the case of an endemic disease with a unique competent host. We discuss the conclusions from our results in section <ref> . The mathematical justification are in Appendix, section <ref>.§ THE MODEL We propose a mathematical model of a vector-borne disease that is spread among a vector V and hosts H_i, i= 1, …,k.We suppose that each population is divided into susceptible individuals (S_V susceptible vectors and S_H_i susceptible hosts) and infectious individuals (I_V infectious vectors and I_H_i infectious hosts). Let N_V and N_H_i represent the total abundances ofvectors and hosts respectively. The dynamics of the disease will be studied by means of the basic reproductive number as we are interested in the strength of a pathogen to spread in an ecosystem. Modification of the ecosystem entails changes in the abundances of the hosts. After these changes are brought, the ecosystem will settle to a stable pattern of constant abundances. We are interested in understanding the basic reproductive number when the ecosystem reaches these steady states. Therefore we will assume the abundance of the vector and hosts are constant in time, i.e. Ṅ_̇V̇ = Ṡ_̇V̇ + İ_̇V̇=0 and Ṅ_̇Ḣ_̇i̇ = Ṡ_̇Ḣ_̇i̇ + İ_̇Ḣ_̇i̇=0 for i=1,…,k. In that way, it suffices to consider as state variablesonly the number of infectious species. We define the total number of hosts as N_H= ∑_i=1^kN_H_i. Our model is a system of ordinary differential equations for the infectious populations of hosts and vectors:İ_̇Ḣ_̇i̇ = β_VH_i I_V S_H_iN_H- δ_H_i I_H_i, i = 1, …, kİ_̇V̇ =∑_i=1^k β_H_iV I_H_iN_H_iN_HS_VN_V - δ_V I_V.We assume frequency-dependent transmission and that the vector does not have preference for a specific host, hence the number of contacts between the vector and the hosts are diluted by the total population of hosts. We also assume that there are no intraspecies infections and that there is no interspecies infection between hosts, or that these are negligible. Therefore,the only mean of infection is through contact with the vectors as Fig. <ref> shows.The parameters of the model are presented in Table <ref>.Note that we could alternatively assume that infected hosts gain immunity after recovering. In such case the model would yield the same next generation matrix (see Appendix <ref>), and since our analysis depends entirely on this matrix we would obtain the same results.§ RESULTS §.§ Properties of the basic reproductive number and the existence of competent hostsWe define the basic reproductive number ℛ_0^H_i of the cycle formed by host H_i and the vector V by(ℛ_0^H_i)^2= β_VH_iδ_Vβ_H_iVδ_H_i.The quantity ℛ_0^H_i is the basic reproductive number of the epidemiological model (<ref>) when N_H=N_H_i. It corresponds to the average number of secondary cases produced by a single infected host H_i in an otherwise susceptible population when the only cycle taken into account is the interaction between V and H_i. In this setting, the infection will spread in the population if ℛ_0^H_i > 1, and it will disappear if ℛ_0^H_i<1. Therefore, we say that a host H_i is competent if ℛ_0^H_i≥ 1 and suboptimal if ℛ_0^H_i<1.In general, taking into account all cycles, if D_i = N_H_i/N_H is the density of the host H_i in the total population of hosts, then the basic reproductive number ℛ_0 of the whole system is given byℛ_0^2 = ∑_i=1^k (ℛ_0^H_i)^2 D_i^2,(see (<ref>) in Appendix). Note that this implies that the combined effect of decoy hosts is less than additive.The quantity ℛ_0 is a convex function of D_1, … D_k. We have D_i ≥ 0 for i = 1, …, k and ∑_i=1^kD_i = 1. Using Lagrange multipliers, we obtain that the minimum value of ℛ_0 is attained in (D_1^* , …, D_k^*), where(ℛ_0^H_1)^2D_1^*= … =(ℛ_0^H_k)^2D_k^*.Therefore, we haveD_i^* = 1/ (ℛ_0^H_i)^2∑_j=1^k 1/ (ℛ_0^H_j)^2 fori=1,…,kand (ℛ_0)_min^2 = 1∑_j=1^k 1/ (ℛ_0^H_j)^2= 1kH((ℛ_0^H_1)^2, …, (ℛ_0^H_k)^2),where H((ℛ_0^H_1)^2, …, (ℛ_0^H_k)^2) is the harmonic mean of (ℛ_0^H_1)^2, …, (ℛ_0^H_k)^2. From the properties of the harmonic mean we havei=1, …, kmin{(ℛ_0^H_i)^2}≤ H((ℛ_0^H_1)^2, …, (ℛ_0^H_k)^2) ≤ ki=1, …, kmin{(ℛ_0^H_i)^2}.Using (<ref>), we obtain1/ki=1, …, kmin{(ℛ_0^H_i)^2}≤ (ℛ_0)_min^2 ≤i=1, …, kmin{(ℛ_0^H_i)^2}.From the last inequalities we can observe the following. First, the presence of a reservoir with ℛ_0^H_i<1 implies that (ℛ_0)_min<1. Hence, in some cases we may have ℛ_0<1. Furthermore, from (<ref>) we obtain that the larger the number of the hosts is, the smaller the basic reproductive number could be. This explains how high biodiversity could lead to the dilution of the disease. On the other hand, if all the reservoirs are effectively transmitting the disease(ℛ_0^H_i≫ 1, i = 1, …, k) and there are few host (k is small), then ℛ_0>1. This explains why in the case when competent host species thrive as a result of biodiversity loss we can expect the amplication of the disease, as discussed in <cit.> for the case of the Lyme disease <cit.> and the Nipah virus <cit.>.Furthermore, as the function (ℛ_0)^2(D_1, …, D_k) is convex, we haveℛ_0^2 ≤i=1, …, kmax{(ℛ_0^H_i)^2}.This inequality implies that the disease can not be amplified beyond the basic reproductive number of the most competent host. We obtain the following theorem.There exist values of D_1, …, D_k for which ℛ_0≥ 1 if and only if(ℛ_0)_min < 1 < ℛ_0^H_i,for some i. In particular, under the assumption of model (<ref>), the endemicity of a disease implies the existence of a competent host. Figure <ref> represents a contour plot of ℛ_0 in the case of two hosts.§.§ Community constraints In this section we will take into consideration host interaction using community constraints. First, we will consider the case when the abundance of hosts follow linear constraints. Secondly, we will show that in the study of small changes in the abundances we can linearize general constraints.§.§.§ Linear case Let us assume that the abundances of the hosts N_H_1,…, N_H_k follow k-1 linear constraints: ∑_j=1^k a_ij N_H_i + b_i = 0, fori =1,…, k-1,for some constants a_ij, b_i.If the matrix (a_ij)_1≤ i,j≤ k-1 is nonsingular, the abundance of all hosts can be explained by the abundance of the host H_k:N_H_i = -A_i N_H_k + B_i,fori=1…, k-1,for some constants A_i, B_i. In particular, if A_i > 0 in (<ref>), then N_H_k increases as N_H_i decreases. Moreover, when A_i=dN_H_idN_H_k>1 the changes in N_H_i are more pronounced than the changes in N_H_k. Therefore, we say that the host k is the resilient if A_i>1 for i=1,…, k-1 and it is non-resilient if 0<A_i < 1 for i=1,…, k-1.We haved ℛ_0/d N_H_k=D_𝐮ℛ_0 = ∑_i=i^k u_i r_i,where 𝐮=(-A_1, …, - A_k-1,1 ) andr_i=∂ℛ_0∂ N_H_i= 1N_H ℛ_0((ℛ_0^H_i)^2 D_i - ℛ_0^2). We define the index Γ_k =N_H_k/ℛ_0d ℛ_0/d N_H_kThe index Γ_k measures the sensitivity of ℛ_0 to changes of the population N_k.§.§.§ General constraints Let us assume that the abundances of the hosts 𝐍=(N_H_1,…, N_H_k) follow the m community constraints: 𝐅(𝐍) = (F_1(𝐍), …, F_m(𝐍)) = (0, …, 0)=0,for some m < k. Here F_1,…,F_m are real-valued differentiable functions defined where the values for 𝐍 have biological sense. Let E be the set of such values of 𝐍 where the community constraints are satisfied and let 𝐍_0∈ E. Under suitable conditions (see subsection <ref> in Appendix), we haveN_i = g_i(N_m+1, … , N_k) fori=1,…,m,for some functions g_1, …, g_m and for 𝐍∈ E close to 𝐍_0. The derivatives ∂ g_i∂ N_j, i = 1, …, m, j=m+1,…,k can be computed in terms of the derivatives of the functions F_1, …, F_m.If m=k-1 and ∂ℛ_0∂ N_k(𝐍_0) 0 in a neighborhood of 𝐍_0, then we haveN_i = g_i(N_k) fori=1, …, k-1.Moreover, for all 𝐍∈ E close to 𝐍_0 we have the approximation N_i = g_i(N_k) ≈-A_i N_k + B_i,for some constants A_i, B_i (see Appendix). Thus, locally we can consider linear restrictions as in (<ref>).§.§ The case of a single competent host In this section we consider the case of an endemic disease. Theorem <ref> implies the existence of a competent host in this setting. We will show that in the case when this competent host is unique the increase in its density implies the amplification of the disease if the densities of the rest of the hosts decrease. This corresponds to the cases when there is a unique host that thrives with biodiversity loss and this host is competent.We assume that ℛ_0^H_i<1 for i =1, …, k-1 and ℛ_0^H_k>1. Let D_1, …,D_k be such that ℛ_0 ≥ 1. Then∂ R_0∂ D_i < 0 fori=1,…,k-1and∂ R_0∂ D_k > 0.In particular, under the assumption of model (<ref>), in the case of an endemic disease with a unique competent host, increase in its density together with decrease in the density of all other hosts implies amplification of the disease.See section <ref> in Appendix.Under the assumption of model (<ref>), in the case of an endemic disease with a unique competent and resilient host, increase in its density implies amplification of the disease. Let us assume N_H_i = -A N_H_k + B_ifori = 1,…, k-1.for some constants A, B_i. If A>1, then the host H_k is resilient and, the greater A is, the more resilient H_k is. We have that Γ_k is an increasing function of A (see subsection <ref> in Appendix). Furthermore, taking D_k and A large, we have Γ_k ≈(k-1)A.Hence Γ_k increases as k, D_k and A increase. This implies that the more resilient the host H_k is, the greater its effect on ℛ_0 is, in the case when this host is abundant. The case k=2 is represented in Fig. <ref>.§ CONCLUSIONSIn this paper we present a mathematical framework that explains how changes of biodiversity can lead to the dilution or amplification of the disease. We show that the square of the basic reproductive number of the whole ecosystem is the weighted average of the squares of the basic reproductive numbers of the cycles between the vector and the hosts, weighted by their densities. Therefore, the accumulative effect of the hosts that buffer the disease is less than additive. Moreover, we obtain that the mininum of the basic reproductive number of the whole system is the harmonic mean of the basic reproductive numbers of the cycles. Hence, we conclude that an increase in biodiversity could dilute the disease and that loss in biodiversity could amplify the disease. Furthermore, we obtain that a necessary condition for the endemicity of a disease is the presence of a competent host. Finally, we study the case of an endemic disease. To explain how changes in the ecosystem affects the density of the hosts we assume that the abundances of the hosts follow a conservation law given by community constraints. We show that in the case when we have small changes in abundances, general constraints can always be linearized, thus it is sufficient to consider only linear constraints. We obtain that in the case of a disease with a unique resilient and competent host increase in its density amplifies the infection.§ APPENDIX§.§ Next generation matrix We will compute ℛ_0 using the NGM method from <cit.>. From model (<ref>) we obtain the matrices F and V that define the NGM: F= [0 β_H_1V D_1 β_H_2V D_2… β_H_kV D_k; β_VH_1 D_10000; β_VH_2 D_20000;⋱⋱⋱…0; β_VH_k D_k0000;], V= [ δ_V 0 0 0 0; 0 δ_H_1 0 0 0; 0 0 δ_H_2 … 0; ⋱ ⋱ ⋱ … 0; 0 0 0 … δ_H_k; ].Hence, the NGM is: G = FV^-1= [0 β_H_1V/δ_H_1 D_1 β_H_2V/δ_H_2 D_2… β_H_kv/δ_H_k D_k; β_VH_1/δ_V D_100…0; β_VH_2/δ_V D_200…0;⋮⋱⋱…0; β_VH_k/δ_V D_k00…0;].Computing the spectral radius of the matrix G, we obtain that the basic reproductive number of the whole system is given byℛ_0=ρ(FV^-1) = √(∑_i=1^k β_VH_i/δ_Vβ_H_iV/δ_H_iD_i^2). The disease free equilibrium (DFE) of model (<ref>) is 𝐈^*=(I_V^*,I_H_1^*,…,I_H_k^*)=0. The following theorem explains how the basic reproductive number is related to the stability of the DFE in model (<ref>) <cit.>. Let 𝐈^* be the DFE of (<ref>). Then, ℛ_0<1 implies that 𝐈^* is locally asymptotically stable and ℛ_0>1 implies that 𝐈^* is unstable.§.§ Community constraints Let F_1,…,F_m and E be as in subsection <ref> and let 𝐍_0∈ E. We assume that the matrix J_1 = ∂(F_1, …, F_m)/∂(N_H_1, …, N_H_m)(𝐍_0) = ( ∂ F_i (𝐍_0)/∂ N_H_j)_1≤ i,j≤ m is invertible and let us define J_2 = ∂(F_1, …, F_m)/∂(N_H_m+1, …, N_H_m)(𝐍_0) = ( ∂ F_i (𝐍_0)/∂ N_H_j)_1≤ i≤ m,m< j≤ k. The implicit function theorem states that there exists a neighborhood in E of 𝐍_0 where we have N_i = g_i(N_m+1, … , N_k) for i = 1, …, m. Furthermore, if 𝐠= (g_1, …, g_m), then ∂𝐠/∂ N_H_j = [ ∂ g_1/∂ N_H_j; ⋮; ∂ g_m/∂ N_H_j ] = - J_1 ^-1[ ∂ F_1/∂ N_H_j; ⋮; ∂ F_m/∂ N_H_j ],for m< j≤ k. We defineJ = ∂(F_1, …, F_m)/∂(N_H_1, …, N_H_k)(𝐍_0) = ( ∂ F_i (𝐍_0)/∂ N_H_j) _1≤ i≤ m, 1≤ j ≤ k.We are interested in computing D_𝐮ℛ_0 for 𝐮∈ T_𝐍_0 E, whereT_𝐍_0 E = {𝐮∈ℛ^k | J 𝐮 = 0}.If 𝐮 = (u_1, …, u_k), using u_m+1, …, u_k as free variables, we have that [ u_1; ⋮; u_m ] = - ∑_j=m+1^k u_j J_1 ^-1[ ∂ F_1/∂ N_H_j; ⋮; ∂ F_m/∂ N_H_j ] = - J_1^-1J_2[ u_m+1; ⋮; u_k ].Therefore, for a given set of values u_m+1, …, u_k, we can obtain the values u_1, …, u_m andD_𝐮ℛ_0 = ∑_i=i^k u_i r_i,where r_i= 1N_H ℛ_0((ℛ_0^H_i)^2 D_i - ℛ_0^2) is evaluated in 𝐍_0.If we assume m=k-1, then there exists a neighborhood in E of 𝐍_0 whereN_H_i = g_i(N_H_k) fori=1, …, k-1. Furthermore,J_2=[∂ F_1(𝐍_0)/∂ N_H_j; ⋮; ∂F_k-1(𝐍_0)/∂ N_H_j ]and[ u_1; ⋮; u_k-1 ] = - u_k J_1 ^-1[∂ F_1(𝐍_0)/∂ N_H_j; ⋮; ∂F_k-1(𝐍_0)/∂ N_H_j ] = -J_1^-1J_2u_k,for (u_1, …, u_k-1, u_k)∈ T_𝐍_0 E. Taking u_k =1, we have[ u_1; ⋮; u_k-1 ] = [ ∂ g_1/∂ N_H_k; ⋮; ∂ g_m/∂ N_H_k ].Therefore, for 𝐍∈ E close to 𝐍_0 we have the approximations N_H_i = g_i(N_H_k) ≈ u_i (N_H_k - N_H_k^0) + N_H_i^0 = -A_i N_H_k + B_i,for i = 1, …, k-1, where A_i= -u_i and B_i = N_H_i^0 - u_i N_H_k^0.§.§ One competent hostWe assume that ℛ_0^H_i<1 for i =1, …, k-1 and ℛ_0^H_k>1. Let D_1, …,D_k be such that ℛ_0 ≥ 1. We will prove that ∂ℛ_0∂ D_i < 0for i=1,…,k-1 and ∂ℛ_0∂ D_k > 0. Using ∑_j=1^kD_j=1, we have∂ℛ_0∂ D_i = 1ℛ_0((ℛ_0^H_i)^2D_i- (ℛ_0^H_k)^2D_k) for i=1,…,k-1.Furthermore, since ℛ_0= ∑_i=1^k (ℛ_0^H_i)^2 D_i^2, we obtainD_k((ℛ_0^H_k)^2 D_k - 1) ≥∑_i=1^k-1 D_i(1-(ℛ_0^H_i)^2 D_i) ≥ 0.Therefore,(ℛ_0^H_k)^2 D_k ≥ 1,hence∂ℛ_0∂ D_i < 0 fori=1,…,k-1and∂ℛ_0∂ D_k=∂ℛ_0∂ D_1∂ D_1∂ D_k>0. We haveΓ_k = D_k/ℛ_0^2∑_i=i^k u_i ((ℛ_0^H_i)^2 D_i - ℛ_0^2). If 𝐮 = (-A, …, -A, 1), then Γ_k =D_k/ℛ_0^2 (Ar + ((ℛ_0^H_k)^2 D_k - ℛ_0^2)),where r = -∑_i=i^k-1((ℛ_0^H_i)^2 D_i - ℛ_0^2). Since the hosts H_1, …, H_k-1 are suboptimal, we have r >0, hence Γ_k is an increasing function of A.If D_k is large, then D_1, …, D_k-1 are small and (ℛ_0^H_i)^2 D_i - ℛ_0^2 ≈ - ℛ_0^2 fori = 1, …, k-1. Therefore,r ≈ (k-1)ℛ_0^2. Furthermore, if (ℛ_0^H_k)^2 D_k - ℛ_0^2 ≈ 0 andA is large, then Γ_k ≈(k-1)A.99 allan2009ecological Allan, B. F., Langerhans, R. B., Ryberg, W. A., Landesman, W. J., Griffin, N. W., Katz, R. S., ... Chase, J. M. (2009). Ecological correlates of risk and incidence of west nile virus in the united states. Oecologia, 158 (4), 699?708. epstein2006nipah Epstein, J. H., Field, H. E., Luby, S., Pulliam, J. R. C.,Daszak, P. (2006). Nipah virus: Impact, origins, and causes of emergence. Current Infectious Disease Reports , 8 (1), 59?65. LoGiudice2008impact LoGiudice, K., Duerr, S. T. K., Newhouse, M. J., Schmidt, K. A., Killilea, M. E.,Ostfeld, R. S. (2008). Impact of host community composition on lyme disease risk. Ecology, 89(10), 2841-2849.ostfeld2000biodiversity Ostfeld, R. S.,Keesing, F. (2000). Biodiversity and disease risk: the case of lyme disease. Conservation Biology, 14 (3), 722?728. keesing2009hosts Keesing, F., Brunner, J., Duerr, S., Killilea, M., LoGiudice, K., Schmidt, K., . . . Ostfeld, R. S. (2009, 11). Hosts as ecological traps for the vector of lyme disease. Proceedings of the Royal Soci- ety B: Biological Sciences , 276 (1675), 3911?3919.cronin2010host Cronin, J. P., Welsh, M. E., Dekkers, M. G., Abercrombie, S. T., and Mitchell, C. E. (2010). Host physiological phenotype explains pathogen reservoir potential. Ecology Letters, 13 (10), 1221-1232. dobson2004population Dobson, A. (2004). Population dynamics of pathogens with multiple host species. the american naturalist , 164 (S5), S64-S78. johnson2010diversity Johnson, P., and Thieltges, D. (2010). Diversity, decoys and the dilution effect: how ecological communities affect disease risk. Journal of Experimental Biology, 213 (6), 961-970. johnson2012parasite Johnson, P. T., and Hoverman, J. T. (2012). Parasite diversity and coinfection determine pathogen infection success and host tness. Proceedings of the National Academy of Sciences, 109 (23), 9006-9011. johnson2009community Johnson, P. T., Lund, P. J., Hartson, R. B., and Yoshino, T. P. (2009). Community diversity reduces schistosoma mansoni transmission, host pathology and human infection risk. Proceedings of the Royal Society of London B: Biological Sciences, 276 (1662), 1657-1663. keesing2010impacts Keesing, Felicia, et al. "Impacts of biodiversity on the emergence and transmission of infectious diseases." Nature 468.7324 (2010): 647-652. martin2006investment Martin Ii, L. B., Hasselquist, D., and Wikelski, M. (2006). Investment in immune defense is linked to pace of life in house sparrows. Oecologia, 147 (4), 565-575. Murray2002Murray, J. D. (2002). Mathematical biology i. an introduction (3rd ed., Vol. 17). New York: Springer. doi: 10.1007/b98868 spivak1965calculusSpivak, M. (1965). Calculus on manifolds (Vol. 1). WA Benjamin New York. swaddle2008increased Swaddle, J. P., and Calos, S. E. (2008). Increased avian diversity is associated with lower incidence of human west nile infection: observation of the dilution effect. PloS one, 3 (6), e2488. van2002reproduction Van den Driessche, P., and Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical biosciences, 180 (1), 29-48. | http://arxiv.org/abs/1705.09426v2 | {
"authors": [
"Camilo Sanabria Malagon",
"Esteban Vargas Bernal"
],
"categories": [
"q-bio.PE"
],
"primary_category": "q-bio.PE",
"published": "20170526035957",
"title": "Competent hosts and endemicity of multi-host diseases"
} |
Matched filtering with 21cm]Matched filtering with interferometric 21cm experiments White and Padmanabhan] Martin White^1,2,3, Nikhil Padmanabhan^4 ^1 Department of Astronomy, University of California, Berkeley, CA 94720, USA ^2 Department of Physics, University of California, Berkeley, CA 94720, USA ^3 Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA ^4 Department of Physics, Yale University, New Haven, CT 06511, USA. firstpage–lastpage [ [ December 30, 2023 ===================== A new generation of interferometric instruments is emerging which aim to use intensity mapping of redshifted 21cm radiation to measure the large-scale structure of the Universe at z≃ 1-2 over wide areas of sky. While these instruments typically have limited angular resolution, they cover huge volumes and thus can be used to provide large samples ofrare objects.In this paper we study how well such instruments could find spatially extended large-scale structures, such as cosmic voids, using a matched filter formalism.Such a formalism allows us to work in Fourier space, the natural space for interferometers, and to study the impact of finite u-v coverage, noise and foregrounds on our ability to recover voids.We find that, in the absence of foregrounds such, instruments would provide enormous catalogs of voids, with high completeness, but that control of foregrounds is key to realizing this goal.gravitation; galaxies: haloes; galaxies: statistics; cosmological parameters; large-scale structure of Universe § INTRODUCTIONIntensity mapping with radio interferometers has emerged as a potentially powerful means of efficiently mapping large volumes of the Universe, albeit at low spatial resolution.Several groups are fielding 21cm intensity mapping experiments using a variety of technical designs <cit.>. Even though such instruments do not have the angular resolution to see individual galaxies, or even large clusters, they are capable of mapping the larger elements of the cosmic web (e.g. protoclusters and cosmic voids). Protoclusters are the progenitors of the most massive systems in the Universe today <cit.>. Cosmic voids make up most of the Universe, by volume <cit.>. As we shall show, in each case the system size is sufficiently large that they can be reliably found with upcoming intensity mapping experiments if foregrounds can be controlled sufficiently well.Cosmic voids, regions almost devoid of galaxies, are intrinsically interesting as the major constituent of the cosmic web by volume, and as an extreme environment for galaxy evolution <cit.>. They may be an excellent laboratory for studying material that clusters weakly like dark energy <cit.> or neutrinos <cit.>) or for testing modified gravity <cit.>. In this paper we show that 21cm instruments aimed at measuring the large-scale power spectrum, either proposed or under construction, should enable a search of enormous cosmic volumes at high redshift for these rare objects, which are large enough to be detected at high significance <cit.>.Voids and protoclusters are inherently “configuration space objects”, in the sense of being highly coherent under- or overdensities in the matter field in configuration space.However most future 21cm experiments are interferometers which naturally work in Fourier space.We will use a matched filter formalism to allow us to work in the interferometer's natural space, where the noise and sampling are easy to understand.This formalism also provides a natural way to combine data sets which live in different domains, e.g. optical imaging data with 21cm interferometry.We will illustrate our ideas by focusing on cosmic voids, though much of what we say could be applied to protoclusters as well. Our goal in this paper will thus be the detection of voids, and the challenges associated with this.We assume that these candidate voids will be appropriately analyzed or followed up for different science applications. It is worth keeping in mind that, for certain applications, the intermediate step of constructing an explicit void catalog and characterizing its purity and completeness may not be necessary.One might be able to construct estimators of the quantities of interest directly from the visibilities. We shall not consider such approaches in this paper.The outline of the paper is as follows. In <ref> we establish our notation and provide some background on interferometry, foregrounds for 21cm experiments and matched filters. Section <ref> describes the numerical simulations that we use to test our matched filter and the profiles of voids in those simulations. Our main results are given in <ref> and we present our conclusions in <ref>. We relegate a number of technical details to a series of appendices. In particular Appendix <ref> discusses instrument noise for an interferometer in the cosmological context, Appendix <ref> describes the formalism of transiting telescopes (including cylinder telescopes) and the flat-sky limit, and Appendix <ref> discusses the manner in which neutral hydrogen might be expected to trace the matter field at intermediate redshift.§ BACKGROUND AND REVIEWIn this section we provide some background information, to set notation and provide an easy reference for our later derivations.§.§ Visibilities In an interferometer the fundamental datum is the correlation between two feeds (or antennae), known as a visibility.For an intensity measurement the visibility is <cit.>V_ij∝∫ d^2n A^2(n) T(n) e^2π in·u⃗_ijwhere T(n) is the brightness temperature in the sky direction n, A(n) is the primary beam (assumed the same for all feeds) and u⃗_ij is the difference in position vectors of the i^ th and j^ th feeds in units of the observing wavelength. It is common to normalize the visibilities so that they return brightness temperature.We convert from brightness temperature to cosmological overdensity throughout, so we omit the exact normalization here. We will work in visibility space, since this is the natural space for the interferometer and has the simplest noise properties. Some useful conversions between common quantities are given in Table <ref>.Visibilities are measured over a range of frequencies, and we shall follow the common procedure in 21cm studies of Fourier transforming in the frequency direction to obtain a data cube in 3D Fourier space, k⃗. The conversion from frequency to distance (and hence Fourier mode) is|dχ/dν| = c/H(z) (1+z)^2/ν_0with ν_0=1420MHz.For small sky areas, the visibility thus measures the Fourier transform of the sky signal, apodized by the primary beam. Approximating the sky as flat and assumingthe signal of interest, τ, is azimuthally symmetric (see Appendix <ref>)τ(k) = 2π∫ω̃ dω̃ J_0(k ω̃) τ(ω̃)where ω̃ is an angular, radial coordinate andV_ij∝[τ⋆ B](2π u_ij)with the ⋆ representing a convolution and τ and B being the Fourier transforms of τ(n) and A^2(n) respectively. The surveys of interest to us here will cover large sky areas.The (very correlated) visibilities from the different pointings can be combined to produce higher resolution in the u-v plane (a process known as mosaicking), entirely analogously to the manner in which the many slits in a diffraction grating sharpen the transmitted lines (e.g. , or see the discussion infor the cosmological context). In such a case, the effective B is determined by the survey area rather than the primary beam (analogous to a survey window in a galaxy survey) and will be very small.Combined with the fact that our signals will be very smooth in u-v this allows us to neglect B to simplify our presentation. Reinstating it does not change any of our conclusions. §.§ 21cm interferometers We shall start by considering an interferometer consisting of an array of dishes (the interesting case of transiting, cylinder telescopes presents only technical modifications and is described in Appendix <ref>). As a concrete example, we use the HIRAX experiment <cit.>. HIRAX will use 1024 6m parabolic dishes in a compact grid covering the frequency range 400<ν<800MHz (i.e. 0.8<z<2.5 for 21cm radiation). HIRAX is a transit telescope: all dishes will be pointed at the meridian with a given declination, and the sky will rotate overhead in a constant drift-scan.Each declination pointing will give access to a 6^∘ wide stripe of the sky and the complete survey will cover 15,000 square degrees.Fig. <ref> plots the circularly averaged distribution of baselines at z=1, given our assumptions for HIRAX. The x-axis is the baseline separation in units of the wavelength, |u⃗|, while the y-axis is the number density of baselines per d^2u, conventionally normalized to integrate to the number of antenna pairs. The upper x-axis shows what k modes these baselines map to. Recalling that the noise scales as the inverse of the baseline density (see Appendix <ref>), we see that a HIRAX-like experiment is sensitive to a broad range of k scales well suited to the detection of voids and protoclusters.§.§ Noise for 21cm experimentsThe major difficulty facing upcoming 21cm experiments is astrophysical foregrounds <cit.>. Foregrounds have been extensively studied in the context of (high z) epoch of reionization studies, i.e. at lower frequencies than of direct interest here.However the amplitudes of the signal and foreground scale in a roughly similar manner with frequency, so many of the lessons hold in our case <cit.>. Since the main (Galactic) foregrounds are relatively smooth in frequency, their removal impacts primarily the slowly varying modes along the line of sight, i.e. the low k_∥ modes.However since the foreground are very bright (compared to the signal) and no instrument can be characterized perfectly, there is also some leaking of foreground power into other parts of the k_⊥-k_∥ plane.The precise range of scales accessible to 21cm experiments after foreground removal is currently a source of debate.We do not attempt to model foreground subtraction explicitly, but take into account its effects by restricting the range of the k⃗_⊥ - k_∥ plane we use.There are two regions of this plane we could lose to foreground removal. The first is low k_∥ modes, i.e. modes close to transverse to the line-of-sight. This boundary is slightly fuzzy and not well known. For instance, <cit.> claim that foreground removal leaves modes with k_∥> 0.02 h Mpc^-1 available for cosmological use, while <cit.> claims k_∥<0.1 h Mpc^-1 modes are unusable.We shall consider the impact of a k_∥ cut within this range and we will see that our ability to find voids is quite sensitive to this cut.In addition to low k_∥, non-idealities in the instrument lead to leakage of foreground information into higher k_∥-k_⊥ modes. This is usually phrased in terms of a foreground “wedge” <cit.>. The wedge does not form a hard boundary, but delineates a region where modes far from the line-of-sight direction can become increasingly contaminated. For a spatially flat Universe we can define the wedge geometrically as <cit.>ℛ= χ H/c(1+z) = E(z)/1+z∫_0^zdz'/E(z')where E(z)=H(z)/H_0 is the evolution parameter, and we assume we cannot access the signal in modes with |k_∥|/k_⊥<ℛ or|k_∥|/k < μ_ min = ℛ/√(1+ℛ^2)≈ 0.6with the last step being for z=1. It is worth emphasizing that this foreground “wedge” does not represent a fundamental loss of information, and may be mitigated with an improved model of the instrument (ideally the wedge can be reduced by sinΘ where Θ is the field of view; ). We bracket these cases by considering cases with and without the foreground wedge, and discuss the impact on our void finder.Finally we must contend with shot-noise and receiver noise in the instrument. <cit.> argue that shot-noise is sub-dominant to receiver noise for upcoming surveys, so we shall neglect it in what follows <cit.>. To simplify our presentation we shall treat the receiver noise as uncorrelated between visibilities and constant for all pairs of receivers. The noise thus scales with the number of baselines that probe a particular scale, and only an overall scaling is required. If the noise is uncorrelated from frequency channel to frequency channel, and only slowly varying with frequency, then the noise level is independent of k_∥. It is convenient to quote the thermal noise power in terms of the linear theory power spectrum, P_L, in much the same way as galaxy surveys specify their shot-noise by giving n̅P at some fiducial scale. Since one of the design goals of all of these surveys is a measurement of the baryon acoustic oscillation (BAO) scale, we follow the standard practice and specify the receiver noise as a fraction of P_L at k_⊥, fid=0.2 hMpc^-1. The surveys should achieve P_L/P_ noise>1 at k_⊥, fid and we shall explore a range of values (see Appendix <ref>). Once P_L/P_ noise>3 the results become very insensitive to the precise value. §.§ Matched filters A matched filter is a convenient means of finding a signal of known shape in a noisy data set.If we write the the data as an amplitude times a template plus Gaussian noise (d=Aτ+n), the maximum-likelihood estimate of A and its scatter is given by = τ N^-1 d/τ N^-1τ , σ^-2 = τ N^-1τWe take the “noise” covariance to include both instrument noise and non-template cosmological signal and shall assume throughout that this noise is diagonal in k-space. The main feature of this expression is that areas of the k-plane which are not sampled or are lost to foregrounds receive zero weight (N^-1=0).We find that our ability to isolate voids is very insensitive to the exact profile chosen for τ.In fact even a top-hat profile produces a highly pure and complete void catalog for low noise and good k-space sampling.Similarly the performance is not particularly sensitive to the particular choice for N(k), but rather to the larger questions of whether there are significant regions of k-space where N^-1=0 or very uneven sensitivity of the instrument due to the spacing of the feeds.We shall use N-body simulations for our signal, and work with a periodic, cubic box.In such situations, given a 3D density field, δ(x⃗), and a template, τ(x⃗), we can implement the flat-sky version of the matched filter very efficiently using FFTs if the noise is diagonal in k-space. Recalling that a shift in configuration space amounts to multiplication by a phase in Fourier space, the matched filter for a void centered at a⃗ isτ N^-1d →∑_k⃗ e^ik⃗·a⃗ τ_0(k⃗)δ^⋆(k⃗)/N(k⃗)where τ_0 is the template for a void centered at the origin. The sum is simply an (inverse) Fourier transform, so we can test for all a⃗ at once. A similar set of steps can be used for the denominator τ N^-1τ, allowing a fast computation of S/N for any position, a⃗. Thus with forward Fourier transforms of the template and data and one inverse transform we can compute the matched filter amplitude, A, everywhere in space and hence its (volume weighted) distribution at random locations and at the positions of voids.There is, in principle, no reason why the matched filter can't be modified to remove spectrally smooth foregrounds at the same time as searching for voids or protoclusters.We choose not to implement this approach, preferring instead to set N^-1=0 for modes which we deem unusable due to foregrounds. § SIMULATIONS §.§ N-body To illustrate our ideas we make use of several N-body simulations, each of the ΛCDM family.Specifically we use the z=1 outputs of 10 simulations run with the TreePM code described in <cit.>. This code has been extensively compared to other N-body codes in <cit.>, and these simulations have been previously used (and described) in <cit.> and <cit.>.Each run utilized 2048^3 particles in a periodic box of side 1380 h^-1Mpc to model a cosmology with Ω_m=0.292, h=0.69 and σ_8=0.82.This gives a particle mass of m_p=2.5× 10^10 h^-1M_⊙. Slices of the redshift-space density field at z=1 from one of the simulations are shown in Fig. <ref>, to illustrate the types of structures we are searching for.There are spatially coherent regions of over- or under-density with scales of 𝒪(10Mpc) clearly visible in the figure.Properly modeling the distribution of HI at z=1-2 is beyond the scope of this paper.Our simulations would need much higher resolution, to resolve the halos likely to host neutral hydrogen at z=1-2, and the halo occupancy is anyway highly uncertain <cit.>. Instead we assume the HI is an unbiased tracer of the matter field, and simply use the dark matter density.In Appendix <ref> we use a halo model of HI in a higher resolution (but smaller volume) simulation to show that this is a conservative approximation for the purposes of establishing how well 21cm experiments can find voids. §.§ Voids in the Simulations We define voids through a spherical underdensity algorithm (for a comparison with other void finders see , for a general comparison of void finders see ).The dark matter particles are binned onto a regular, Cartesian grid of 1380^3 points.Around each density minimum with 1+δ<0.2 we grow a sphere until the mean enclosed density is 1+δ̅<0.4.Visually such an underdensity gives voids which match expectations (see Fig. <ref>). The voids are then ordered by their radius R_V and overlapping voids with smaller radii are removed from the list. As is the case for the large overdensities (protoclusters) these large underdensities (voids) are very rare, necessitating surveys of large volumes. The number density of redshift-space voids at z=1 is 10^-5 h^-3 Mpc^3 for 10<R_v<15 h^-1Mpc and 6× 10^-7 for 20<R_v<25 h^-1Mpc and falls quickly with redshift.The matched filter essentially performs a “weighted convolution” of the density field with a profile, and thus requires some knowledge of the shape of the object it is trying to `match'. While the performance of the filter is relatively insensitive to the precise profile we use, we describe the choices we have made based on the N-body simulations described above.To begin we note that a void has an extent 𝒪(10Mpc), and thus covers only a small region of sky (< 1 arcminute) and a small portion of the frequency coverage of the telescope. We are thus justified in treating the sky as locally flat and the k⃗_⊥ coverage as approximately wavelength independent [Recall that the conversion from u⃗ to k⃗_⊥ depends on the frequency of the observation.].We expect the profile to have significant power at k∼ 0.1 hMpc^-1, well within the band of sensitivity of 21cm interferometers aimed at large-scale structure observations (see Fig. <ref> before).Fig. <ref> shows the averaged real space profile of voids with 10 < R_V < 15 h^-1Mpcin our N-body simulations at z=1. There are numerous analytic void profiles in the literature <cit.>. Most of these do not fit our N-body results particularly well, which is most likely due to the different choices of void finder employed.In particular our void profile approaches 0 smoothly from below at large radius, i.e. we do not find a prominent “compensation wall” at the void edge. This echoes the findings of <cit.>, who also found no compensation wall for voids which are not part of a larger overdensity. A simple, 2-parameter form which does provide a good fit to our N-body data isδ = δ_0/1+(r/r_s)^6where δ_0 and r_s are the interior underdensity and void scale radius respectively.This is shown in Fig. <ref> as the solid line.In Fourier space this profile becomesτ(k)=4π∫ r^2 dr δ(r) j_0(kr) = 2π^2/3 κδ_0 r_s^3 e^-κ/2[e^-κ/2 + √(3)sin y-cos y]where κ=k r_s and y=√(3) κ/2.For large k the profile is exponentially suppressed. Since the profile is not compensated, τ(k≪ 1)≃ (2π^2/3)(1-κ^2/3+⋯)δ_0 r_s^3 does not go to zero as k→ 0. This is clearly only an approximation, since on sufficiently large scales the profile must go to zero due to mass conservation, but it does not seem to adversely affect our filter. We remind the reader that it is this Fourier space form that is input into the matched filter.The above was all in real space. An analytic model for a void in redshift space could simply use the linear theory analysis of <cit.>.A better alternative would be to make use of the Gaussian streaming model <cit.>.<cit.> have shown that this model works well if linear theory expressions for the mean pairwise velocity and dispersion are computed from the assumed profile. We have taken a simpler approach, using the simulations to measure the anisotropy. Fig. <ref> shows the Fourier transform of the same voids shown in Fig. <ref>, except now in redshift space.Material which is outflowing causes the void to appear deeper and wider in the line-of-sight direction <cit.>, enhancing the profile along k_∥.In principle, this makes the redshift-space profile less sensitive to loss of modes in the “wedge” than would be anticipated from the real-space profile (though the filter will tend to downweight the line-of-sight modes more due to the enhanced cosmic clustering close to the line of sight).However, for intermediate scales k ∼ 0.2 h Mpc^-1, the void profiles are remarkably close to spherical, with only a very mild quadrupole.Given this small anisotropy, we shall continue to use a spherically symmetric void profile even in redshift space. This choice was motivated purely for the simplicity of the presentation and does not represent a limitation of the method, and we expect these choices to be revisited in future work. § RESULTS §.§ Filter amplitude distributions We now turn to the performance of the matched filter. Recall that we can evaluate the matched filter at an arbitrary point - ideally positions centered on voids would have significantly larger values of  that a randomly chosen point. [Since our input void profile has a negative central underdensity, we expect voids to have positive values of Â.] The left panel of Fig. <ref> plots the distribution of  in the ideal case of an effectively noiseless n̅ P=10 survey. The distribution is close to Gaussian with a width of 0.86; this compares with the analytically predicted value (Eq. <ref>) of 0.90. The Gaussianity of this distribution is easily understood by observing that the matched filter simply smoothes the (configuration space) density field with a kernel that is O(10) Mpc wide; on these scales, the density field is very close to Gaussian. We do see evidence of non-Gaussianity from collapsed objects in a slight skew towards negative values of Â. Although the matched filter has the void radius as an input parameter, we find that the shapes of the distributions (after scaling out the variance) are very similar. We therefore simply standardize all our distributions by the appropriate variance.We can now compare the above distribution with the matched filter evaluated at void centers. We consider two sets of voids : 10 < R_V < 15 h^-1 Mpc and 20 < R_V < 25 h^-1 Mpc and we set the filter radius to 12.5 h^-1Mpc and 22.5 h^-1Mpc respectively. We find that the distribution of  evaluated at the void centers is clearly separated from the full distribution of the matched filter. Approximately 80% of the smaller voids are detected at > 1 σ from zero while ∼ 90% of the larger voids are detected at > 1.5 σ. It is also worth noting that our reference distribution includes points that are in voids. Indeed ∼ 8% of the simulation volume is contained in voids larger than 10 h^-1 Mpc, which would correspond to a threshold choice of ∼ 1σ. Note that this is somewhat different from the Gaussian expectation of ∼ 1.5 σ; this difference can be traced to non-Gaussianity in the tails of the distribution of Â.We now consider how survey non-idealities impact the efficiency of the matched filter. There are two aspects relevant to the 21cm interferometer case. The first is that the instrument only samples particular k-modes and that this sampling is modulated by the number of baselines in the interferometer. The second is that, as discussed in Sec. <ref>, astrophysical foregrounds and instrumental imperfections can contaminate both low k_|| modes and the so-called “wedge”, further restricting the accessible k-space. The impact of these are summarized in the middle and right panels of Fig. <ref>. The relatively wide and dense coverage in k-space of our HIRAX-like survey implies that the filter's performance does not degrade significantly compared to the ideal case. Removing modes contaminated by foregrounds has a more significant effect. While we still see a separation between voids and randomly chosen points, only 50% of the voids are now above the thresholds discussed above.While the detailed performance of the void finder will depend on the details of the interferometer, the principal conclusion of the above discussion is that for the designs that are being considered voids are relatively easily detected in the absence of foregrounds but the loss of low k_∥ modes is a serious matter and some foreground mitigation strategy is necessary. Fig. <ref> shows similar performance plots for a idealization of the CHIME experiment (see Appendix <ref> for details). As with our HIRAX example, we find a clear separation between the distribution of voids and random points with similar recovered fractions of voids for the cases without any foregrounds, and a loss of separation when foregrounds become important.As with all matched filter applications, there are a number of input choices. The choice of the void profile is the most notable example in this case. We experimented with different choices of void shapes and sizes and find that the results above are quite robust. A different complication arises from the fact that our void profiles are estimated from the dark matter. Appendix <ref> explores the shapes of voids with a more realistic modeling of the 21cm density field. We find that shapes of the voids here are very similar (and possibly more pronounced) to those in the dark matter. We therefore expect our results to be qualitatively unchanged with more realistic modeling of the 21cm field.Another choice in our matched filter is the power spectrum used in the noise covariance matrix to account for large-scale structure noise. While different choices here change the exact width of the distribution of Â, it does not change our basic result that voids are detected with very high significance in the absence of foregrounds. §.§ An Example Application : A Void Catalog As an example application, we discuss how to use such a matched filter to construct a void catalog. Our intention here is not to attempt to quantify (or optimize) the purity and completeness of such an algorithm, since this will be data and instrument specificand so much depends upon the manner in which foregrounds are subtracted. Instead, we outline the steps of a possible algorithm and perform some simple calculations with it, and defer detailed discussions to future work.For this demonstration, we choose a single simulation box from our suite of ten simulations. We run the matched filter on this box with the void radius R_V varying from 33.3 h^-1Mpc to 20 h^-1Mpc in 10% steps.We keep a list of all points where the matched filter amplitude, A, exceeds 2 σ. Starting from the largest void(s) and working down in radius we eliminate any voids which overlap.If two overlapping voids have the same radius the one with the smallest A is removed.The result of this procedure is our “void catalog”.In a single 1380 h^-1Mpc box the largest 1,000 voids have radius above about 20 h^-1Mpc. With full u-v coverage and low noise we find that all but 1 of the 32 largest “true” voids contain a match in our catalog within 0.75 R_V and these matches are all in the upper 5^ th percentile of the A distribution. Just under half of them (14 of the 32 voids) show significant (>R_V/3) mis-centering, i.e. the detected void center is > R_V/3 away from the center of the closest true void. For the u-v coverage of our HIRAX-like experiment, and n̅P=3, four of the 32 largest “true” voids do not have a match within 0.75 R_V and again all are highly significant. The situation changes dramatically as we include a k_∥, min and μ cut.Fork_∥, min=0.05 hMpc^-1 and μ>0.56 we find only 5 of the top 32 voids in our catalog, though these voids are in the extreme tails of the A distribution.Most of this effect is driven by the k_∥ cut. If we relax the cut to 0.02 hMpc^-1 then we recover 10 of the 32 largest voids and for a cut of 0.01 hMpc^-1 we recover 20 of them.We can recast the results of this and the previous section into the more traditional forms of the completeness and purity of the sample. In the absence of foregrounds, our detected void catalog is both pure (only ∼ 10% of detected voids do not correspond to true voids) and complete (> 90% of true voids are detected at better than 1.5σ) for large (∼ 20 h^-1Mpc) voids. However, both of these numbers are sensitive to foregrounds. For our most conservative case of foregrounds contaminating all modes with k_|| < 0.05 hMpc^-1 and μ < 0.56, the majority of the most prominent detections do not correspond to true underlying voids and only ∼ 50% of true voids are detected at high significance.It is possible that some of the low k_∥ information lost to the interferometer by foregrounds could be replaced by another experiment. As an example, modern photometric surveys can achieve high photometric redshift precision for certain types of galaxies, and thus can map the low k_∥ modes of the 3D density field.In fact, such surveys have been used to search for voids <cit.>. Including the photometric survey in our matched filter presents no problem in principle — one simply augments the data vector and includes a model for the void in configuration space — but could be difficult in practice. Assuming the combination recovers all of the k_∥ range, we recover our no-foreground forecasts. If there is a gap in coverage the results are adversely affected. To take a pessimistic example: if we lose modes 0.02<k_∥<0.05 hMpc^-1 we are able to recover 12 of our top 32 voids.For 0.03<k_∥<0.05 hMpc^-1 it is half of our top 32 voids.These lost k_∥ modes potentially could be reconstructed from higher-point information in the 21cm field itself <cit.>. There is considerable interest in developing these reconstruction schemes for 21cm surveys to enable cross-correlations with photometric surveys or CMB lensing maps.Initial results <cit.> suggest that modes k_∥ < 0.01 hMpc^-1 and k_⊥ < 0.05 hMpc^-1 could be recovered.As with the example above, the efficiency of the void finder will depend on the details of the performance of these reconstructions.We can visualize this information in another way. Fig. <ref> shows the stacked matter profile around our top 1,000 void candidates for various choices of k_∥, min and μ_ min.With full u-v coverage there is a clear, coherent underdensity at the locations of the void candidates.The shallower inner profile in Fig. <ref> when compared to Fig. <ref> arises due to mis-centering. While some voids were well centered, a significant fraction had offsets. After visually inspecting these voids, we find that, in most cases, the true void center was a significant detection in the matched filter, but happened not to be the most significant detection and was removed by our relatively simple pruning algorithm. Fig. <ref> shows an example of such a case. Both the fraction of found voids and the degree of mis-centering get worse when modes are lost to foregrounds, as the other curves in Fig. <ref> show. For our pessimistic scenario of k_∥>0.05 hMpc^-1 and μ>0.56 there is barely any underdensity detected at all. This suggests that doing science with voids selected via 21cm experiments will be difficult unless the foregrounds can be brought under control. We however note that the algorithm used to construct our void catalog is relatively simple; e.g. a more robust algorithm might use multi-scale information to get more robust measurements and there is significant potential for complementarity between optical imaging surveys and 21cm measurements. § CONCLUSIONSRecent advances in technology have made it feasible to study the 21cm emission from objects at cosmological distances.A new generation of telescopes is being designed and built which aim to survey enormous volumes of the Universe with modest resolution at redshifts z≃ 1-2. A primary focus of these facilities is the measurement of the power spectrum of large-scale structure, as traced by neutral hydrogen, which will hopefully improve constraints on our cosmological model. While these instruments do not have sufficient angular resolution to resolve the emission from individual objects, we point out that they should be able to make catalogs of the largest members of the cosmic web – protoclusters and voids – if they are able to control foregrounds sufficiently.We have considered instruments which measure the sky interferometrically, which means they naturally operate in Fourier space.The finite sampling of the Fourier plane, and the loss of sensitivity in some modes due to foregrounds, make it difficult to generate a real-space, 3D map from the data and hence to search for exotica whose properties are not known in advance.However, our understanding of the cosmic web allows us to specify in advance what sort of objects we are interested in finding and searches for objects of known shape do not need to go through the map-making step: a matched filter provides a natural method for finding such objects. The matched filter formalism also allows us to mix multiple data sets, each of which is provided in its own domain.The cosmic web contains voids on a variety of scales, and voids which touch or merge.We have only studied the simplest matched filter.The algorithm can be modified to iteratively add voids to an existing catalog, always adding the void which leads to the largest increase in the likelihood given the already-found voids <cit.>. This involves a scan over void (or protocluster) sizes, and increases the complexity of the algorithm. A multiprobe approach could use deep, optical imaging data in conjunction with 21cm data in much the same way as multifrequency information is sometimes used for cluster finding <cit.>. As our main aim was to assess the feasibility of void detection with 21cm surveys, we defer further consideration of such a process to future work.Throughout we have focused our discussion on voids as exemplars of large structures in the cosmic web.Of course, the matched filter algorithm is more general and the huge volume and sensitivity of upcoming experiments can be used to search for a number of exotic objects.At the other end of the density distribution from voids are the large, coherent overdensities associated with protoclusters.Despite keen interest in the community in how clusters form and evolve, and years of observational and numerical efforts, the study of early cluster formation (at high z) remains observationally limited. Protoclusters are rare, present only modest overdensities and lack many of the features used to discover clusters (e.g. a hot ICM or a red sequence). Observations of protoclusters at high z would provide important clues into cluster assembly and the processes of galaxy formation <cit.>. Given the diversity of protoclusters, having large samples with well understood selection is important.Like voids, protoclusters form large coherent structures amenable to discovery in upcoming 21cm experiments. Assuming a mean interior density of 200 times the background, the linear size of the mean-density region from which material accretes into a present day cluster is several (comoving) Mpc.The progenitors of large clusters should thus be identifiable in relatively low resolution maps that can cover large volumes <cit.>.Slices through the density field in one of our simulations are shown in Fig. <ref> where the large extended mass profile of the protoclusters is evident.In fact, the most massive clusters in the mature Universe form not from the most overdense regions at high z but from large, possibly only moderately overdense regions such as shown in Fig. <ref> <cit.>. While we do not show it here, the typical protocluster covers a larger volume at z≃ 2, rendering it potentially easier to see while still being well within the redshift reach of HIRAX or CHIME.The abundance of such protoclusters is identical to the abundance of the clusters at z=0: for a mass threshold of 3× 10^14 h^-1M_⊙ it is 4× 10^-6 h^3 Mpc^-3. This emphasizes the need for a survey to cover a large volume in order to properly sample the heterogeneous population of protoclusters. As an example, if it covered 15,000 deg^2 between z=1 and 2 HIRAX would survey 50(h^-1 Gpc)^3 encompassing ∼ 200,000 protoclusters[Almost by definition the number density of protoclusters is redshift independent.].CHIME is anticipated to cover a similar volume in the northern hemisphere. In some models the star formation associated with halos in protoclusters makes up a significant fraction of the ionizing photon budget for reionization <cit.> at z≃ 6-7.If foregrounds could be controlled, using interferometers designed for studying reionization to search for protoclusters could provide an interesting synergy.We thank Emanuele Castorina and Richard Shaw for useful discussions and comments on an early version of this paper. N.P. thanks Laura Newburgh for useful conversations. We thank the referee for detailed comments on the paper. M.W. is supported by DOE. N.P. is supported in part by DOE DE-SC0008080.This work made extensive use of the NASA Astrophysics Data System and of the astro-ph preprint archive at arXiv.org.The analysis made use of the computing resources of the National Energy Research Scientific Computing Center.§ SIGNAL TO NOISE RATIOWe present a self-contained derivation of the instrument noise power spectrum, converted to cosmological units. Our derivation is similar to that in <cit.>, but related expressions have also appeared in <cit.>.The brightness temperature, T_b, is defined in terms of the intensity at frequency ν as I_ν=2k_BT_b(ν/c)^2=2k_BT_b/λ^2. We begin by noting that if we normalize our visibilities in terms of temperature (rather than intensity) the power can be written in terms of the brightness temperature power spectrum and window function as⟨| V_i|^2⟩ = ∫ d^2u P_T(u⃗)W(u⃗) ≈ P_T(u⃗) ∫ d^2u W(u⃗)with the last approximation holding if the window function is compact and P_T is smooth.Conventionally the beam is normalized to unity at peak, so its area in the u-v plane integrates to unity and thus the window function integrates to the inverse area∫ d^2u W(u⃗) ∼1/d^2 u ,which gives⟨| V_i|^2⟩≈P_T(u⃗)/d^2 u . It may be helpful to derive Eq. (<ref>) differently. If we treat the visibility as measuring a single Fourier mode of the 2D brightness temperature field, we can relate this to the 2D power spectrum of this field⟨ V(ℓ⃗) V^⋆(ℓ⃗') ⟩ = (2π)^2 δ^D(ℓ⃗ - ℓ⃗') P_T(ℓ⃗=2πu⃗) ≈δ^K_ℓ⃗,ℓ⃗'P_T(u⃗)/d^2 u ,where δ^D,K are the Dirac and Kronecker δ functions. The above equation explicitly relates u⃗ to the 2D wavevector ℓ⃗, and the last approximation comes from assuming a discretized set of wavevectors[For instance, this is exactly what happens on an FFT grid in a simulation.].In the same units the visibility noise is diagonal <cit.>⟨| N_i|^2⟩ = [2k_B/λ^2]^-2[2k_B T_ sys/A_e]^2 1/Δν t_ p = [λ^2 T_ sys/A_e]^2 1/Δν t_ pper baseline.Here T_ sys is the system temperature, A_e the effective area of the telescope (equal to the aperture efficiency times the physical area), Δν is the bandwidth, t_ p is the observing time per pointing and we have assumed a single polarization.The above is all that is needed to implement a matched filter on the data, where we can work at the level of the visibilities. It is however useful to translate this into the cosmological units used in the paper. We start by defining the number of baselines per unit area in the u-v plane, n(u⃗), normalized such that∫ d^2u n(u⃗) = N_ pairs = N_a (N_a-1)/2where N_a is the number of antennae, and N_ pairs is the number of pairs (i.e. instantaneous baselines). Averaging over the number of baselines, the noise becomes ⟨ N_i^2 ⟩/(n(u⃗) d^2u). Using Eq. (<ref>) we obtainP_N(u⃗) = [λ^2 T_ sys/A_e]^2 1/n(u⃗)1/Δν t_ p=[λ^2 T_ sys^2/A_e] 1/n(u⃗)4π f_ sky/Δν t_ obs .The last equality follows from N_pΩ_p = 4 π f_ sky where N_p=t_ obs/t_ p is the number of pointings and t_ obs is the total observing time. The area covered by each pointing Ω_p is approximately given by λ^2/A_e. Physically, the above equations assume that each pointing yield a disjoint set of modes.To convert this visibility noise into a cosmological power spectrum, we divide by the mean cosmological brightness temperature <cit.>T̅ = 188 x_HI(z) Ω_H,0h(1+z)^2/H(z)/H_0mK ,with x_HI the neutral hydrogen fraction, and convert from u⃗ to k⃗_⊥ in comoving coordinates and similarly for frequency to k_∥ to obtainP_N = ( T_ sys/T̅)^2 ( λ^2/A_ e) 4 π f_ sky/t_ obs n(u⃗)d^2V/dΩ dνwhere in a spatially flat modeld^2V/dΩ dν = χ^2 dχ/dz dz/dν = χ^2 c (1+z)^2/H(z) ν_0with ν_0=1420MHz. Unfortunately the value of Ω_H,0 h is quite uncertain and it enters quadratically in the noise power spectrum. <cit.> measure 10^3Ω_H,0≃ 0.9± 0.3 at z≈ 1 through the abundance of damped Lyman-α systems (see also the compilations of data in ). The measurement of Ω_H,0b through 21cm auto-correlations by <cit.> has a similar value and fractional error. We will consider the range (0.6-1.2)× 10^-3 or Ω_H,0 h=(4-9)× 10^-4. For Ω_H,0 h=4× 10^-4 and the HIRAX-like interferometer described in the text operating for 3 years we obtain P_N≈ 600 h^-3Mpc^3 at z=1 and k_⊥=0.2 h^-1Mpc. Comparing to the linear matter spectrum, and assuming b=1, we have (P_L/P_N)(k_⊥)≈ 1. If Ω_H,0 h=9× 10^-4 we obtain P_N≈ 150 h^-3Mpc^3 and have (P_L/P_N)(k_⊥)≈ 4.While this is similar in spirit to nP in galaxy surveys, it is worth emphasizing that this quantity is intrisincally 2D, while nP is spherically symmetric. In particular, at fixed k, the average value of k_⊥ is (π/4)k. § TRANSIT TELESCOPES AND THE M-MODE FORMALISMThe interferometers for 21cm intensity mapping experiments are designed to be transit telescopes, using the Earth's rotation to map large areas of the sky. This mapping process simultaneously performs two operations that are traditionally treated separately - filling in the u-v plane[As we discuss later in this section, a more appropriate basis for discussing these telescopes are spherical harmonics. We use the u-v plane here to mean an appropriate “Fourier” transform of the sky.], and improving the resolution in the u-v plane[Recall that a single visibility measurement is smeared in the “u-v” plane by the Fourier transform of the primary beam and “mosaicking”combines observations of different areas of the sky to make this window function more compact.] by “mosaicking”. Furthermore, some upcoming experiments, notably CHIME[http://chime.phas.ubc.ca/] and Tianlai[http://tianlai.bao.ac.cn], use a close-packed array of cylinders rather than traditional dishes. In the CHIME configuration, 4 cylinders (each 20m in diameter and ∼ 100m long, oriented north-south) are placed adjacent in the east-west direction <cit.>. The primary beam from such a configuration is highly extended in the north-south direction, while being focussed by the cylinders in the east-west direction.Both of these features naturally cover large angles on the sky. The natural basis for describing these telescopes is not the usual Fourier basis, but rather spherical harmonics. However, most astrophysical signals (including the voids discussed here) cover small areas in the sky and are easily described in a flat-sky limit. The goal of this Appendix is to make the connection between the wide-angle and flat sky formalism explicit. We start with a review of the m-mode formalism, following <cit.> who state the fundamental visibility measurements in a spherical harmonic basis. We then take the flat-sky limit of this result and show that we recover the traditional u-v plane interpretation. Making this connection also allows us to explicitly see how the Earth's rotation fills in the u-v plane. We then develop the matched filter formalism in this basis. We conclude with a worked example of the m-mode formalism, to help build intuition. §.§ Review of the m-mode formalism Following <cit.>, if the beam transfer function pointed at azimuth ϕ isB_ij(n;ϕ)∝ A^2(n;ϕ)exp[2π i n·u⃗_ij(ϕ)]thenV_ij(ϕ) = ∫ dn T(n) B_ij(n;ϕ)(plus noise, of course). We remind the reader to distinguish between the pointing center of the beam (the azimuth of which is ϕ) and the coordinate that integrates over the beam (n). Expanding T and B_ij(ϕ) into spherical harmonicsT(n)= ∑_ℓ m a_ℓ m Y_ℓ m(n) B_ij(n;ϕ)= ∑_ℓ m B_ℓ m^ij Y_ℓ m^⋆(n)we obtainV_ij(ϕ) = ∑_ℓ m B_ij^ℓ m(ϕ) a_lm .The rotation of the Earth in ϕ causes the beam to transform as B^ℓ m(ϕ) = B^ℓ m(0) e^i m ϕ. Defining V_ij^m = ∫dϕ/2π e^-imϕV_ij(ϕ)we obtainV_ij^m = ∑_ℓ B_ij^ℓ m a_ℓ mwhere B_ij^ℓ m without an explicit argument is understood to be at ϕ=0 (the phase factor cancels out its conjugate in the definition of V_ij^m.) These V_ij^m (or their Fourier conjugate V_ij(ϕ) are the fundamental observables of the telescope. §.§ The Flat-Sky Approximation It is illuminating to show that the above expression recovers the usual flat-sky Fourier representation for small areas of the sky. We will use ℓ⃗ to represent the 2D Fourier wavevector, with magnitude ℓ and polar angle φ_ℓ (not to be confused with the pointing center ϕ). The correspondence between a_ℓ m and a(ℓ⃗) is <cit.>a(ℓ⃗) = √(4 π/2 ℓ +1)∑_m i^-m a_ℓ m e^i m φ_ℓanda_ℓ m = √(2 ℓ +1/4 π) i^m ∫dφ_ℓ/2π a(ℓ⃗)e^-i m φ_ℓ .with a similar expansion for B_ij^ℓ m. Substituting into the visibility equation, V_ij(ϕ) = ∑_ℓ m B_ij^ℓ m(ϕ) a_ℓ m, we obtain, for large ℓ,V_ij(ϕ) ≈1/(2π)^3∑_ℓ m∫ dφ_ℓ dφ_ℓ' ℓ a(ℓ⃗) B(ℓ⃗', ϕ) e^i m (φ_ℓ - φ_ℓ') ,where ℓ⃗ and ℓ⃗' have the same magnitude. Doing the sum over m yields a δ-function that collapses one of the azimuthal integrals to yieldV_ij(ϕ) ≈∫ℓ dℓ dφ_ℓ/(2π)^2 a(ℓ⃗) B(ℓ⃗, ϕ)where we have approximated the sum over ℓ by an integral. The above shows that the visibilities approximately measure a mode ℓ⃗, smeared by the Fourier transform of the beam function. We can use the above results to understand how the rotation of the Earth fills in the u-v plane. In the flat-sky limit, the Fourier transform of the beam is B(ℓ⃗)∼∑_m i^-mB_ℓ mexp[imφ_ℓ]. Rotating about the z axis by α scales the B_ℓ m by exp[imα], which is clearly equivalent to rotating ℓ⃗ by α. The u-v coverage of the telescope traces out circles in the u-v plane as the Earth rotates. We note that this is different from the usual result for interferometers, and reflects the transit nature of these telescopes.§.§ Matched Filters In order to define the matched filter, we need to express the signal in terms of the observable quantities, in this case the visibilities.Since all of the objects of interest in this study are 𝒪(10Mpc) in size, at a distance of >1Gpc, they subtend small angles on the sky, allowing us to express the signal using the same flat-sky Fourier representation used in the main paper.To begin, consider a single frequency, corresponding to a fixed redshift or (redshift-space) distance.Suppose our template, τ, is centered at θ=0, is ϕ-independent and non-zero only when θ≪ 1. We expandτ_ℓ m(z)= ∫ dn Y_ℓ m^⋆(n) τ(θ) =2π δ^K_m0 √(2ℓ+1/4π)∫ d(cosθ) P_ℓ(cosθ) τ(θ) ≃ δ^K_m0 √(2ℓ+1/4π) [2π∫ω̃ dω̃ J_0(ℓω̃) τ(ω̃)]where in the last line we have defined ω̃=2sin(θ/2)≃θ and used P_ℓ(cosθ)≈ J_0(ℓθ) for θ≪ 1. The √((2ℓ+1)/4π) is just Y_ℓ 0(z). If we extend the upper limit of ω̃-integration to infinity, we recognize in the brackets on the last line the Hankel transform of τ or the 2D Fourier transform of τ with spherical symmetry (e.g. ).Now we can rotate the template from the north pole (z) to an arbitrary n using Wigner functions, 𝒟^ℓ_m'm. However, in our case τ_ℓ m∝δ^K_m0 and Y_ℓ m∝𝒟^ℓ_0m so that the spherical harmonic coefficients for a template centered on n areτ_ℓ m(n)= √(4π/2ℓ+1) Y_ℓ m^⋆(n) τ_ℓ 0(z) = [2π∫ω̃ dω̃ J_0(ℓω̃)τ(ω̃)] Y_ℓ m^⋆(n)(with no implied sum over ℓ).These τ_ℓ m can now be inserted into our formula for the m-mode visibility to obtainV_ij^m (n) = ∑_ℓ B_ij^ℓ m Y_ℓ m^⋆(n) [2π∫ω̃ dω̃ J_0(ℓω̃)τ(ω̃)]This is the central relation needed for the matched filter, as it expresses a linear relationship between the observable and the template. We recognize the the combination B^ℓ m_ij Y_ℓ m^⋆ as the beam transfer function, B_ij(n;ϕ), evaluated at the position of the object but now modulated by the Fourier transform of τ.The above expressions are all for a single frequency. If we now perform the Fourier transform in frequency, the term in square brackets becomes the 3D Fourier transform for an azimuthally symmeteric function in cylindrical coordinates: τ(k_⊥,k_∥) with ℓ≃ |k_⊥|. For a narrow range of frequencies (corresponding to an astrophysical object such as a void or protocluster for example) the k_⊥ probed by the interferometer are almost constant.For a wide range of frequencies one must account for the shifting of u⃗_ij, and ℓ, with wavelength at fixed baseline separation. This represents no difficulty in principle, since we need only evaluate our template where there is data, but it formally breaks the Fourier transform property. It is important to note that this Fourier transform is not necessary for the matched filter, which can be written in visibility-frequency space.As before, the matched filter is defined byÂ(n) = V_ij^m N^-1 V_ij^m (n)/ V_ij^m(n) N^-1 V_ij^m(n) ,where the noise covariance matrix both includes the visibility noise and projects out contaminated modes. There are a few practical differences between this treatment and the flat-sky Fourier version we discuss in the main text. In the simplified flat sky treatment, shifting the matched filter to an arbitrary position x⃗ was simply a multiplication of  by exp(i k⃗r⃗), which allowed us to efficiently evaluate the matched filter at all possible void positions with inverse FFTs. In particular, the denominator of  is translation-invariant. While these simplifications remain true in the azimuthal direction, they no longer hold for the polar or radial directions. Therefore, one must explicitly evaluate the matched filter at all possible void positions. It may be possible to reduce the computational burden by using the Fourier versions of the expressions about more sparsely sampled central void positions. Since the precise implementation will be survey dependent, we do not pursue more detailed implementations here. §.§ A Worked Example We conclude with an analytic example to make this formalism more concrete. Our discussion here parallels that in <cit.>. Consider the interferometer situated at the equator (θ_0=π/2, ϕ_0=0) and looking directly overhead. The baselines, u⃗_ij, lie in the y-z plane.We will consider two cases, a north-south baseline (u⃗=uz) and an east-west baseline (u⃗=uy).For a small field of view, we approximate the sky as flat with Cartesian coordinates ϕ, δ, where δ≡π/2-θ is the latitude. A Gaussian beam, normalized to unit peak, then hasB(n)=B(ϕ,δ) = exp[-ϕ^2+δ^2/2σ^2]×{[ exp[2π i u ϕ]for y (EW); exp[2π i u δ]for z (NS) ].where we have suppressed the ij indices labeling the visibility for convenience.The visibility for this baseline isV(n) = ∫ dn B(n) T(n).Instead of immediately going to the spherical harmonic expansion, it is algebraicly illuminating and amusing to imagine the sky as a torus. The appropriate orthogonal basis is then the usual Fourier basisV = ∫ dn∑ B_nm e^-i n δ e^-i m ϕ∑ T_n'm' e^i n' δ e^i m' ϕwhich collapses toV^m = (2π)^2 ∑_nm B_nm T_nmwhere we have also implicitly gone to the m-mode basis (to account for the Earth's rotation). This expression is analogous to the spherical harmonic version. The beam multipole moments are then given byB_nm = ∫dϕ dδ/(2π)^2 B(ϕ, δ) e^i n δ e^i m ϕ .Since we assume the beams are compact in both ϕ and δ, we are free to extend the limits of integration to ±∞.For the specific case of our Gaussian beam, these integrals are then just Gaussian integrals and can be easily evaluated. For an EW baseline, we getB_nm∝exp[-σ^2 n^2/2]exp[-σ^2 (m ± 2π u)^2/2]while for the NS baseline, we findB_nm∝exp[-σ^2 (n ± 2π u)^2/2]exp[-σ^2 m^2/2]where the ± cases come from the two possible choices for the sign of u. These have a clear physical interpretation - the EW baseline probes modes centered around (n=0, m=± 2π u) while the NS baseline is centered on (n=2π u, m=0).Note that these expressions indicate that it is the baseline distribution and the primary beam which delineate the range of (ℓ m) modes which need to be kept in the sums of the previous section.Returning to a spherical sky, we will adopt a similar strategy to understand what modes a given baseline probes.Since the beam is compact, we will approximate the spherical harmonics by a Fourier series, in which case the algebra proceeds as in the case of the torus. All that will remain will be to understand the correspondence between mode coefficients n on the torus and (ℓ, m) on the sphere[Note that in the ϕ direction, both the sphere and the torus have Fourier expansions.].For our specific case, the multipole moments then becomeB_ℓ m = ∫ dϕ dsinδ Y_ℓ m(π/2-δ, ϕ)B(ϕ,δ) ≃ ∫_-∞^∞ dϕ dδ Y_ℓ m(π/2-δ, ϕ)B(ϕ,δ)where we assume δ≪ 1 in the second line. Near the equator, we have [The approximation agrees to the first two terms in the Taylor series. For completeness, we note thatN_ℓ m = 2^m√(π)√(2ℓ +1/4π)√((ℓ - m)!/(ℓ + m)!) 1/Γ(1/2-(l+m)/2) Γ(1+ (l-m)/2) .]Y_ℓ m≃ N_ℓ m e^i m ϕ{[cos n_ℓ mδ for ℓ+meven; -sin n_ℓ mδfor ℓ+modd ].,where N_ℓ m is a constant andn_ℓ m^2 = ℓ (ℓ+1) -m^2 - {[ 0 for ℓ+meven; 1for ℓ+modd ].. Since we have reduced the problem to the toroidal sky case, we proceed as before and find that EW baselines measure modes centered on (n_ℓ m=0, m=2 π u). In the limit that ℓ≫ 1, this implies that these baselines measure modes with m ∼ 2 π u, ℓ∼ m. As one might expect, ℓ and m are coupled together by the spherical geometry. For NS baselines, the m-mode visibilities probe (n_ℓ m=2 π u, m=0) or ℓ∼ 2π u, m ∼ 0. The azimuthal symmetry of the baseline configuration is reflected in the visibilities isolating the m ∼ 0 modes. These two cases represent the two limiting cases; baselines with components in both the EW and NS directions will probe more general ℓ, m modes.For this particular case, this also completes the correspondence with the usual flat-sky treatment where a baseline measures a particular ℓ⃗ Fourier mode.Here, the visibility m modes measure particular ℓ, m modes.§ MODELING THE 21CM SIGNALIn the main text we have assumed that neutral hydrogen traces the mass field in an unbiased manner for the purposes of testing our matched filter on simulations.In this appendix we present a more refined model and argue that this assumption is conservative (for our purposes).At low z most of the hydrogen in the Universe is ionized, and the 21cm signal comes only from self-shielded regions such as galaxies[Most likely between the outskirts of disks until where the gas becomes molecular within star-forming regions.].Unfortunately there are not many observational constraints on the manner in which HI traces galaxies and halos in the high-z Universe.There have been a large number of approaches to modeling this uncertain signal. Some approaches work directly at the level of the density field. For example, <cit.> use Gaussian density fields. <cit.> assumes a constant bias times the matter power spectrum (this is implicitly what we do in the main text, with b=1). The CRIME code by <cit.> uses lognormal realizations. <cit.> selected dark matter particles based on a density threshold to mock up self-shielded regions.An alternative is to use a halo-based approach, specifying the mass of HI to assign to a dark matter halo of a given mass, M_h.A popular model was introduced by <cit.>, which populated halos with circular velocities above 30km/s with HI such that the HI mass saturates at high halo mass.A similar model was proposed by <cit.>, who modeled the low-M cutoff as an exponential. <cit.> use abundance matching between blue galaxies in the HI mass function at z≈ 0. <cit.> employ a double power-law model. <cit.> propose a form with an exponential cut-off at both low and high halo masses. <cit.> allow a non-unity slope in addition to the high and low mass cut-offs. The model we shall follow is due to <cit.>, which assumesM_HI∝ M_h^α e^-M_ cut/M_hwith the constant of proportionality adjusted to match the observed value of Ω_HI.Aside from the normalization, this model has two free parameters, α and M_ cut, which control the behavior at high and low halo masses.There is evidence from simulations that α<1 <cit.> with α≈ 3/4 a reasonable estimate. We shall use this value.Note that in contrast to some of the other models this assumption puts significant HI mass in higher mass halos. There is some evidence at z≃ 0 that HI is depleted in galaxies within clusters <cit.>, but the behavior at z∼ 1 is unknown. In the simulations of <cit.> the trend of M_HI with M_h is different at high and low redshift. The remaining free parameter, M_ cut, then adjusts the bias[For α=3/4 at z≈ 1 the bias ranges from 1.4 to 1.7 as lgM_ cut runs from 10.5 to 11.5 in h^-1M_⊙ units. This is consistent with the amplitude of the measured clustering at z∼ 1 by <cit.> but those measurements are not precise enough to place strong limits on the bias.] of the HI. While a range of values are allowed within the observational constraints, typical values for the low-mass cut-off, M_ cut, are around 10^11 h^-1M_⊙.We shall explore a range around this value (lgM_ cut=10.5, 11 and 11.5 with masses in h^-1M_⊙) to illustrate the effects.The simulations used in the main body of this paper do not have sufficient resolution to track the halos expected to host much of the HI at z∼ 1. Thus in this appendix we use a different simulation, run with the same code, which employed 2560^3 particles in a box of side 256 h^-1Mpc. This is the same simulation as used in <cit.>, to which the reader is referred for more details. We generate a mock HI field from the z≃ 1 halo catalog using the mapping of Eq. (<ref>).We find voids in this simulation using the same technique as described in the main text.For completeness we also find protoclusters, in a manner similar to <cit.>: starting from a friends-of-friends halo catalog (with a linking length of 0.168 times the mean interparticle spacing) we select each z=0 halo more massive than 10^14 h^-1M_⊙. We then track the particles within a few hundred kpc of the most bound particle back to z=1.The center of mass of these is taken to be the protocluster position at z=1.A comparison of the (real-space) profiles of protoclusters and voids in the dark matter and mock HI at z≃ 1 is shown in Fig. <ref> for three values of M_ cut. The curves are noisier than from the larger volume simulations, due to the poorer statistics, however we see that the protoclusters in the HI have just as much broad, distributed emission as the matter profiles. The voids in the HI have a qualitatively similar “bucket shaped” profile to the mass density, but are notably more empty. As noted by <cit.>, the halo mass function shifts dramatically to lower masses in underdense regions.Thus we expect to see voids in the massive halo and HI distributions be “more empty” than in the mass. Given the greater contrast in HI than in the matter, our approximation in the main text is conservative from the point of view of finding protoclusters and voids with 21cm experiments.mn2e | http://arxiv.org/abs/1705.09669v2 | {
"authors": [
"Martin White",
"Nikhil Padmanabhan"
],
"categories": [
"astro-ph.CO"
],
"primary_category": "astro-ph.CO",
"published": "20170526180111",
"title": "Matched filtering with interferometric 21cm experiments"
} |
[ [ Received ; accepted ======================= We propose a novel adaptive approximation approach for test-time resource-constrained prediction. Given an input instance at test-time, a gating function identifies a prediction model for the input among a collection of models. Our objective is to minimize overall average cost without sacrificing accuracy. We learn gating and prediction models on fully labeled training data by means of a bottom-up strategy. Our novel bottom-up method first trains a high-accuracy complex model. Then a low-complexity gating and prediction model are subsequently learnt to adaptively approximate the high-accuracy model in regions where low-cost models are capable of making highly accurate predictions. We pose an empirical loss minimization problem with cost constraints to jointly train gating and prediction models. On a number of benchmark datasets our method outperforms state-of-the-art achieving higher accuracy for the same cost. § INTRODUCTIONResource costs arise during test-time prediction in a number of machine learning applications. Feature costs in Internet, Healthcare, and Surveillance applications arise due to to feature extraction time <cit.>, and feature/sensor acquisition <cit.>.The goal in such scenarios is to learn models on fully annotated training data that maintains high accuracy while meeting average resource constraints during prediction-time. There have been a number of promisingapproaches that focus on methods for reducing costs while improving overall accuracy <cit.>. These methods are adaptive in that, at test-time, resources (features, computation etc) are allocated adaptively depending on the difficulty of the input. Many of these methods train models in a top-down manner, namely, attempt to build out the model by selectively adding the most cost-effective features to improve accuracy.In contrast we propose a novel bottom-up approach. We train adaptive models on annotated training data by selectively identifying parts of the input space for which high accuracy can be maintained at a lower cost. The principle advantage of our method is twofold. First, our approach can be readily applied to cases where it is desirable to reduce costs of an existing high-cost legacy system. Second, training top-down models leads to fundamental combinatorial issues in multi-stage search over all feature subsets (see Sec. <ref>). In contrast, we bypass many of these issues by posing a natural adaptive approximation objective to partition the input space into easy and hard cases. Our key insight is that reducing costs of an existing high-accuracy system (bottom-up approach) is generally easier by selectively identifying redundancies using L1 or other group sparse norms. In particular, when no legacy system is available, our method consists of first learning a high-accuracy model that minimizes the empirical loss regardless of costs. The resulting high prediction-cost model (HPC) can be readily trained using any of the existing methods. Next, we then jointly learn a low-cost gating function as well as a low prediction-cost (LPC) model so as to adaptively approximate the high-accuracy model by identifying regions of input space where a low-cost gating and LPC model are adequate to achieve high-accuracy. At test-time, for each input instance, the gating function decides whether or not the LPC model is adequate for accurate classification. Intuitively, “easy” examples can be correctly classified using only an LPC model while “hard” examples require HPC model. By identifying which of the input instances can be classified accurately with LPCs we bypass the utilization of HPC model, thus reducing average prediction cost. Figure <ref> is a schematic of our approach, where x is feature vector and y is the predicted label; we aim to learn g and an LPC model to adaptively approximate the HPC.r0.5< g r a p h i c s > Single stage schematic of our approach. We learn low-cost gating g and a low-prediction cost (LPC) model to adaptively approximate a high prediction cost (HPC) model. The problem would be simpler if our task were to primarily partition the input space into regions where LPC models would suffice. The difficulty is that we must also learn a low gating-cost function capable of identifying input instances for which LPC suffices.Since both prediction and gating account for cost, we favor design strategies that lead to shared features and decision architectures between the gating function and the LPC model. We pose the problem as a discriminative empirical risk minimization problem that jointly optimizes for gating and prediction models in terms of a joint margin-based objective function.The resulting objective is separately convex in gating and prediction functions. We propose an alternating minimization scheme that is guaranteed to converge since with appropriate choice of loss-functions (for instance, logistic loss), each optimization step amounts to a probabilistic approximation/projection (I-projection/M-projection) onto a probability space. While our method can be recursively applied in multiple stages to successively approximate the adaptive system obtained in the previous stage, thereby refining accuracy-cost trade-off, we observe that on benchmark datasets even a single stage of our method outperforms state-of-art in accuracy-cost performance.§ RELATED WORK Learning decision rules to minimize error subject to a budget constraint during prediction-time is an area of active interest <cit.>.Pre-trained Models: In one instantiation of these methods it is assumed that there exists a collection of prediction models with amortized costs <cit.> so that a natural ordering of prediction models can be imposed. In other instances, the feature dimension is assumed to be sufficiently low so as to admit an exhaustive enumeration of all the combinatorial possibilities <cit.>. These methods then learn a policy to choose amongst the ordered prediction models. In contrast we do not impose any of these restrictions. Top-Down Methods: For high-dimensional spaces, many existing approaches focus on learning complex adaptive decision functions top-down <cit.>. Conceptually, during training, top-down methods acquire new features based on their utility value. This requires exploration of partitions of the input space together with different combinatorial low-cost feature subsets that would result in higher accuracy. These methods are based on multi-stage exploration leading to combinatorially hard problems. Different novel relaxations and greedy heuristics have been developed in this context. Bottom-up Methods: Our work is somewhat related to <cit.>, who propose to prune a fully trained random forests (RF) to reduce costs. Nevertheless, in contrast to our adaptive system, their perspective is to compress the original model and utilize the pruned forest as a stand-alone model for test-time prediction. Furthermore, their method is specifically tailored to random forests. Another set of related work includes classifier cascade <cit.> and decision DAG <cit.>, both of which aim to re-weight/re-order a set of pre-trained base learners to reduce prediction budget. Our method, on the other hand, only requires to pre-train a high-accuracy model and jointly learns the low-cost models to approximate it; therefore ours can be viewed as complementary to the existing work. The teacher-student framework <cit.> is also related to our bottom-up approach; a low-cost student model learns to approximate the teacher model so as to meet test-time budget. However, the goal there is to learn a better stand-alone student model. In contrast, we make use of both the low-cost (student) and high-accuracy (teacher) model during prediction via a gating function, which learns the limitation of the low-cost (student) model and consult the high-accuracy (teacher) model if necessary, thereby avoiding accuracy loss.Our composite system is also related to HME <cit.>, which learns the composite system based on max-likelihood estimation of models. A major difference is that HME does not address budget constraints. A fundamental aspect of budget constraints is the resulting asymmetry, whereby, we start with an HPC model and sequentially approximate with LPCs. This asymmetry leads us to propose a bottom-up strategy where the high-accuracy predictor can be separately estimated and is critical to posing a direct empirical loss minimization problem.§ PROBLEM SETUPWe consider the standard learning scenario of resource constrained prediction with feature costs. A training sample S={(x^(i),y^(i)):i=1,…,N} is generated i.i.d. from an unknown distribution,where x^(i)∈^K is the feature vector with an acquisition cost c_α≥ 0 assigned to each of the features α=1,…,K and y^(i) is the label for the i^ example. In the case of multi-class classification y ∈{1,…,M}, where M is the number of classes. Let us consider a single stage of our training method in order to formalize our setup. The model, f_0, is a high prediction-cost (HPC) model, which is either a priori known, or which we train to high-accuracy regardless of cost considerations. We would like to learn an alternative low prediction-cost (LPC) model f_1.Given an example x, at test-time, we have the option of selecting which model, f_0 or f_1, to utilize to make a prediction.The accuracy of a prediction model f_z is modeled by a loss function ℓ(f_z(x),y),z∈{0,1}. We exclusively employ the logistic loss function inbinary classification: ℓ(f_z(x),y)=log(1+exp(-yf_z(x)), although our framework allows other loss models. For a given x, we assume that once it pays the cost to acquire a feature, its value can be efficiently cached; its subsequent use does not incur additional cost. Thus, the cost of utilizing a particular prediction model, denoted by c(f_z,x), is computed as the sum of the acquisition cost of unique features required by f_z.Oracle Gating: Consider a general gating likelihood function q(z|x) with z ∈{0, 1}, that outputs the likelihood of sending the input x toa prediction model, f_z.The overall empirical loss is:𝔼_S_n𝔼_q(z | x) [ℓ(f_z(x),y)] =𝔼_S_n [ℓ(f_0(x),y)]+ 𝔼_S_n[q(1 | x) (ℓ(f_1(x),y) - ℓ(f_0(x),y))]_ExcessLossThe first term only depends on f_0, and from our perspective a constant. Similar to average loss we can write the average cost as (assuming gating cost is negligible for now):𝔼_S_n𝔼_q(z|x) [c(f_z,x)] = 𝔼_S_n [c(f_0,x)]-𝔼_S_n [q(1|x)(c(f_0,x)-c(f_1,x))_CostReduction],where the first term is again constant. We can characterize the optimal gating function (see <cit.>) that minimizes the overall average loss subject to average cost constraint: ℓ(f_1,x)-ℓ(f_0,x)^Excesslossq(1|x)=0q(1|x)=1η(c(f_0,x)-c(f_1,x))^Costreductionfor a suitable choice η∈ℝ. This characterization encodes the important principle that if the marginal cost reduction is smaller than the excess loss, we opt for the HPC model. Nevertheless, this characterization is generally infeasible. Note that the LHS depends on knowing how well HPC performs on the input instance. Since this information is unavailable, this target can be unreachable with low-cost gating. Gating Approximation: Rather than directly enforcing a low-cost structure on q, we decouple the constraint and introduce a parameterized family of gating functions g ∈ G that attempts to mimic (or approximate) q. To ensure such approximation, we can minimize some distance measure D(q(· |x),g(x)).A natural choice for an approximation metric is the Kullback-Leibler (KL) divergence although other choices are possible. The KL divergence between q and g is given by D_KL(q(· |x)g(x)) = ∑_z q(z |x) log(q(z |x)/σ((0.5-z)g(x))), where σ(s)=1/(1+e^-s) is the sigmoid function. Besides KL divergence, we have also proposed another symmetrized metric fitting g directly to the log odds ratio of q. See Suppl. Material for details.Budget Constraint: With the gating function g, the cost of predicting x depends on whether the example is sent to f_0 or f_1. Let c(f_0, g,x) denote the feature cost of passing x to f_0 through g. As discussed, this is equal to the sum of the acquisition cost of unique features required by f_0 and g for x. Similarly c(f_1, g,x) denotes the cost if x is sent to f_1 through g. In many cases the cost c(f_z,g,x) is independent of the example x and depends primarily on the model being used. This is true for linear models where each x must be processed through the same collection of features. For these cases c(f_z,g,x) ≜ c(f_z,g). The total budget simplifies to: 𝔼_S_n[q(0|x)]c(f_0,g)+(1-𝔼_S_n[q(0|x)])c(f_1,g)=c(f_1,g)+𝔼_S_n[q(0|x)] (c(f_0,g)-c(f_1,g)).The budget thus depends on 3 quantities: 𝔼_S_n[q(0|x)], c(f_1,g) and c(f_0,g). Often f_0 is a high-cost model that requires most, if not all, of features so c(f_0,g) can be considered a large constant.Thus, to meet the budget constraint, we would like to have (a) low-cost g and f_1 (small c(f_1,g)); and (b) small fraction of examples being sent to the high-accuracy model (small 𝔼_S_n[q(0|x)]). We can therefore split the budget constraint into two separate objectives: (a) ensure low-cost through penalty Ω(f_1,g)=γ∑_αc_αV_α+W_α_0, where γ is a tradeoff parameter and the indicator variables V_α,W_α∈{0,1} denote whether or not the feature α is required by f_1 and g, respectively. Depending on the model parameterization, we can approximate Ω(f_1,g) using a group-sparse norm or in a stage-wise manner as we will see in Algorithms <ref> and <ref>. (b)Ensure only P_full fraction of examples are sent to f_0 via the constraint 𝔼_S_n[q(0|x)] ≤ P_full.Putting Together: We are now ready to pose our general optimization problem:OPTmin_f_1 ∈ F, g ∈ G, q 𝔼_S_n∑_z[ q(z|x)ℓ(f_z(x),y)]^Losses + D(q(·|x),g(x))^GatingApprox +Ω(f_1,g)^FeatureCosts subject to:𝔼_S_n[q(0|x)] ≤ P_full.(Fractiontof_0)The objective function penalizes excess loss and ensures through the second term that this excess loss can be enforced through admissible gating functions. The third term penalizes the feature cost usage of f_1 and g. The budget constraint limits the fraction of examples sent to the costly model f_0.Remark 1:We presented the case for a single stage approximation system. However, it is straightforward to recursively continue this process. We can then view the composite system f_0 ≜ (g,f_1, f_0) as a black-box predictor and train a new pair of gating and prediction models to approximate the composite system. Remark 2:To limit the scope of our paper, we focus on reducing feature acquisition cost during prediction as it is a more challenging (combinatorial) problem. However, other prediction-time costs such as computation cost can be encoded in the choice of functional classes F and G in (<ref>). Surrogate Upper Bound of Composite System:We can get better insight for the first two terms of the objective in (<ref>) if we view z ∈{0,1} as a latent variable and consider the composite system (y|x) = ∑_z (z|x;g) (y|x,f_z). A standard application of Jensen's inequality reveals that,-log((y|x)) ≤𝔼_q(z|x)ℓ(f_z(x),y) + D_KL(q(z|x)(z|x;g)). Therefore, the conditional-entropy of the composite system is bounded by the expected value of our loss function (we overload notation and represent random-variables in lower-case format):H(y | x) ≜𝔼[-log((y|x))] ≤𝔼_x× y[𝔼_q(z|x)ℓ(f_z(x),y) + D_KL(q(z|x)(z|x;g))]. This implies that the first two terms of our objective attempt to bound the loss of the composite system; the third term in the objective together with the constraint serve to enforce budget limits on the composite system.Group Sparsity: Since the cost for feature re-use is zero we encourage feature re-use among gating and prediction models. So the fundamental question here is: How to choose a common, sparse (low-cost) subset of features on which both g and f_1 operate, such that g can effective gate examples between f_1 and f_0 for accurate prediction? This is a hard combinatorial problem. The main contribution of our paper is to address it using the general optimization framework of (<ref>). § ALGORITHMSTo be concrete, we instantiate our general framework (<ref>) into two algorithms via different parameterizations of g,f_1: Adapt-lin for the linear class and Adapt-Gbrt for the non-parametric class. Both of them use the KL-divergence as distance measure. We also provide a third algorithm Adapt-Lstsq that uses the symmetrized distance in the Suppl. Material. All of the algorithms perform alternating minimization of (<ref>) over q,g,f_1.Note that convergence of alternating minimization follows as in <cit.>. Common to all of our algorithms, we use two parameters to control cost: P_full and γ. In practice they are swept to generate various cost-accuracy tradeoffs and we choose the best one satisfying the budget B using validation data.Adapt-lin: Let g(x) = g^Tx and f_1(x)=f_1^Tx be linear classifiers. A feature is used if the corresponding component is non-zero: V_α=1 if f_1,α≠ 0, and W_α=1 if g_α≠ 0.r0.5 .49 The minimization for q solves the following problem:[OPT1min_q 2l1/N∑_i=1^N [(1-q_i)A_i+q_iB_i - H(q_i) ]; s.t.1/N∑_i=1^N q_i ≤P_full, ]where we have used shorthand notations q_i = q(z=0|x^(i)), H(q_i)=-q_ilog(q_i)-(1-q_i)log(1-q_i), A_i=log(1+e^-y^(i)f_1^Tx^(i))+log(1+e^g^Tx^(i)) and B_i=-log p(y^(i)|z^(i)=0;f_0)+log(1+e^-g^Tx^(i)). This optimization has a closed form solution: q_i = 1/(1+e^B_i-A_i+β) for some non-negative constant β such that the constraint is satisfied. This optimization is also known as I-Projection in information geometry because of the entropy term <cit.>. Having optimized q, we hold it constant and minimize with respect to g,f_1 by solving the problem (<ref>), where we have relaxed the non-convex cost ∑_αc_αV_α+W_α_0 into a L_2,1 norm for group sparsity and a tradeoff parameter γ to make sure the feature budget is satisfied. Once we solve for g,f_1, we can hold them constant and minimize with respect to q again. Adapt-Lin is summarized in Algorithm <ref>. OPT2 min_g,f_11/N∑_i=1^N [(1-q_i)(log(1+e^-y^(i)f_1^Tx^(i))+log(1+e^g^Tx^(i)))+q_i log(1+e^-g^Tx^(i)) ] + γ∑_α√(g_α^2+f_1,α^2). Adapt-Gbrt: We can also consider the non-parametric family of classifiers such as gradient boosted trees <cit.>: g(x) = ∑_t=1^T g^t(x) and f_1(x)=∑_t=1^T f_1^t(x), where g^t and f_1^t are limited-depth regression trees. Since the trees are limited to low depth, we assume that the feature utility of each tree is example-independent: V_α,t(x)≊ V_α,t, W_α,t(x)≊ W_α,t,∀ x. V_α,t=1 if feature α appears in f_1^t, otherwise V_α,t=0, similarly for W_α,t. The optimization over q still solves (<ref>). We modify A_i=log(1+e^-y^(i)f_1(x^(i)))+log(1+e^g(x^(i))) and B_i=-log p(y^(i)|z^(i)=0;f_0)+log(1+e^-g(x^(i))). Next, to minimize over g,f_1, denote loss:ℓ(f_1,g) =1/N∑_i=1^N[(1-q_i)· (log(1+e^-y^(i)f_1(x^(i)))+log(1+e^g(x^(i))))+q_i log(1+e^-g(x^(i)))],which is essentially the same as the first part of the objective in (<ref>).Thus, we need to minimize ℓ(f_1,g)+Ω(f_1,g) with respect to f_1 and g. Since both f_1 and g are gradient boosted trees, we naturally adopt a stage-wise approximation for the objective. In particular, we define an impurity function which on the one hand approximates the negative gradient of ℓ(f_1,g) with the squared loss, and on the other hand penalizes the initial acquisition of features by their cost c_α.To capture the initial acquisition penalty, we let u_α∈{0,1} indicates if feature α has already been used in previous trees (u_α=0), or not (u_α=1). u_α is updated after adding each tree. Thus we arrive at the following impurity for f_1 and g, respectively:1/2∑_i=1^N(-∂ℓ(f_1,g)/∂ f_1(x^(i)) - f_1^t(x^(i)))^2+γ∑_α u_α c_α V_α,t, 1/2∑_i=1^N(-∂ℓ(f_1,g)/∂ g(x^(i)) - g^t(x^(i)))^2+γ∑_α u_α c_α W_α,t.Minimizing such impurity functions balances the need to minimize loss and re-using the already acquired features. Classification and Regression Tree (CART) <cit.> can be used to construct decision trees with such an impurity function. Adapt-GBRT is summarized in Algorithm <ref>.Note that a similar impurity is used in GreedyMiser <cit.>. Interestingly, if P_full is set to 0, all the examples are forced to f_1, then Adapt-Gbrt exactly recovers the GreedyMiser. In this sense, GreedyMiser is a special case of our algorithm. As we will see in the next section, thanks to the bottom-up approach, Adapt-Gbrt benefits from high-accuracy initialization and is able to perform accuracy-cost tradeoff in accuracy levels beyond what is possible for GreedyMiser. § EXPERIMENTSBaseline Algorithms: We consider the following simple L1 baseline approach for learning f_1 and g: first perform a L1-regularized logistic regression on all data to identify a relevant, sparse subset of features; then learn f_1 using training data restricted to the identified feature(s); finally, learn g based on the correctness of f_1 predictions as pseudo labels (i.e. assign pseudo label 1 to example x if f_1(x) agrees with the true label y and 0 otherwise). We also compare with two state-of-the-art feature-budgeted algorithms: GreedyMiser<cit.> - a top-down method that builds out an ensemble of gradient boosted trees with feature cost budget;and BudgetPrune<cit.> - a bottom-up method that prunes a random forest with feature cost budget. A number of other methods such as ASTC <cit.> and CSTC <cit.> are omitted as they have been shown to under-perform GreedyMiser on the same set of datasets <cit.>. Detailed experiment setups can be found in the Suppl. Material.We first visualize/verify the adaptive approximation ability of Adapt-Lin and Adapt-Gbrt on the Synthetic-1 dataset without feature costs. Next, we illustrate the key difference between Adapt-Lin and the L1 baseline approach on the Synthetic-2 as well as the Letters datasets. Finally, we compare Adapt-Gbrt with state-of-the-art methods on several resource constraint benchmark datasets. Power of Adaptation: We construct a 2D binary classification dataset (Synthetic-1) as shown in (a) of Figure <ref>. We learn an RBF-SVM as the high-accuracy classifier f_0 as in (d).To better visualize the adaptive approximation process in 2D, we turn off the feature costs (i.e. set Ω(f_1,g) to 0 in (<ref>)) and run Adapt-Lin and Adapt-Gbrt. The initializations of g and f_1 in (b) results in wrong predictions for many red points in the blue region. After 10 iterations of Adapt-Lin, f_1 adapts much better to the local region assigned by g while g sends about 60% (P_full) of examples to f_0. Similarly, the initialization in (e) results in wrong predictions in the blue region. Adapt-Gbrt is able to identify the ambiguous region in the center and send those examples to f_0 via g. Both of our algorithms maintain the same level of prediction accuracy as f_0 yet are able to classify large fractions of examples via much simpler models.r0.4< g r a p h i c s >A 2-D synthetic example for adaptive feature acquisition. On the left: data distributed in four clusters. The two features correspond to x and y coordinates, respectively. On the right: accuracy-cost tradeoff curves. Our algorithm can recover the optimal adaptive system whereas a L1-based approach cannot. Power of Joint Optimization: We return to the problem of prediction under feature budget constrains.We illustrate why a simple L1 baseline approach for learning f_1 and g would not work using a 2D dataset (Synthetic-2) as shown in Figure <ref> (left). The data points are distributed in four clusters, with black triangles and red circles representing two class labels. Let both feature 1 and 2 carry unit acquisition cost. A complex classifier f_0 that acquires both features can achieve full accuracy at the cost of 2. In our synthetic example, clusters 1 and 2 are given more data points so that the L1-regularized logistic regression would produce the vertical red dashed line, separating cluster 1 from the others. So feature 1 is acquired for both g and f_1. The best such an adaptive system can do is to send cluster 1 to f_1 and the other three clusters to the complex classifier f_0, incurring an average cost of 1.75, which is sub-optimal. Adapt-Lin, on the other hand, optimizing between q,g,f_1 in an alternating manner, is able to recover the horizontal lines in Figure <ref> (left) for g and f_1. g sends the first two clusters to the full classifier and the last two clusters to f_1. f_1 correctly classifies clusters 3 and 4. So all of the examples are correctly classified by the adaptive system; yet only feature 2 needs to be acquired for cluster 3 and 4 so the overall average feature cost is 1.5, as shown by the solid curve in the accuracy-cost tradeoff plot on the right of Figure <ref>.This example shows that the L1 baseline approach is sub-optimal as it doesnot optimize the selection of feature subsets jointly for g and f_1.r7cmDataset Statistics0.5!Dataset #Train #Validation #Test #Features Feature Costs Letters 12000 4000 4000 16 Uniform MiniBooNE 45523 195106503150 Uniform Forest36603 156885810154 Uniform CIFAR10 19761 8468 10000400Uniform Yahoo!141397146769 184968 519CPU unitsReal Datasets: We test various aspects of our algorithms and compare with state-of-the-art feature-budgeted algorithms on five real world benchmark datasets: Letters, MiniBooNE Particle Identification, Forest Covertype datasets from the UCI repository <cit.>, CIFAR-10 <cit.> and Yahoo! Learning to Rank<cit.>. Yahoo! is a ranking dataset where each example is associated with features of a query-document pair together with the relevance rank of the document to the query. There are 519 such features in total; each is associated with an acquisition cost in the set {1,5,20,50,100,150,200}, which represents the units of CPU time required to extract the feature and is provided by a Yahoo! employee. The labels are binarized into relevant or not relevant. The task is to learn a model that takes a new query and its associated documents and produce a relevance ranking so that the relevant documents come on top, and to do this using as little feature cost as possible. The performance metric is Average Precision @ 5 following <cit.>. The other datasets have unknown feature costs so we assign costs to be 1 for all features; the aim is to show Adapt-Gbrt successfully selects sparse subset of “usefull” features for f_1 and g. We summarize the statistics of these datasets in Table <ref>. Next, we highlight the key insights from the real dataset experiments.Generality of Approximation: Our framework allows approximation of powerful classifiers such as RBF-SVM and Random Forests as shown in Figure <ref> as red and black curves, respectively. In particular, Adapt-Gbrt can well maintain high accuracy while reducing cost. This is a key advantage for our algorithms because we can choose to approximate the f_0 that achieves the best accuracy.Adapt-Lin Vs L1: Figure <ref> shows that Adapt-Lin outperforms L1 baseline method on real dataset as well. Again, this confirms the intuition we have in the Synthetic-2 example as Adapt-Lin is able to iteratively select the common subset of features jointly for g and f_1. r0.37< g r a p h i c s >Compare the L1 baseline approach, Adapt-Lin and Adapt-Gbrt based on RBF-SVM and RF as f_0's on the Letters dataset. Adapt-Gbrt Vs Adapt-Lin: Adapt-Gbrt leads to significantly better performance than Adapt-Lin in approximating both RBF-SVM and RF as shown in Figure <ref>. This is expected as the non-parametric non-linear classifiers are much more powerful than linear ones.Adapt-Gbrt Vs BudgetPrune: Both are bottom-up approaches that benefit from good initializations. In (a), (b) and (c) of Figure <ref> we let f_0 in Adapt-Gbrt be the same RF that BudgetPrune starts with. Adapt-Gbrt is able to maintain high accuracy longer as the budget decreases. Thus, Adapt-Gbrt improves state-of-the-art bottom-up method. Notice in (c) of Figure <ref> around the cost of 100, BudgetPrune has a spike in precision. We believe this is because the initial pruning improved the generalization performance of RF. But in the cost region of 40-80, Adapt-Gbrt maintains much better accuracy than BudgetPrune. Furthermore, Adapt-Gbrt has the freedom to approximate the best f_0 given the problem. So in (d) of Figure <ref> we see that with f_0 being RBF-SVM, Adapt-Gbrt can achieve much higher accuracy than BudgetPrune.Adapt-Gbrt Vs GreedyMiser: Adapt-Gbrt outperforms GreedyMiser on all the datasets. The gaps in Figure <ref>, (b) (c) and (d) of Figure <ref> are especially significant. Significant Cost Reduction: Without sacrificing top accuracies (within 1%), Adapt-Gbrt reduces average feature costs during test-time by around 63%, 32%, 58%, 12% and 31% on MiniBooNE, Forest, Yahoo, Cifar10 and Letters datasets, respectively. § CONCLUSIONSWe presentedan adaptive approximation approach to account for feature acquisition costs that arise in various applications.At test-time our method uses a gating function to identify a prediction model among a collection of models that is adapted to the input. The overall goal is to reduce costs without sacrificing accuracy. We learn gating and prediction models by means of a bottom-up strategy that trains low prediction-cost models to approximate high prediction-cost models in regions where low-cost models suffice. On a number of benchmark datasets our method leads to an average of 40% cost reduction without sacrificing test accuracy (within 1%). It outperforms state-of-the-art top-down and bottom-up budgeted learning algorithms, with a significant margin in several cases.§.§.§ Acknowledgments Feng Nan would like to thank Dr Ofer Dekel for ideas and discussions on resource constrained machine learning during an internship in Microsoft Research in summer 2016. Familiarity and intuition gained during the internship contributed to the motivation and formulation in this paper. We also thank Dr Joseph Wang and Tolga Bolukbasi for discussions and helps in experiments. plain§ APPENDIX§.§ Adapt-Lstsq Other Symmetrized metrics: KL divergence is not symmetric and leads to widely different properties in terms of approximation. We also consider a symmetrized metric: D(r(z),s(z)) =(logr(0)/r(1) - logs(0)/s(1) )^2This metric can be viewed intuitively as a regression of g(x)=log((1|g;x)/(0|g;x) against the observed log odds ratio of q(z|x).The main advantage of using KL is that optimizing w.r.t. q can be solved in closed form. The disadvantage we observe is that in some cases, the loss for minimizing w.r.t. g, which is a weighted sum of log-losses of opposing directions, becomes quite flat and difficult to optimize especially for linear gating functions. The symmetrized measure, on the other hand, makes the optimization w.r.t. g better conditioned as the gating function g fits directly to the log odds ratio of q. However, the disadvantage of using the symmetrized measure is that optimizing w.r.t. q no longer has closed form solution; furthermore, it is even non-convex. We offer an ADMM approach for q optimization.We still follow an alternating minimization approach. To keep the presentation simply, we assume g, f_1 to be linear classifiers and there is no feature costs involved. To minimize over q, we must solve [ OPT5min_q_i∈ [0,1] 2l1/N∑_i=1^N [(1-q_i)A_i+(logq_i/1-q_i-g(x^(i)))^2 ]; s.t.1/N∑_i=1^N q_i ≤P_full, ]where q_i=q(z=0|x^(i)), A_i=log(1+e^-y^(i)f_1^Tx^(i))+log p(y^(i)|z^(i)=1;f_0). Unlike (OPT3), this optimization problem no longer has a closed-form solution. Fortunately, the q_i's in the objective are decoupled and there is only one coupling constraint. We can solve this problem using an ADMM approach <cit.>. To optimize over g, we simply need to solve a linear least squares problem:OPT6min_g1/N∑_i=1^N (logq_i/1-q_i-g^T(x^(i)))^2.To optimize over f_1, we solve a weighted logistic regression problem:OPT7min_f_11/N∑_i=1^N (1-q_i)log(1+e^-y^(i)f_1^Tx^(i)).We shall call the above algorithm Adapt-Lstsq, summarized in Algorithm <ref>.§.§ Experimental DetailsWe provide detailed parameter settings and steps for our experiments here. §.§ Synthetic-1 ExperimentWe generate the data in Python using the following command:For Adapt-Gbrt we used 5 depth-2 trees for g and f_1.§.§ Synthetic-2 Experiment:We generate 4 clusters on a 2D plane with centers: (1,1), (-1,1), (-1,-1), (-1, -3) and Gaussian noise with standard deviation of 0.01. The first two clusters have 20 examples each and the last two clusters have 15 examples each. We sweep the regularization parameter of L1-regularized logistic regression and recover feature 1 as the sparse subset, which leads to sub-optimal adaptive system. On the other hand, we can easily train a RBF SVM classifier to correctly classify all clusters and we use it as f_0. If we initialize g and f_1 with Gaussian distribution centered around 0, Adapt-Lin with can often recover feature 2 as the sparse subset and learn the correct g and f_1. Or, we could initialize g=(1,1) and f_1=(1,1) then Adapt-Lin can recover the optimal solution. §.§ Letters Dataset <cit.> This letters recognition dataset contains 20000 examples with 16 features, each of which is assigned unit cost. We binarized the labels so that the letters before "N" is class 0 and the letters after and including "N" are class 1.We split the examples 12000/4000/4000 for training/validation/test sets. We train RBF SVM and RF (500 trees) with cross-validation as f_0. RBF SVM achieves the higher accuracy of 0.978 compared to RF 0.961.To run the greedy algorithm, we first cross validate L1-regularized logistic regression with 20 C parameters in logspace of [1e-3,1e1]. For each C value, we obtain a classifier and we order the absolute values of its components and threshold them at different levels to recover all 16 possible supports (ranging from 1 feature to all 16 features). We save all such possible supports as we sweep C value. Then for each of the supports we have saved, we train a L2-regularized logistic regression only based on the support features with regularization set to 1 as f_1. The gating g is then learned using L2-regularized logistic regression based on the same feature support and pseudo labels of f_1 - 1 if it is correctly classified and 0 otherwise. To get different cost-accuracy tradeoff, we sweep the class weights between 0 and 1 so as to influence g to send different fractions of examples to the f_0. To run Adapt-Lin, we initialize g to be 0 and f_1 to be the output of the L2-regularized logistic regression based on all the features. We then perform the alternative minimization for 50 iterations and sweep γ between [1e-4,1e0] for 20 points and P_full in [0.1,0.9] for 9 points. To run Adapt-Gbrt, we use 500 depth 4 trees for g and f_1 each. Weinitialize g to be 0 and f_1 to be the GreedyMiser output of 500 trees. We then perform the alternative minimization for 30 iterations and sweep γ between [1e-1,1e2] for 10 points in logspace and P_full in [0.1,0.9] for 9 points. In addition, we also sweep the learning rate for GBRT for 9 points between [0.1,1].For fair comparison, we run GreedyMiser with 1000 depth 4 trees so that the model size matches that of Adapt-Gbrt. The learning rate is swept between [1e-5,1] with 20 points and the λ is swept between [0.1, 100] with 20 points.Finally, we evaluate all the resulting systems from the parameter sweeps of all the algorithms on validation data and choose the efficient frontier and use the corresponding settings to evaluate and plot the test performance.§.§ MiniBooNE Particle Identification and Forest Covertype Datasets <cit.>: The MiniBooNE data set is a binary classification task to distinguish electron neutrinos from muon neutrinos. There are 45523/19510/65031 examples in training/validation/test sets. Each example has 50 features, each with unit cost. The Forest data set contains cartographic variables to predict 7 forest cover types. There are 36603/15688/58101 examples in training/validation/test sets. Each example has 54 features, each with unit cost. We use the unpruned RF of BudgetPrune <cit.> as f_0 (40 trees for both datasets.) The settings for Adapt-Gbrt are the following.For MiniBooNE we use 100 depth 4 trees for g and f_1 each. We initialize g to be 0 and f_1 to be the GreedyMiser output of 100 trees. We then perform the alternative minimization for 50 iterations and sweep γ between [1e-1,1e2] for 20 points in logspace and P_full in [0.1,0.9] for 9 points. In addition, we also sweep the learning rate for GBRT for 9 points between [0.1,1]. For Forest we use 500 depth 4 trees for g and f_1 each. We initialize g to be 0 and f_1 to be the GreedyMiser output of 500 trees. We then perform the alternative minimization for 50 iterations and sweep γ between [1e-1,1e2] for 20 points in logspace and P_full in [0.1,0.9] for 9 points. In addition, we also sweep the learning rate for GBRT for 9 points between [0.1,1].For fair comparison, we run GreedyMiser with 200 depth 4 trees so that the model size matches that of Adapt-Gbrt for MiniBooNE. We run GreedyMiser with 1000 depth 4 trees so that the model size matches that of Adapt-Gbrt for Forest. Finally, we evaluate all the resulting systems from the parameter sweeps on validation data and choose the efficient frontier and use the corresponding settings to evaluate and plot the test performance.§.§ Yahoo! Learning to Rank<cit.>:This ranking dataset consists of 473134 web documents and 19944 queries. Each example is associated with features of a query-document pair together with the relevance rank of the document to the query. There are 519 such features in total; each is associated with an acquisition cost in the set {1,5,20,50,100,150,200}, which represents the units of CPU time required to extract the feature and is provided by a Yahoo! employee. The labels are binarized into relevant or not relevant. The task is to learn a model that takes a new query and its associated documents and produce a relevance ranking so that the relevant documents come on top, and to do this using as little feature cost as possible. The performance metric is Average Precision @ 5 following <cit.>.We use the unpruned RF of BudgetPrune <cit.> as f_0 (140 trees for both datasets.) The settings for Adapt-Gbrt are the following. we use 100 depth 4 trees for g and f_1 each. Weinitialize g to be 0 and f_1 to be the GreedyMiser output of 100 trees. We then perform the alternative minimization for 20 iterations and sweep γ between [1e-1,1e3] for 30 points in logspace and P_full in [0.1,0.9] for 9 points. In addition, we also sweep the learning rate for GBRT for 9 points between [0.1,1].For fair comparison, we run GreedyMiser with 200 depth 4 trees so that the model size matches that of Adapt-Gbrt for Yahoo. Finally, we evaluate all the resulting systems from the parameter sweeps on validation data and choose the efficient frontier and use the corresponding settings to evaluate and plot the test performance. §.§ CIFAR10 <cit.>: CIFAR-10 data set consists of 32x32 colour images in 10 classes.400 features for each image are extracted using technique described in <cit.>. The data are binarized by combining the first 5 classes into one class and the others into the second class. There are 19,761/8,468/10,000 examples in training/validation/test sets.BudgetPrune starts with a RF of 40 trees, which achieves an accuracy of 69%. We use an RBF-SVM as f_0 that achieves a test accuracy of 79.5%. The settings for Adapt-Gbrt are the following. we use 200 depth 5 trees for g and f_1 each. Weinitialize g to be 0 and f_1 to be the GreedyMiser output of 200 trees. We then perform the alternative minimization for 50 iterations and sweep γ between [1e-4,10] for 15 points in logspace and P_full in [0.1,0.9] for 9 points. In addition, we also sweep the learning rate for GBRT for 10 points between [0.01,1].For fair comparison, we run GreedyMiser with 400 depth 5 trees so that the model size matches that of Adapt-Gbrt. Finally, we evaluate all the resulting systems from the parameter sweeps on validation data and choose the efficient frontier and use the corresponding settings to evaluate and plot the test performance. | http://arxiv.org/abs/1705.10194v1 | {
"authors": [
"Feng Nan",
"Venkatesh Saligrama"
],
"categories": [
"stat.ML",
"cs.LG"
],
"primary_category": "stat.ML",
"published": "20170526122842",
"title": "Adaptive Classification for Prediction Under a Budget"
} |
Mining Process Model Descriptions of Daily Life through Event AbstractionNiek Tax Eindhoven University of Technology, [email protected] Natalia Sidorova Eindhoven University of Technology [email protected] Reinder Haakma Philips Research [email protected] Wil M.P. van der Aalst Eindhoven University of Technology [email protected]* N. Tax, N. Sidorova, R. Haakma, W.M.P. van der Aalst December 30, 2023 ========================================================Process mining techniques focus on extracting insight in processes from event logs. Process mining has the potential to provide valuable insights in (un)healthy habits and to contribute to ambient assisted living solutions when applied on data from smart home environments. However, events recorded in smart home environments are on the level of sensor triggers, at which process discovery algorithms produce overgeneralizing process models that allow for too much behavior and that are difficult to interpret for human experts. We show that abstracting the events to a higher-level interpretation can enable discovery of more precise and more comprehensible models. We present a framework for the extraction of features that can be used for abstraction with supervised learning methods that is based on the XES IEEE standard for event logs. This framework can automatically abstract sensor-level events to their interpretation at the human activity level, after training it on training data for which both the sensor and human activity events are known. We demonstrate our abstraction framework on three real-life smart home event logs and show that the process models that can be discovered after abstraction are more precise indeed.§ INTRODUCTIONProcess mining is a fast growing discipline that combines methods from computational intelligence, data mining, process modeling and process analysis <cit.>. Process discovery, the task of extracting process models from logs, plays an important role in process mining. There are many different process discovery algorithms (<cit.>), which can discover many different types of process models, including BPMN models, Petri nets, process trees, UML activity diagrams, and statecharts. While originally the scope of process mining has been on business processes, it has broadened in recent years towards other application areas, including the analysis of human behavior <cit.>. Process model descriptions of human behavior can be used amongst others to aid lifestyle coaching for healthy living, or to asses the ability of independent living of elderly or people with illness.Events in the event log are generated by e.g. motion sensors placed in the home, power sensors placed on appliances, open/close sensors placed on closets and cabinets, etc. This clearly distinguishes process mining for smart homes from the traditional application domain of business processes, where events in the log are logged by IT systems when an business tasks are performed. In event logs from business processes the event labels generally have a clear semantic meaning, like register mortgage request. In the smart home domain the events are on the sensor level, while the human expert is interested in analyzing the behavior in terms of activities of daily life. Additionally, simply using the sensor that generated the event as the event label has been shown to result in non-informative process models that overgeneralize the event log and allow for too much behavior <cit.>. In the field of process mining such overgeneralizing process models are generally referred to as being imprecise.In our earlier work <cit.> we showed how to discover more precise process models by taking the name of the sensor as a starting point for the event label and then refine the labels using the time of the day at which the event occurred. However, labels in such process models still represent sensors, and they have no direct interpretation on the human activity level. In this paper we leverage diary style annotations of the activities performed on a human activity level and use them learn a mapping from sensor-level events to human activity events. This enables discovery of process models that describe the human activities directly, leading to more comprehensible and more precise descriptions of human behavior. Often it is infeasible or simply too expensive to obtain such diaries for periods of time longer than a couple of weeks. To mine a process model of human behavior more than a couple of weeks of data is needed. Therefore, there is a need to infer human level interpretations of behavior from sensors.With supervised learning techniques the mapping from sensor-level events to human activity level events can be learned through examples, without requiring a hand-made ontology of how human activities relate to sensors. Similar approaches have been explored in the activity recognition field, where continuous-valued time series from sensors are mapped to time series of human activity. Change points in these time series are triggered by sensor-level events like opening/closing the fridge door, and the annotations of the higher level events (e.g. cooking) are often obtained through manual activity diaries. However, in contrast to techniques from the activity recognition field, we operate on discrete events on the sensor-level instead of continuous time series.In this paper we extend the work started in <cit.>. We describe a framework for supervised abstraction of events that enables the discovery of more precise process models from smart home event logs. Additionally, the process models obtained represent human activity directly, thereby enabling direct analysis of human behavior itself, instead of indirect analysis through sensor-level models. In Section <ref> we give an overview of the related work from the activity recognition field. Basic concepts, notations, and definitions that we use throughout the rest of the paper are introduced in Section <ref>. In Section <ref> we explain conceptually why abstraction from sensor-level to human activity level events can help to the process discovery step to find more precise process models. In Section <ref> we describe a framework for retrieving useful features for abstraction from event logs using specific concepts of the IEEE XES standard for event logs <cit.>. Section <ref> demonstrates the added value of supervised event abstraction for process mining in the smart home domain and show that it enables discovery of more precise models on three real life smart home event logs. Section <ref> concludes the paper and identifies some areas of future work.§ RELATED WORK Event abstraction based on supervised learning is an unexplored problem in process mining. Most related work for abstracting from sensor-level to human activity level events can be found in the field of activity recognition, which focuses on the task of detecting different types of human activity from either passive sensors <cit.>, wearable sensors <cit.>, or cameras <cit.>.Activity recognition methods generally operate on discrete time windows over the time series of continuous-valued sensor values and aim to map each time window onto the correct type of human activity, e.g. eating or sleeping. Activity recognition methods can be classified into probabilistic approaches <cit.> and ontological reasoning approaches <cit.>. The advantage of probabilistic approaches over ontological reasoning approaches is their ability to handle noisy, uncertain and incomplete sensor data <cit.>.Tapia <cit.> was the first to explore supervised learning for inference of human activities from passive sensors, using a naive Bayes classifier. Many more recent activity recognition approaches use probabilistic graphical models <cit.>: Van Kasteren et al. <cit.> explored Conditional Random Fields <cit.> and Hidden Markov Models <cit.>, and Van Kasteren and Kröse <cit.> applied Bayesian Networks <cit.> on the activity recognition task. Kim et al. <cit.> found Hidden Markov Models to be unable to capture long-range or transitive dependencies between observations, which results in difficulties recognizing multiple interacting activities (concurrent or interwoven). Conditional Random Fields do not possess these limitations.Our work differentiates itself from existing activity recognition work in the form of the input data on which they operate and in the goal that it aims to achieve. On the input side, activity recognition techniques consider the data to be a multidimensional time series of the sensor values over time, based on which time windows are mapped onto human activities. An appropriate time window size is determined using domain knowledge of the data set. Instead, we aim for a generic method that does not require this domain knowledge, and that works in general for any event log. An approach based on time windows contrasts with our aim for generality, as no single time window size exists that is suitable for all event logs. The durations of the events within a single event log might differ drastically (e.g. one event might take seconds, while another event takes months), in which case time window based approaches will either miss short events in case of larger time windows or resort to very large numbers of time windows resulting in very long computational time. Therefore, we map each individual sensor-level event to a human activity level event and do not use time windows. In a smart home environment context with passive sensors, each change in a binary sensor value can be considered to be a low-level event. A second difference with existing activity recognition techniques is that our framework aims to find an abstraction of the data that enables discovery of more precise process models, where classical activity recognition methods do not have a link with the application of process mining.Other related work can be found in the area of process mining, where several techniques address the challenge of abstracting low-level (e.g. sensor-level) events to high level (e.g. human activity level) events (<cit.>). Most existing event abstraction methods rely on clustering methods, where each cluster of low-level events is interpreted as one single high-level event. However, using unsupervised learning introduces two new problems. First, it is unclear how to label high-level events that are obtained by clustering low-level events. Current techniques require the user / process analyst to provide high-level event labels themselves based on domain knowledge. Secondly, unsupervised learning gives no guidance with respect to the desired level of abstraction. Many existing event abstraction methods contain one or more parameters to control the size the clusters, and finding the right level of abstraction providing meaningful results is often a matter of trial-and-error.One abstraction technique from the process mining field that does not rely on unsupervised learning is proposed by Mannhardt et al. <cit.>. This approach relies on domain knowledge to abstract low-level events into high-level events, requiring the user to specify a low-level process model for each high-level activity. However, in the context of human behavior it is unreasonable to expect the user to provide the process model in sensor terms for each human activity from domain knowledge. § PRELIMINARIES In this section we introduce basic concepts and notation used throughout the paper.We use the standard sequence definition, and denote a sequence by listing its elements, e.g. we write ⟨ a_1,a_2,…,a_n⟩ for a (finite) sequence s:{1,…,n}→ S of elements from some alphabet S, where s(i)=a_i for any i ∈{1,…,n}. §.§ Petri netsA process modeling notation that is commonly used in process mining is the Petri net. Petri nets are directed bipartite graphs consisting of transitions and places, connected by arcs. Transitions represent activities, while places represent the state of the system before and after execution of a transition. Labels are assigned to transitions to indicate the type of activity that they represent. A special label τ is used to represent invisible transitions, which are only used for routing purposes and do not represent any real activity. A labeled Petri net is a tuple N=(P,T,F,R,ℓ) where P is a finite set of places, T is a finite set of transitions such that P ∩ T = ∅, and F ⊆ (P × T) ∪ (T × P) is a set of directed arcs, called the flow relation, R is a finite set of labels representing event types, with τ∉ R is a label representing an invisible action, and ℓ:T→ R∪{τ} is a labeling function that assigns a label to each transition. The state of a Petri net is defined w.r.t. the state that a process instance can be in during its execution. A state of a Petri net is captured by the marking of its places with tokens. In a given state, each place is either empty, or it contains a certain number of tokens. A transition is enabled in a given marking if all places with an outgoing arc to this transition contain at least one token. Once a transition fires (i.e. is executed), a token is removed from all places with outgoing arcs to the firing transition and a token is put to all places with incoming arcs from the firing transition, leading to a new state. A marked Petri net is a pair (N,M), where N=(P,T,F,L,ℓ) is a labeled Petri net and M ∈ℕ^P denotes the marking of N. For n ∈ (P ∪ T) we use ∙ n and n ∙ to denote the set of inputs and outputs of n respectively. Let C(s,e) indicate the number of occurrences (count) of element e in multiset s. A transition t∈ T is enabled in a marking M of net N if ∀ p ∈∙ t : C(M,p)>0. An enabled transition t may fire, removing one token from each of the input places ∙ t and producing one token for each of the output places t∙.Figure <ref> shows four Petri nets, with the circles representing places, the squares representing transitions. The gray rectangles represent (invisible) τ-transitions. Places depicted as [node distance=1.4cm, on grid,>=stealth', bend angle=20, auto, every place/.style= minimum size=3mm, ][place,pattern=custom north west lines,hatchspread=1.5pt,hatchthickness=0.25pt,hatchcolor=gray] ; belong to the final marking, indicating that the process execution can terminate in this marking.The Petri net shown in Figure <ref> initially has one token in the place p1, indicated by the dot. Firing the enabled silent transition takes the token from p1 and puts a token in both p2 and p3, enabling both MC and DCC. When MC fires, it takes the token from p2 and puts a token in p4. When DCC fires, it takes the token from p3 and puts a token in p5. After MC and DCC have both fired, resulting in a token in both p4 and p5, W is enabled. Executing W takes the token from both p4 and p5, and puts a token in p6, which is a place that belongs to the final marking, indicates that the process execution can stop here. Alternatively, it can fire the silent transition, taking the token from p6 and placing a token in p2 and p5, which allows for execution of MC and W to reach the marking consisting of p6 again. We refer the interested reader to <cit.> for an extensive review of Petri nets. §.§ Conditional Random Field We consider the recognition of human activity level events as a sequence labeling task in which each sensor-level event is classified into one of the human activity level events. Linear-chain Conditional Random Fields (CRFs) <cit.> are a type of probabilistic graphical model which has shown to perform well on many sequence labeling tasks in the fields of language processing and computer vision. Conceptually CRFs can be regarded as a sequential version of multiclass logistic regression, i.e., the predictions in the prediction sequence are dependent on each other. A CRF models the conditional probability distribution of the label sequence given an observation sequence using a log-linear model. Linear-chain CRFs take the following form: p(y|x) = 1/Z(x)exp(∑_t=1∑_kλ_k f_k(t,y_t-1,y_t,x)) where Z(x) is the normalization factor which makes sure that the values of the probability distribution range from zero to one. X=⟨ x_1,…,x_n⟩ is an observation sequence (the sensor-level events), Y=⟨ y_1,…,y_n⟩ is the associated label sequence (the human activity level events), f_k and λ_k respectively are feature functions and their weights. Feature functions, which can be binary or real valued, are defined on the observations and are used to compute label probabilities. In contrast to Hidden Markov Models <cit.>, CRFs do not assume the feature functions to be mutually independent.§ MOTIVATING EXAMPLEFigure <ref> shows a simplistic example demonstrating how a process can seem unstructured at the sensor level of events, while being structured at a human behavior level. Petri net <ref> shows the process at the human activity level. The Taking medicine human activity level activity is itself defined as a process, which is shown in Figures <ref> to <ref>. Eating is also defined as a process, which is shown in Figure <ref>.When we apply the Inductive Miner process discovery algorithm <cit.> to sensor-level traces generated by the hierarchical process of Figure <ref>, we obtain the process model shown in Figure <ref>. This process model allows for almost all possible sequences over the alphabet {CD,D,DCC,MC,W}, with the only constraint introduced by the model being that if a W occurs, then it has to be preceded by a DCC event. Firing of all other transitions in the model can be skipped. Behaviorally this model is very close to the so called "flower" model <cit.>, the model that allows for all behavior over its alphabet. The alternating structure between Taking medicine and Eating that was present in the human activity level process in Figure <ref> cannot be observed in the process model in Figure <ref>. This is caused by high variance in start and end events of the sensor-level subprocesses of Taking medicine and Eating as well as by the overlap in types of activities between these two subprocesses. Both subprocesses contain DCC, and the miner cannot see that there are actually two different contexts for the DCC activity to split the label in the model. Abstracting the sensor-level events to their respective human activity level events before applying process discovery to the resulting human activity log unveils the alternating structure between Eating and Taking medicine as shown in Figure <ref>.§ EVENT ABSTRACTION AS A SEQUENCE LABELING TASK In this section we describe the framework for supervised abstraction of events based on Conditional Random Fields (CRFs). Additionally, we describe feature functions for event logs in a general way by using the IEEE XES standard <cit.>. XES, which is an abbreviation for eXtensible Event Stream, is the IEEE standard for process mining event logs. An overview of the XES file structure which is shown in Figure <ref>. An event log is defined as a set of traces, which in itself are a sequences of events. The log, traces and events can all contain one or more attributes, which consist of a key and a value of a certain type. Event or trace attributes may be global, which indicates that the attribute needs to be defined for each event or trace respectively. A log contains one or more classifiers, which can be seen as labeling functions on the events of a log, defined on global event attributes. Extensions define a set of attributes on log, trace, or event level, in such a way that the semantics of these attributes are clearly defined. One can view XES extensions as a specification of attributes that events, traces, or event logs themselves frequently contain. XES defines the following standard extensions: Concept Specifies the generally understood name of the event/trace/log (attribute 'Concept:name'). Lifecycle Specifies the lifecycle phase (attribute 'Lifecycle:transition') that the event represents in a transactional model of their generating activity. The Lifecycle extension also specifies a standard transactional model for activities. OrganizationalSpecifies three attributes for events, which identify the actor having caused the event (attribute 'Organizational:resource'), his role in the organization (attribute 'Organizational:role'), and the group or department within the organization where he is located (attribute 'Organizational:group'). TimeSpecifies the date and time at which an event occurred (attribute 'Time:timestamp'). We introduce a special attribute of type String with key label, which represents the human activity level activity. The concept name of an event is then considered to be a sensor-level name of an event. The label attribute specifies the human activity level label for each event individually, allowing for example one sensor-level event of type Dishes & cups cabinet to be of human activity level type Taking medicine, and another sensor-level event of the same type to be of human level activity type Eating. Note that for some traces human level activity annotations might be available, in which case its events contain the label attribute, while other traces might not be annotated. Human activity level interpretations of unannotated traces, obtained by inferring the label attribute based on information that is present in the annotated traces, allow the use of unannotated traces for process discovery and conformance checking on a human activity level. Figure <ref> provides a conceptual overview of the supervised event abstraction method. The approach takes two inputs: 1) a set of annotated traces, which are traces where the human activity level event that each sensor level event belongs to (the label attribute of the sensor-level event) is known, and 2) a set of unannotated traces, which are traces where only the sensor-level events known. Conditional Random Fields (CRFs) are trained of the annotated traces to create a probabilistic mapping from sensor-level events to human activity level events. This mapping, once obtained, can be applied to the unannotated traces in order to estimate the corresponding human activity level event for each sensor-level event (its label attribute). Often multiple consecutive sensor-level events will have the same label attribute. We assume that multiple human activity level events cannot occur in parallel. This enables us to interpret a sequence of events with identical label values as a single human activity level event. To obtain a final human activity level log, we collapse sequences of events with the same value for the label attribute into two events with this value as concept name, where the first event has a lifecycle 'start' and the second has the lifecycle 'complete'. Table <ref> and Table <ref> illustrate this collapsing procedure through an input and output event log.The method described in this section is implemented and available for use as package AbstractEventsSupervised in the ProM 6 <cit.> process mining toolkit and makes use of the GRMM <cit.> implementation of CRFs.We now show for each XES extension how it can be translated into useful feature functions for event abstraction. Note that we do not limit ourselves to XES logs that contain all XES extensions; when a log contains a subset of the extensions, a subset of the feature functions will be available for the supervised learning step. This approach leads to a feature space of unknown size, potentially causing problems related to the curse of dimensionality. To address this we use L1-regularized CRFs. In the training phase we search for values of weight vector λ that minimize the cross entropy between the ground truth target and the predicted label on the training data. L1-regularization adds a λ penalty terms to this minimization function that is proportionate to the size of the weight vector, giving the model an incentive not to use all of the available features (i.e., setting some features to zero weight). This results in prediction models that are sparse and therefore simpler, which helps to prevent overfitting.=-1 §.§ From a XES Log to a Feature SpaceWe now discuss per XES extension how feature functions can be obtained.Concept extension The sensor-level labels (concept names) of the preceding events in a trace can contain useful contextual information for classification into the correct human activity level event type. Based on the n-gram ⟨ a_1, a_2, …, a_n ⟩ consisting of the sensor-level labels of the n last-seen events in a trace, we can estimate a categorical probability distribution over the classes of human activity level activities from the training log, such that the probability of class l is equal to the number of times that the n-gram was observed while the n-th event was annotated with class l, divided by the total number of times that the n-gram was observed. The CRF model requires feature functions with numerical range. A feature function based on the concept extension has two parameters, n and l, and is valued with the estimated categorical probability density of the current sensor-level event having human activity level label l given the n-gram with the last n sensor-level event labels. It can be useful to combine multiple features that are based on the concept extension, where the features have different values for n and l.Organizational extension Similar to the concept extension feature functions, categorical probability distributions can be estimated on the training set for n-grams of resource, role, or group attributes of the last n events. Likewise, an organizational extension based feature function with three parameters, n-gram size n, o∈{resource,role,group}, and label l, is valued with the probability density according to the estimated categorical probability distribution of label l given the n-gram of the last n event resources/roles/groups.Time extensionIn terms of time, there are several potentially existing patterns. A certain type of human activity might for example be concentrated in a certain parts of a day, of a week, or of a month. This concentration can however not be modeled with a single Gaussian distribution, as it might be the case that a type of human activity has high probability to occur in the morning or in the evening, but low probability to occur in the afternoon in-between. A mixture distribution consisting of multiple components is therefore needed to model the probability distribution over timestamps. The most well-known mixture distribution is the Gaussian Mixture Model (GMM), where each component of the mixture is defined by a normal distribution. The circular, non-Euclidean, nature of the data space of time-of-the-day, time-of-the-week, or time-of-the-month however introduces problems for the GMM, as, using time-of-the-day as an example, 00:00 is actually very close to 23:59. Figure <ref> illustrates this problem. The Gaussian component with a mean around 10 o'clock has a standard deviation that is much higher than what one would expect when looking at the histogram, as the GMM tries to explain the data points just after midnight with this component. These data points just after midnight would however have been much better explained with the Gaussian component with the mean around 20 o'clock, which is much closer in time. Alternatively, we use a mixture model with components of the von Mises distribution, which is a close approximation of a normal distribution wrapped around the circle. To determine the correct number of components of such a von Mises Mixture Model (VMMM) we use Bayesian Information Criterion (BIC) <cit.>, choses the number of components which explains the data with the highest likelihood, while adding a penalty for the number of model parameters. A VMMM is estimated on training data, modeling the probabilities of each type of human activity based on the time passed since the start of the day, week or month. A time extension feature function with two parameters, t∈{day,week,month,…} and label l, is valued with the VMMM-estimated probability of label l given the t view on the event timestamp. An alternative approach to estimate the probability density on data that lies on a manifold, such as a circle, is described by Cohen and Welling <cit.>.Lifecycle extension & Time extension The XES standard <cit.> defines several lifecycle stages of process activities, which represent the transactional model of their generating activity. Lifecycle values that are commonly found in real life logs are start and complete which respectively represent when this activity started and ended However, a larger set of lifecycle values is defined in the XES standard, including schedule, suspend, and resume. The time differences between different stages of an activity lifecycle can be calculated when an event log possesses both the lifecycle extension and the time extension. For example, when observing the complete of an activity, the time between this complete and the corresponding start of this activity can contain useful information for predicting the correct human activity label. Finding the associated start event becomes nontrivial when multiple instances of the same activity are in parallel, as it is then unknown which complete event belongs to which start event. We assume consecutive lifecycle steps of activities running in parallel to occur in the same order as the preceding lifecycle step. For example, when we observe two start events of an activity of type A in a row, followed by two complete events of type A, we assume the first complete to belong to the first start, and the second complete to belong to the second start.The XES standard defines an ordering over the lifecycle values. For each type of human activity, we fit a Gaussian Mixture Model (GMM) to the set of time differences between each two consecutive lifecycle steps. A feature based on both the combination of the lifecycle and the time extension with activity label parameter l and lifecycle c is valued the probability density of activity l as estimated by the GMM given the time between the current event and lifecycle value c. Bayesian Information Criterion (BIC) <cit.> is again used to determine the number of components of the GMM. Note that while these features are time-based, regular GMMs can be used instead of VMMMs since time duration is a Euclidean, non-circular, space. §.§ Evaluating Human Activity Level Event Predictions for Process Mining ApplicationsExisting approaches in the field of activity recognition take as input time windows where each time window is represented by a feature vector that describes the sensor activity or status during that time window. Hence, evaluation methods in the activity recognition field are window-based, using evaluation metrics like the percentage of correctly classified time slices <cit.>. There are two reasons to deviate from this evaluation methodology in a process mining setting. First, our method operates on events instead of time windows. Second, the accuracy of the resulting high level sequences is much more important for many process mining techniques (e.g. process discovery, conformance checking) than the accuracy of predicting each individual minute of the day.A well-known metric for the distance of two sequences is the Levenshtein distance <cit.>. However, Levenshtein distance is not suitable to compare sequences of human actions, as human behavior sometimes includes branches in which it does not matter in which order two activities are performed. For example, most people shower and have breakfast after waking up, but people do not necessarily always perform the two in the same order. Indeed, when ⟨ a, b ⟩ is the sequence of predicted human activities, and ⟨ b, a ⟩ is the actual sequence of human activities, we consider this to be only a minor error, since it is often not relevant in which order two parallel activities are executed. Levenshtein distance would assign a cost of 2 to this abstraction, as transforming the predicted sequence into the ground truth sequence would require one deletion and one insertion operation. For example, most people shower and have breakfast after waking up, but people do not necessarily always perform the two in the same order. An evaluation measure that better reflects the prediction quality of event abstraction is the Damerau-Levenstein distance <cit.>, which adds a swapping operation to the set of operations used by Levenshtein distance. Damerau-Levenshtein distance would assign a cost of 1 to transform ⟨ a, b⟩ into ⟨ b, a⟩. To obtain comparable numbers for different numbers of predicted events we normalize the Damerau-Levenshtein distance by the maximum of the length of the ground truth trace and the length of the predicted trace and subtract the normalized Damerau-Levenshtein distance from 1 to obtain Damerau-Levenshtein Similarity (DLS).§ CASE STUDIES In this section we evaluate the supervised event abstraction framework on three case studies on real life smart home data sets.§.§ Experimental setupWe include three real life smart home event logs in the evaluation: the Van Kasteren event log <cit.>, and two event logs from a smart home experiment conducted by MIT <cit.>. All three event logs used in for the evaluation consist of multidimensional time series data with all dimensions binary, where each binary dimension represents the state of one in-home sensor. These sensors include motion sensors, open/close sensors, and power sensors (discretized to 0/1 states). We transform the multidimensional time series data from sensors into events by regarding each sensor change point as an event. Cases are created by grouping events together that occurred in the same day, with a cut-off point at midnight. High-level labels are provided for the event logs.The following XES extensions can be used for these event logs: ConceptThe sensor that generated the event. TimeThe time stamp of the sensor change point. LifecycleStart when the event represents a sensor value change from 0 to 1 and Complete when it represents a sensor value change from 1 to 0. Note that human activity level annotations are provided for all traces in the three event logs. To evaluate how well the supervised event abstraction method generalized to unannotated traces, we artificially use a part of the traces to train the abstraction model and apply them on a test set where we regard the annotations to be non-existent. We evaluate the obtained human activity labels against the ground truth labels in a Leave-One-Trace-Out-Cross-Validation setup where we iteratively leave out one trace to evaluate how well this mapping generalizes to unseen events and cases. We measure the accuracy of the human activity level traces compared to the ground truth human activity level traces in terms of Damerau-Levenshtein similarity <cit.>.Additionally, we evaluate the quality of the process model that can be discovered from the human activity level traces. To discover a process model from the human activity level event log we use the Inductive Miner <cit.>. There are several criteria to express the fit between a process model and an event log in the area of process mining. Two of those criteria are fitness <cit.>, which measures the degree to which the behavior that is observed in the event log can be replayed on the process model, and precision <cit.>, which measures the degree to which the behavior that was never observed in the event log cannot be replayed on the process model. Low precision typically is indicates an overly general process model, that allows for too much behavior. We compare the fitness and precision of the models produced by the Inductive Miner on the sensor-level log and the human activity level log. §.§ Case Study 1: Van Kasteren Event Log For the first case study we use the smart home environment log described in Van Kasteren et al. <cit.>. The Van Kasteren log contains 1285 events divided over fourteen different sensors. The log contains 23 days of data.The average Damerau-Levenshtein similarity between the predicted human activity level traces in the Leave-One-Trace-Out-Cross-Validation experimental setup and the ground truth human activity level traces is 0.7516, which shows that the supervised event abstraction method produces traces which are fairly similar to the ground truth.Figure <ref> shows the result of the Inductive Miner <cit.> for the sensor-level events in the Van Kasteren data set. The resulting process model starts with a choice between four activities: hall-toilet door, hall-bedroom door, hall-bathroom door, and frontdoor. After this choice the model branches into three parallel blocks, where the upper block consists of a large choice between eight different activities. The other two parallel blocks respectively contain a loop of the cups cupboard and the fridge. This model closely resembles the flower model, which allows for all behavior in any arbitrary order. There seems to be very little structure on this sensor level of event granularity. Figure <ref> shows the result of the Inductive Miner on the aggregated set of predicted test traces. We can see that the main daily routine starts with breakfast, after which the subject leaves the house to go to work. After work the subject prepares dinner and goes to bed. The activities use toilet and take shower are put in parallel to this sequence of activities, indicating that they occur at different places in the sequences of activities.Table <ref> shows the effect of the abstraction on the fitness and precision of the models discovered by the Inductive Miner. It shows that the precision of the model discovered on the abstracted log is much higher than the precision of the model discovered on the sensor data, indicating that the abstraction helps discovering a model that is more behaviorally constrained and more specific. §.§ Case Study 2: MIT Household A Event Log For the second case study we use the data of household A of a smart home experiment conducted by MIT <cit.>. Household A contains data of 16 days of living, 2701 sensor-level events registered by 26 different sensors. The human level activities are provided in the form of a taxonomy of activities on three levels, called heading, category and subcategory. On the heading level the human activities are very general in nature, such as the activity personal needs. The eight different activities on the heading level branch into 19 different activities on the category level, where personal needs branches into e.g. eating, sleeping, and personal hygiene. The 19 categories are divided over 34 subcategories, which contain very specific human activities. At the subcategory level the category meal cleanup is for example divided into washing dishes and putting away dishes. At the subcategory level there are more types of human activities than there are sensors-level activities, which makes the abstraction task very hard. Therefore, we set the target label to the category level.Figure <ref> shows the model discovered with the Inductive Miner on the sensor-level events of the MIT household A log. The model obtained allows for too much behavior, as it contains two large choice blocks. We found a Damerau-Levenshtein similarity of 0.6348 in the Leave-One-Trace-Out-Cross-Validation experiment. Note that the abstraction accuracy on this log is lower than the abstraction accuracy on the Van Kasteren event log. However, the MIT household A log contains more different types of human activity, resulting in a more difficult prediction task with a higher number of possible target classes. Figure <ref> shows the process model discovered from the human activity level traces that we predicted from the sensor-level events. Even though the model is too large to print in a readable way, from its shape it is clear that the abstracted model is much more behaviorally constrained than the sensor-level model. The precision and fitness values in Table <ref> show that indeed the process model after abstraction has become behaviorally more specific while the portion of behavior of the data that fits the process model remains more or less the same.§.§ Case Study 3: MIT Household B Event Log For the third case study we use the data of household B of the MIT smart home experiment <cit.>. Household B contains data of 17 days of living, 1962 sensor-level events registered by 20 different sensors. Identically to MIT household A the human-level activities are provided as a three-level taxonomy. Again, we use the subcategory level of this taxonomy as target activity label.The model discovered with the Inductive Miner <cit.> from the sensor-level events is shown in Figure <ref>. The model obtained allows for too much behavior, as it contains two large choice blocks. We found a Damerau-Levenshtein similarity of 0.5865 in the Leave-One-Trace-Out-Cross-Validation experiment, which is lower than the similarity found on the MIT A data set while the target classes of the abstraction are the same for the two data sets. This can be explained by the fact that there is less training data for this event log, as household B contains 1932 sensor-level events where household A contains 2701 sensor-level events. Figure <ref> shows the process model discovered from abstracted log. Again this model is not readable due to its size, but its shape shows it to be behaviorally quite specific. The precision and fitness values in Table <ref> also that process model after abstraction has indeed become behaviorally more specific while the portion of behavior of the data that fits the process model decreased only slightly.§ CONCLUSION In this paper we presented a framework to abstract events using supervised learning which has been implemented in the ProM process mining toolkit. An important part of the framework is a generic way to extract useful features for abstraction from the extensions defined in the XES IEEE standard for event logs. We propose the Damerau-Levenshtein Similarity for evaluation of the abstraction results, and motivate why it fits the application of process mining. Finally, we showed on three real life smart home data sets that application of the supervised event abstraction framework enables us to mine more precise process model description of human life compared to what could be mined from the original data on the sensor-level. Additionally, these process models contain interpretable event labels on the human behavior activity level.spbasic | http://arxiv.org/abs/1705.10202v1 | {
"authors": [
"Niek Tax",
"Natalia Sidorova",
"Reinder Haakma",
"Wil M. P. van der Aalst"
],
"categories": [
"cs.LG",
"cs.AI",
"cs.DB"
],
"primary_category": "cs.LG",
"published": "20170525203256",
"title": "Mining Process Model Descriptions of Daily Life through Event Abstraction"
} |
[email protected] [url]http://www.grupolys.org/ cgomezr Universidade da Coruña. FASTPARSE Lab, LyS Research Group, Departamento de Computación, Facultade de Informática, Elviña, 15071 A Coruña, Spain dependency distance natural language parsing dependency parsing crossing dependencies non-projectivity^© 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license (<http://creativecommons.org/licenses/by-nc-nd/4.0/>). This is the accepted manuscript (final peer-reviewed manuscript) accepted for publication in Physics of Life Reviews, and may not reflect subsequent changes resulting from the publishing process such as copy-editing, formatting or pagination. The published journal article can be found at <https://doi.org/10.1016/j.plrev.2017.05.007>.Liu et al. <cit.> provide a comprehensive account of research on dependency distance in human languages.While the article is a very rich and useful report on this complex subject, here I will expand on a few specific issues where research in computational linguistics (specifically natural language processing) can inform DDM research, and vice versa. These aspects have not been explored much in <cit.> or elsewhere, probably due to the little overlap between both research communities, but they may provide interesting insights forimproving our understanding of the evolution of human languages, the mechanisms by which the brain processes and understands language, and the construction of effective computer systems to achieve this goal. § CROSSINGS, DEPENDENCY DISTANCE AND THE PARALLELISM BETWEEN EXCEPTIONS TO PROJECTIVITY AND EXCEPTIONS TO DDM As mentioned in <cit.>, there is a close relation between DDM and the scarcity of crossing dependencies in natural languages. This low frequency of crossings has long been observed <cit.>, later quantified <cit.>, and recently statistically tested <cit.>, in a wide range of human languages.It is worth noting, however, that strict projectivity (a prohibition of crossing dependencies) is not an adequate model of the syntax of real sentences. One the one hand, it fails to explain a number of relevant linguistic phenomena present in various languages <cit.>. On the other hand, an overwhelming majority of syntactic corpora in recent multilingual collections have been observed to contain non-projectivity <cit.>: crossing dependencies are scarce, but far from absent <cit.>. For these reasons, while projectivity can be useful from an engineering point of view, in the context of a tradeoff between coverage and efficiency in natural language parsers <cit.>; taking it for granted when investigating DDM or using it as an assumption of models to explain DDM <cit.> can lead to a methodological pitfall: the scarcity of crossing dependencies is likely not an independent constraint of language that contributes to DDM, but rather a consequence of it <cit.>.To account for the limited amount of crossing dependencies that arise in language and to be able to build parsers that can handle non-projective syntactic phenomena in an efficient manner, researchers have explored various classes of so-called mildly non-projective dependency structures <cit.>. These are sets of trees that allow a limited degree of non-projectivity, permitting crossing dependencies only if they follow certain conditions or patterns. Various such classes have been defined that claim very high coverage over the trees in a variety of corpora, allowing a large majority of the non-projective dependencies that appear in practice. However, the reasons for this success (and especially, the reasons why some of the proposed sets have more coverage than others) are currently not very well known. Namely, the reasons for the high coverage of each given mildly non-projective class could include either being an adequate description of the particular situations where crossing dependencies can arise, or yielding a large enough class of syntactic trees to provide high coverage by sheer brute force, or a mix of both. The observation that a given class can be more or less adequate depending on the criteria used to annotate the syntactic dependencies <cit.> suggests at least some influence of the first factor.The research on syntactic patterns involving long-distance dependencies, reviewed by Liu et al. in Section 5 of <cit.>, could help clarify this question. DDM and the scarcity of crossing dependencies are closely related, as the latter is motivated by short dependency distance <cit.>. Furthermore, in both cases, there is a predominant trend (the majority of dependencies are short and do not cross) but there are exceptions that escape the trend (long-distance dependencies and crossing dependencies), which are in turn related (longer dependencies are more likely to cross <cit.>). Liu et al. <cit.> review some possible reasons for the minority of long dependencies observed in language, and explanations of how they can survive the pressure for DDM. This raises two questions: (1) do these explanations also apply to the presence of crossing dependencies, and are they related with the adequacy of mildly non-projective classes of trees? (e.g., do the most effective such classes work well because they favor crossing dependencies from words at peripheral positions, or structures that can be easily chunked?); and (2) can we in turn re-use some of what we know about long dependencies for crossing dependencies and their parsing, and employ it to define classes of mildly non-projective structures that will more closely adjust to the kinds of non-projectivity found in language? Both questions are interesting avenues for research, and can advance our knowledge both on DDM and natural language parsing. § THE SURPRISING EFFECTIVENESS OF TRANSITION-BASED PARSERS AND THEIR BIAS TOWARDS DDM Liu et al. <cit.> cite some work in computational linguistics that achieved improvements in parsing accuracy by purposefully introducing dependency distance as a constraint in a parser <cit.>. Additionally, it is worth noting that many state-of-the-art natural language parsers use algorithms with an implicit bias towards short dependency distances, even if they do not introduce it as an explicit restriction.In particular, a popular framework for dependency parsers is the transition-based (or shift-reduce) approach <cit.>, under which a parser is defined with a non-deterministic state machine, a statistical or machine-learning-based model to score transitions, and a search strategy to obtain the optimal sequence of transitions that will yield a parse. Many, if not most, of the current state-of-the-art parsing systems are based on this framework <cit.>, and all the different algorithm variants that are at its core have in common that they build short dependencies before (and requiring fewer transitions than) long ones, be it because building a long dependency requires removing intervening nodes from a stack (as in the popular arc-standard and arc-eager <cit.>, arc-hybrid <cit.> or swap <cit.> algorithms) or because it requires to navigate a list (as in the systems based on the Covington <cit.> algorithm). This bias towards favoring short dependencies can be part of the reason why these systems are so effective in practice, and is an example of the trend pointed out in <cit.> by which purely engineering-oriented parsing models are converging with cognitive theories of language understanding, even when they do not have psycholinguistic modeling among their goals.A quick verification and quantification of the mentioned bias can be undertaken by implementing transition systems and obtaining random trees by taking a random transition at each state. Focusing on sentences of length 20 as an example, the expected mean dependency distance for a uniformly random tree is 7 <cit.>, contrasting with real averages observed in corpora, e.g. 2.52 for Arabic, 3.18 for German, 2.59 for English or 2.51 for Spanish in the sentences of length 20 of the Stanford HamleDT 2.0 treebanks <cit.>. On the other hand, by simulating 10^5 random parses of sentences of length 20, we obtain an average distance of 2.44 with the arc-standard algorithm, and 2.06 with the arc-eager parser with the tree constraint <cit.>. These two parsers are projective, but using an algorithm that can generate arbitrary non-projective trees (the swap parser), the obtained average is 2.38, even smaller than for arc-standard. While further investigation is needed that would be outside the scope of this comment, the data seem to suggest that transition-based parsers implicitly favor dependency lengths that are equal, or even smaller, than the natural ones that appear in language as a consequence of DDM; and this could be an important factor in the practical adequacy of these parsers. § ACKNOWLEDGEMENTSThis research has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 714150 - FASTPARSE), and from the TELEPARES-UDC project (FFI2014-51978-C2-2-R) from MINECO. I thank Ramon Ferrer-i-Cancho for helpful comments. elsarticle-num | http://arxiv.org/abs/1705.09837v2 | {
"authors": [
"Carlos Gómez-Rodríguez"
],
"categories": [
"cs.CL",
"68T50, 91F20, 97C30",
"I.2.7; J.5; F.4.2"
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"primary_category": "cs.CL",
"published": "20170527161937",
"title": "On the relation between dependency distance, crossing dependencies, and parsing. Comment on \"Dependency distance: a new perspective on syntactic patterns in natural languages\" by Haitao Liu et al"
} |
=1 =1 | http://arxiv.org/abs/1705.09270v2 | {
"authors": [
"Alvise Bastianello",
"Andrea De Luca"
],
"categories": [
"cond-mat.stat-mech",
"cond-mat.quant-gas",
"quant-ph"
],
"primary_category": "cond-mat.stat-mech",
"published": "20170525173058",
"title": "Non-Equilibrium Steady State generated by a moving defect: the supersonic threshold"
} |
KUNS-2684 Department of Physics, Kyoto University, Kyoto 606-8502, JapanWe have investigated structures of the ground and excited states of ^16O with the method of variation after spin-parity projection in the antisymmetrized molecular dynamics model combined with the generator coordinate method of ^12C+α cluster. The calculation reasonably reproduces the experimental energy spectra,E2, E3, E4, IS1 transitions, and α-decay properties. The formation of 4 α clusters has been confirmed from nucleon degrees of freedom in the AMD model without assuming existence of any clusters. They form “tetrahedral” 4α- and ^12C+α-cluster structures. The ^12C+α structure constructsthe K^π=0^+ band consisting of the 0^+_2, 2^+_1, and 4^+_1 states and the K^π=0^- band of the 1^-_2, 3^-_2, and 5^-_1 states.The 0^+_1, 3^-_1, and 4^+_2 states are assigned to the ground band constructed fromthe tetrahedral 4α structure. The 0^+_1 and 3^-_1 are approximately interpreted asT_d band members with the ideal tetrahedral configuration.The ground state 4α correlation plays an important role inenhancement of the E3 transition strengthto the 3^-_1. The 4^+_2 state is not the ideal T_d member but constructed froma distorted tetrahedral 4α structure. Moreover,significant state mixing of the tetrahedral 4α and^12C+α cluster structuresoccurs between 4^+_1 and 4^+_2 states, indicating that the T_d configuration of 4α israther fragile at J^π=4^+. Tetrahedral 4α and ^12C+α cluster structures in ^16O Yoshiko Kanada-En'yo December 30, 2023 ====================================================§ INTRODUCTION Nuclear deformation is one of typical collective motions in nuclear systems. Not only axial symmetric quadrupole deformationsbut also triaxial and octupole deformations have been attracting interests. In light nuclear systems, further exotic shapes have been expected because of cluster structures.For instance, a triangle shape in ^12C and a tetrahedral one in ^16O have been discussedby assuming 3α- and 4α-cluster structures. In old days, non-microscopic α-cluster models have been applied in orderto understand energy spectra of ^12C and ^16O <cit.>.Wheeler has suggested a low-lying 3^- state of ^16O as vibration of the tetrahedral configuration of 4 α particles <cit.>. This state has been assigned tothe lowest negative-parity state ^16O(3^-_1, 6.13 MeV), which has been experimentally established later.Since 1970's, microscopic and semi-microscopiccluster modelshave been applied in order to investigatecluster structures of ^16O <cit.>.The T_d symmetry of the tetrahedral4α-cluster structurehas been discussed for a long time to understand energy spectra of^16O <cit.>.The ideal tetrahedral 4α configuration with the T_d symmetry constructs a rotational band of 0^+, 3^-, 4^+, …, states. The ground state and the 3^-_1 at 6.13 MeV have been assigned to theT_d-invariant 4α band. This assignment is supportedby the observed strong E3 transition <cit.>and α-transfer cross sections on ^12C <cit.>.However, the assignment of the 4^+ state in the tetrahedral 4αband has notbeen confirmed yet. Robson assigned the 4^-_1 at 10.36 MeVas the T_d band <cit.>. This assignment describes the significantly strong E4 transition. However, it contradicts tothe large α-decay width of the 4^-_1. Alternatively,the 4^-_1(10.36 MeV) has been considered to belong to a ^12 C+α cluster band starting fromthe 0^+_2(6.05 MeV) <cit.>. The strong α-transfer and weaktwo-nucleon transfer cross sections for the 4^+_1(10.36 MeV) supportthe ^12 C+αcluster structure with the predominant 4p4h component <cit.>.Elliott has discussed α-transfer cross sectionsand assigned the 0^+_1, 3^-_1(6.13 MeV), and 4^+_2(11.10 MeV) to the T_d band constructed froma tetrahedrally deformed intrinsic state <cit.>. Very recently, algebraic approaches for the 4α system has been revived by Bijker and Iachello<cit.> to describe the experimental energy spectra of ^16O based on the T_d symmetry and its excitation modes,which has been proposed by Wheeler. In their works, the 4^+_1(10.36 MeV) was assigned again to theT_d band. Although the 4α models in Refs. <cit.> describethe experimental B(E4) for 0^+_1 → 4^+_1(10.36 MeV), the calculated form factors disagree to the experimentaldata measured by (e,e') scattering as already pointed out in Ref. <cit.>. In addition to the tetrahedral 4α structure,^12 C+α cluster states appear in a similar energy region.The lowest positive-parity ^12 C+α band is the K^π=0^+ band and its counterpart of the parity doublet is the K^π=0^- band <cit.>. The 0^+_2(6.06 MeV), 2^+_1 (6.92 MeV), 4^+_1(10.36 MeV), and 6^+_2(16.23 MeV) are assigned to the K^π=0^+ band, and the 1^-_2(9.59 MeV), 3^-_2(11.60 MeV),and 5^-_1(14.66 MeV) are assigned to the K^π=0^- band because these states have similar features of α-transfer andα-decay properties.In spite of rich cluster phenomena in ^16O, there is no microscopic calculation that sufficiently describes low-energy spectra and cluster structures of ^16O. Firstly, most of microscopic cluster model calculations fail to reproduceexcitation energies of the ^12C+α cluster states except for the case usinga particularly strong exchange nuclear interaction <cit.>. For instance, in ^12C+α and 4α cluster model calculations, the band-head energy of theK^π=0^+ ^12C+αband is calculated to be about E_x(0^+_2)=16 MeV,which largely overestimates the experimental value E_x(0^+_2)=6.06 MeV <cit.>.Therefore, it was difficult to solve the problem of possible coexistence of ^12C+α andtetrahedral 4α states in a similar energy region. Secondly, both non-microscopic and microscopic cluster models a prioriassume existence of α clusters and/or a ^12C cluster, and are not able to check whether clusters areactually formed from nucleon degrees of freedom. In mean-field calculations,the spherical p-shell closed state is usually obtained for the ground state solution except for the cases using particularly strong exchange nuclear interactions <cit.>.Even though significant mixing ofhigher-shell components in the ground state of ^16O hasbeen suggested in recent works with extended mean-field and shell-model approaches<cit.>, neither intrinsic shape nor cluster structure has been discussed explicitly. Moreover, it is generally difficult for mean-field approaches to describe well developed cluster structures in excited states.Recently, we applied the antisymmetrized molecular dynamics (AMD) method<cit.> to ^16O andfound a tetrahedral shape with the 4α-cluster structure in a fully microscopic calculationbased on nucleon degrees of freedomwithout assuming existence of clusters <cit.>.More recently,in a first principlecalculation using the chiral nuclear effective field theory,the tetrahedral 4α structure has been foundin the ground state of^16O<cit.>.Our aim is to investigate cluster structures of low-lying states of ^16O.We focus on the possible coexistence of thetetrahedral 4α and ^12 C+α states and discuss, in particular,4^+ states in the tetrahedral 4α and ^12 C+α bands. To answer to the questions whether 16 nucleons form 4 α clusters and whether theyare arranged in a tetrahedral configuration, we first apply the method of variation after spin-parity projection (VAP)in the framework of AMD, which we call the AMD+VAP <cit.>.Then we combine the AMD+VAP with the generator coordinate method (GCM) <cit.> of ^12 C(AMD)+α, in whichwe use the ^12 C(AMD) cluster wave functions obtained by the AMD+VAP for^12C.The AMD+VAP method has been proved to be useful to describe structures of light nuclei and succeeded to reproduce properties of the ground and excited statesof ^12C <cit.>.The ^12 C(AMD)+α GCM has been applied in our previous workto investigate positive-parity states of ^16O <cit.>.By combining the AMD+VAP with the ^12 C(AMD)+α GCM, we obtainbetter description of asymptotic behavior and excitation energies of ^12 C+α states.We calculate the positive- and negative-parity levels, transition strengths, andα-decay widths, and discuss cluster structures of ^16O.The paper is organized as follows. In the next section, the framework of thepresent calculation is explained. Section <ref> describes the adopted effective interactions.The results of ^16O are shown in Sec. <ref>. Finally, we give a summary and outlooks in Sec. <ref>.§ FORMULATION The present method is the AMD+VAP combined with the^12 C(AMD)+α GCM. The details ofthe AMD+VAP are described in Refs. <cit.>.For the formulation of the ^12 C(AMD)+α GCM,the reader is referred to Ref. <cit.>. §.§ AMD wave function and VAP We define the AMD model wave function and performenergy variation to obtain the energy-minimum solution in the AMD model space. An AMD wave function for an A-nucleon systemis given by a Slater determinant of Gaussian wave packets,Φ_ AMD(Z) = 1/√(A!)A{φ_1,φ_2,...,φ_A },where the ith single-particle wave function is written by a product of the spatial, intrinsic spin, and isospin wave functions asφ_i = ϕ_X_iχ^σ_i χ^τ_i, ϕ_X_i(r_j) = (2ν/π)^4/3exp{-ν(r_j-X_i/√(ν))^2},χ^σ_i= (1/2+ξ_i)χ_↑+ (1/2-ξ_i)χ_↓.Here, ϕ_X_i and χ^σ_i are the spatial and intrinsic spin functions, andχ^τ_i is the isospin function fixed to be proton or neutron. The width parameter νis chosen to be a common value.Thus, the AMD wave function is specified by a set of variational parameters, Z≡{X_1,X_2,…, X_A,ξ_1,ξ_2,…,ξ_A } for Gaussian center positions (X_1,…, X_A) and intrinsic spin orientations(ξ_1,…,ξ_A) of all single-nucleonwave functions, which are independentlytreated as variational parameters.In the AMD framework,existence of neither clusters nor a core nucleus is assumed, but nuclear structures are described based onnucleon degrees of freedom. Nevertheless, the AMD model space covers various cluster structuresas well as shell-model structures owing to flexibility of spatial configurations of single-nucleon Gaussian wave functions,which are fully antisymmetrized. Therefore, if a cluster structure is favored in a system,the cluster structure isautomatically obtained in the energy variation. To express a J^π state, an AMD wave function is projected onto the spin-parity eigenstate, Φ^Jπ(Z)=P^Jπ_MKΦ_ AMD(Z),where P^Jπ_MK is the spin-parity projection operator.To obtain the wave function for the J^π state, the VAP is performed for the J^π-projected AMD wave function,δ⟨Φ^Jπ(Z)|H|Φ^Jπ(Z) ⟩/⟨Φ^Jπ(Z)|Φ^Jπ(Z) ⟩=0,with respect to variation δZ.After the VAP, we obtain the optimum parameter set Z^ opt_J^π for the J^π state.This method is called the AMD+VAP.The obtained AMD wave function Φ_ AMD(Z^ opt_J^π) expressed by a single Slater determinant is regarded as the intrinsic state of the J^π state.Note that the J^π-projected AMD wave function is no longer a Slater determinant and, in principle,contains higher correlations beyond the Hartree-Fock approach.When a local minimum solution is obtained by the VAP for J^π, it is regarded as the second (or higher) J^π state. Another way to obtainthe AMD configuration Z^ opt_J^π_2 optimized for the J^π_2 state isVAP for the component orthogonal to the obtained J^π_1 state,Φ^Jπ_ exc(Z)= (1- |Φ^Jπ(Z^ opt_J^π_1) ⟩⟨Φ^Jπ(Z^ opt_J^π_1)| /⟨Φ^Jπ(Z^ opt_J^π_1)|Φ^Jπ(Z^ opt_J^π_1) ⟩) Φ^Jπ(Z). In the AMD+VAP method, all the AMD wave functions obtained by VAP for various J'^π'_n'are superposed to obtain the final wave function for the J^π_n state, Ψ^^16O(J^π_n)_VAP =∑_K∑_β=J'^π'_n' c_vap(J^π_n;K,β) P^Jπ_MKΦ_AMD(Z^ opt_β),where coefficients c_vap(J^π_n;K,β) for the J^π_n state are determined bydiagonalization ofthe norm and Hamiltonian matrices.§.§ ^12 C(AMD)+α GCM In the ^12 C(AMD)+α GCM,the ^12C-α distance is treated asa generator coordinate. For the description of the^12C cluster,we use ^12C wave functions Φ^^12C_AMD(Z^opt_β_C)obtained by the AMD+VAP for ^12C.Here the label β_C=J^π_n is used for the ^12C(J^π_n) state. In the present calculationwe use three configurations, β_C=0^+_1, 0^+_2, and 1^-_1, corresponding to^12C(0^+_1), ^12C(0^+_2), and ^12C(1^-_1), respectively.These three configurations describe wellenergy spectra of ^12C as shown in Ref. <cit.>.To describe inter-cluster motion between ^12C and α clusters,we superposethe ^12C+α wave functions with various distance dusing the ^12C-cluster wave function Φ^^12C_AMD (-d/4;Z^opt_β_C)localized at a mean center-of-mass position -d/4 (d=(0,0,d)) andthe (0s)^4 α-cluster wave function Φ_α(3d/4) at 3d/4. A ^12C-cluster configuration, Φ^^12C_AMD(Z^opt_β_C), has a cluster structure with an intrinsic deformation oriented in a specific direction.To take into account angular momentum projection of the subsystem ^12C, we considerrotation R̃(Ω) of theΦ^^12C_AMD (-d/4;Z^opt_β_C) around -d/4.The total wave function for ^16O(J^π_n) of the ^12C(AMD)+α GCM modelis written asΨ_α GCM^J^π_n =∑_K,i,j,β_Cc_gcm(J^π_n;K,i,j,β_C) Φ^Jπ K_^12 C+α(d_i,Ω_j,Z^opt_β_C),Φ^Jπ K_^12 C+α(d,Ω_j,Z^opt_β_C) ≡ P^Jπ_MK A{R̃(Ω) Φ_^12 C^ AMD(-S/4;Z^opt_β_C)·Φ_α(3d/4) },where coefficients c_gcm(J^π_n;K,i,j,β_C) are determined bysolving the Hill-Wheeler equation <cit.>, i.e. diagonalizing the norm and Hamiltonian matrices. The superposition of rotated ^12C-cluster wave functions is equivalent tolinear combination of various spin states of the ^12C cluster projected from the intrinsic state.In addition to ^12C-cluster rotation, excitation of the ^12C cluster is taken into account by superposing configurations, β_C=0^+_1, 0^+_2, 1^-_1.Moreover, 3α breaking isalready taken into account inΦ^^12C_AMD(Z^opt_β_C). §.§ AMD+VAP combined with ^12C(AMD)+α GCMWe combine the AMD+VAP method with the ^12C(AMD)+α GCM by superposing all basis wave functions,Ψ^^16O(J^π_n)_VAP+αGCM =∑_K,β c_vap(J^π_n;K,β) P^Jπ_MKΦ_AMD(Z^ opt_β) +∑_K,i,j,β_Cc_gcm(J^π_n;K,i,j,β_C) Φ^Jπ K_^12 C+α(d_i,Ω_j,Z^opt_β_C),where coefficients, c_vap and c_gcm, are determined by the diagonalization of the norm and Hamiltonian matrices. We call this method “VAP+αGCM,”.§ EFFECTIVE NUCLEAR INTERACTIONS In the present calculation, we use the effective nuclear interactions with the parametrization same as that used for ^12C in the AMD+VAP calculation <cit.>.They are the MV1 central force <cit.> and the G3RS spin-orbit force <cit.>. The MV1 force contains finite-range two-body and zero-range three-body terms. We use the case-1 parametrization of the MV1 force and setthe Bartlett (b), Heisenberg (h), and Majorana (m) parameters as b=h=0 and m=0.62. As for strengths of the two-range Gaussian of the G3RS spin-orbit force, we use u_I=-u_II≡ u_ls=3000 MeV to reproduce the 2^+_1 excitation energy of ^12C with the MV1 force. The Coulomb force is approximated using a seven-range Gaussian form. With these interactions,properties of the ground and excited statesof ^12C are described well by the AMD+VAP calculation <cit.>. As for a symmetric nuclear matter,the MV1 force with the present parametrization gives the saturation densityρ_s=0.192 fm^-3, the saturation energy E_s=-17.9 MeV,the effective nucleon mass m^*_SNM=0.59m, and the incompressibility K=245 MeV. It is known that usual two-body effective nuclear interactions withmass-independent parameters have an overshooting problem of nuclear binding and density with increase of the mass number and are not able to describe the saturation property.The overshooting problem is improved with the use of the MV1 force, because it contains a zero-range three-body force, which is equivalentto a density-dependent force for spin and isospin saturated systems. In the sense that the MV1 force consists of finite-range two-body and “density-dependent” zero-range forces, it can be categorized to a similar type interaction to Gogny forces. The present interaction parameters gives reasonable result forthe α, ^12C, and ^16O bindingscompared with the experimental binding energies (B.E.) of α (28.30 MeV), ^12C (92.16 MeV), and ^16O (127.62 MeV): the calculated B.E. of the(0s)^4 α particle is27.8 MeV, that of^12C obtained by the AMD+VAP with 3 configurations(^12 C(0^+_1), ^12 C(0^+_2), and ^12 C(1^-_1))is 87.6 MeV, and that of ^16O with the AMD+VAP (VAP+αGCM) is 123.0 (123.5) MeV.The calculated α-decay threshold of ^16O is about 8 MeV,which is in reasonable agreement with the experimental value 7.16 MeV. § RESULTS §.§ Procedure and parameter setting The width parameter ν for all wave functions ofα,^12C, and ^16O is chosen to be a common value so thatthe center of mass motion can be exactly removed.In the present calculation, we useν=0.19 fm^-2, which minimizesthe energies of ^12C and ^16O.In the AMD+VAP calculation of ^16O, we obtain9 configurations (β) for J^π_n=0^+_1,2,2^+_1, 4^+_1,2, 1^-_1, 2^-_1, 3^-_1, and5^-_1. First we obtain the 0^+_1 configuration Z^ opt_0^+_1 with the VAP without the orthogonal condition, and next obtain the 0^+_2 configuration Z^ opt_0^+_2 with the VAP with the condition orthogonal to the 0^+_1. For other J^π, we iteratively achieve the VAP without the orthogonal conditionby starting from Z^ opt_0^+_1 and Z^ opt_0^+_2as initial configurations.In the VAP for J^π=4^+, we found minimum and local minimum solutions for the 4^+_1 and 4^+_2 configurations. For J^π other than 4^+,we did not obtain local minimum solutions but obtained only a minimum solution in two cases ofinitial configurations.In the ^12C(AMD)+α GCM, we use inter-cluster distances d_i=1.2, 2.4, 3.6, …,8.4 fm (7 points with 1.2 fm interval)for β_C=0^+_1 and 0^+_2 of the ^12C configurations.For β_C=1^-_1, we adoptd_i=1.2,2.4,3.6⋯,6.0 fm (5 points with 1.2 fm interval)to save computational costs.For Euler angles Ω_j of the ^12C-cluster rotation R̃(Ω_j) we use seventeen points (j=1,…,17), as described in Ref. <cit.>.In the K-mixing, we truncate |K| ≥ 5 components to save computational resources.As described previously, we combine the AMD+VAP and^12 C(AMD)+α GCM to obtain final result.In the AMD+VAP,each ^16O wave function is essentially expressed by theJ^π state projected from a Slater determinant, and therefore,it is useful to discuss an intrinsic shape of the state.In other words, strong-coupling cluster structures areobtained within the AMD+VAP.On the other hand, the ^12 C(AMD)+α GCMis essential to describe weak-coupling ^12 C+α cluster states,for which the angular momentumprojection of the subsystem ^12 C is necessary. In the present paper, we start from the AMD+VAP result (hereafter we call it the VAP result) and then analyze the VAP+αGCM result to discuss how theVAP result is affected by mixing of^12 C(AMD)+α configurations. Note that the obtained VAP states show predominantly 4α structures, which areapproximately included by the ^12 C(AMD)+α model space. §.§ Energies, radius, and transitionsThe calculated and experimental values forB.E., root-mean-square(r.m.s.) radius, and ^12C+α threshold are listed in Table <ref>.The ground state properties calculated by the VAP and VAP+αGCM are similar to each other, and they are in reasonable agreement with the experimental data. Energy levels are shown in Fig. <ref>. In the figure,the energy levels in the ground and ^12C+α bands are connected by dashed and solid lines, respectively.In a usual assignment, the experimental 0^+_2, 2^+_1, 4^+_1, 1^-_2, 3^-_2, and 5^-_1 states are considered to belong to theK^π=0^± ^12C+α cluster bands fromα-decay and α-transfer properties of these states. For the ground band,we tentatively assign the experimental 0^+_1, 3^-_1, and 4^+_2 states as band members following theassignment of Ref. <cit.>.In the VAP and VAP+αGCM results, we cancategorize calculated energy levelsinto the ground and ^12C+α bandsbased onE2 transition properties as well as analysis of intrinsic structures.In the VAP calculation, the 4^+, 1^-, and 3^- states of the ^12C+α bandare not obtained as local minimum solutions, but they are constructed by the J^π projection from the ^12C+α cluster structure obtained for 0^+_2, 2^+_1, and 5^-_1. Excitation energies of the ^12C+α states are much overestimated bythe VAP calculation compared with the experimental data. The K^π=0^+ band-head energy of the VAP is E_x(0^+_2)=13.1 MeV, which is about twice higher than the experimental value (7.16 MeV). The VAP+αGCM calculation gives a better result forthe ^12C+α band energies owing to rotation and internal excitation of the^12C cluster. The VAP+αGCM gives E_x(0^+_2)=9.7 MeV of the band-head energy.The energyis still higher than the experimental value, but the overestimationis significantly improved by the present VAP+αGCM calculation compared with thetheoretical value E_x(0^+_2)∼ 16 MeV ofmicroscopic cluster model calculations with the Volkov interaction. As a result of the significant energy reduction of the ^12C+α states, the ordering of the groundand ^12C+α bands is reversed at J^π=4^+ from the VAP to the VAP+αGCM. The 4^+_1 state is the ground band member in the VAP, whereas it belongs to the K^π=0^+^12C+α band in the VAP+αGCM consistently to the usual assignment of the experimental levels and also that of Ref. <cit.>. Strictly speaking, state mixing occurs between the 4^+_1 and 4^+_2 states as discussed later. In Table <ref>,the calculated E2, E3, E4, and isoscalar dipole (IS1) transition strengths areshown compared with experimental data. We also show the theoretical values ofRef. <cit.> calculated bya semi-microscopic ^12C+α cluster model with theorthogonal condition model (OCM) <cit.>. In the VAP result, the E2 transition for 4^+_1→ 2^+_1 is weak because these states belong to different bands, which contradicts to the experimental strong E2 transition.On the other hand,the B(E2) value calculated by the VAP+αGCM is as large as the experimental data. Relatively large B(E2) values for the in-band transitions in the ^12C+α bands are consistent with experimental data. The E3 transition strength for 3^-_1→ 0^+_1 is considerably large in both calculations because of the dominant tetrahedral 4α component in the 0^+_1 and 3^-_1 states.The calculated B(E3) is in good agreement with the experimental data.It should be pointed out thatthe ground state 4α correlation gives important contribution to the enhancement of B(E3;3^-_1→ 0^+_1). Indeed, if we assume the p-shell closed configuration of the final 0^+ state,the strength from the 3^-_1 state of the VAP+αGCM becomes as small as B(E3)=26 e^2fm, indicating that higher shell components contribute to the dominant part of the E3 strength. The ^12C+α OCM calculation fails to reproduce the large B(E3) value becausespatially developed 3α configurations in the ^12C cluster are ignored in the calculation.The calculated values of B(E4) for 4^+_1→ 0^+_1 are consistent withthe experimental value in both of the VAP and VAP+αGCM calculations. Naively, it seems to contradict to the different assignments of the 4^+_1 state in two calculations. In the VAP result, the 4^+_1 state belongs to the ground band with a dominant tetrahedral 4α component.The tetrahedral intrinsic state gives B(E4)=260 e^2fm^8 in the VAP.On the other hand, in the VAP+αGCM, the 4^+_1 is dominated by the ^12C+α component different from the dominant tetrahedral 4α component of the 0^+_1. However, in the VAP+αGCM result,mixing of the tetrahedral 4α and ^12C+α components occurs in the 4^+_1 and4^+_2 states and enhances the B(E4;4^+_1→ 0^+_1) value.Moreover, slight mixing of the ^12C+α component in the 0^+_1 also increases the E4 strength in the VAP+αGCM. Consequently, the calculated B(E4;4^+_1→ 0^+_1) is B(E4;4^+_1→ 0^+_1)=360e^2fm^8 in the VAP+αGCM. Note that the ground state 4α correlation gives significant contributionto the E4 transition. If we assume the p-shell closed 0^+ state, the B(E4) values for the 4^+_1 state of the VAP is reduced to be B(E4)=17 e^2fm^8and that of the VAP+αGCMis B(E4)=4 e^2fm^8. It indicates again thathigher shell components enhance the E4 strength. In the experimental measurement of E4 transitions by (e,e') scattering <cit.>, it has been reported thatthe E4 transition strength for 0^+_1 → 4^+_2 is the same order as that for 0^+_1 → 4^+_1.It may support the strong state mixing between two 4^+ states.For the IS1 strength, it has been experimentally known that the low-energy IS1 strength exhausts significant fraction of the energy weighted sum rule (EWSR) <cit.>. The present calculation gives the significant IS1 strength for 0^+_1→ 1^-_1 with 4-5 % of the EWSR,which is consistent with the experimental data.§.§ Intrinsic structures Since a single AMD wave function is given by a Slater determinant, the AMD wave function Φ_AMD(Z^ opt_J^π_n) optimized for J^π_n in the VAPis regarded as the intrinsic state of the corresponding state. Density distribution of the intrinsic states obtained by the VAP is shownin Figs. <ref> and <ref>.The intrinsic density shows that four α clusters are predominantly formed inthe ground and excited states of ^16O. The 0^+_1, 4^+_1, 1^-_1, 2^-_1, and3^-_1 states show tetrahedral 4α structures. The shapes are not an ideal tetrahedral configuration with the T_d symmetrybut somewhat distorted tetrahedral ones. On the other hand, the 0^+_2, 2^+_1, 4^+_2, and 5^-_1 states show ^12C+α cluster structures, in whichan α cluster is located far from the ^12C cluster with 3α structures. In particular, in the4^+_2 state, the 3α structure of the ^12C cluster is clearly seen and the last α cluster is aligned almost on the 3α plane. It is a similar structure to the planer 4α configuration suggestedin Ref. <cit.> for the excited K^π=0^+ band.In the VAP+αGCM, the ^16O wave functions are expressed bysuperposition of the VAP and ^12C(AMD)+α wave functions.In general, a strong-coupling cluster having a specific intrinsic shape such as the tetrahedral or planar 4α structures may have a large overlap with the dominantconfiguration, whereas a weak-coupling cluster state such as the^12C(0^+_1)+α structure should contain components of various configurations. For the 0^+_1,1^-_1,2^-_1, and 3^-_1 states, the VAP+αGCM wave functions contain significant components of the corresponding VAP states with more than 80% overlap, meaning that these states are understood asstrong-coupling cluster states with the specific intrinsic shapes.For each state, the dominant VAP wave function can be approximately regarded as the intrinsic state. On the other hand, the ^12C+α band members of the VAP+αGCM contain the VAP component with less than 55% and somewhat showweak-coupling cluster features.As discussed later, the states in the K^π=0^±^12C+α bands contain significant ^12C(0^+_1)+α componentwith a rather large inter-cluster distance indicating that they are understood as weak-coupling^12C(0^+_1)+α states. For 4^+ states,the 4^+_1 of the VAP+αGCM has minor (30%)VAP 4^+_1 component and major (50%) VAP 4^+_2 component, whereas the 4^+_2 of the VAP+αGCMhasdominant (55%) VAP 4^+_1 and minor (30%) VAP 4^+_2 components. It indicates thatthe level inversion occurs between the VAP and VAP+αGCM calculations:the 4^+_1 and 4^-_2 states in theVAP+αGCM are approximatelyassigned to the ^12C+α and tetrahedral 4α bands, respectively.However, they still contain minor VAP components by 30%indicating significant state mixing between two 4^+ states. As discussed in 1937 by Wheeler <cit.>,the ideal tetrahedral 4α configuration with the T_d symmetry and its vibrationconstruct specificband structures. We here discuss how much components of the T_d modes are contained in the 0^+, 3^-, 4^+, and 1^- states obtained in the present calculation. We introduce the Brink-Bloch(BB) 4α-cluster wave function <cit.> with the T_d symmetry and that forthe vibration mode and calculate overlap with the VAP and VAP+αGCM wave functions. The BB 4α-cluster wave function is written asΦ^4α(S_1,S_2,S_3,S_4) = A{Φ_α(S_1)Φ_α(S_2) Φ_α(S_3) Φ_α(S_4) }.For the BB 4α-cluster wave function Φ^4α_T_d(d)with the T_d symmetry,we choose the 4α configuration S_iS_1=d/√(3)(1, 1, -1), S_2=d/√(3)(1, -1, 1), S_3=d/√(3)(-1,-1, -1), S_4=d/√(3)(-1, 1, 1).Here d is the size parameter of the T_d-invariant tetrahedral configuration of 4α. For the wave function Φ^4α_T_d(F)(d) corresponding tothe mode “F”, which is a vibration mode for the 1^- stateon the T_d-invariant tetrahedral configuration,we set S_1=d/√(3)(1, 1+ϵ, -(1+ϵ)), S_2=d/√(3)(1, -1+ϵ), 1+ϵ), S_3=d/√(3)(-1,-(1-ϵ), -(1-ϵ)), S_4=d/√(3)(-1, (1-ϵ), (1-ϵ)),where ϵ is taken to be an enough small value.The VAP wave functions for the 0^+_1, 3^-_1, and4^+_1 stateshave maximum overlaps with the J^π-projected T_d wave functionand that for the 1^-_1 state has maximum overlap withthe J^π-projected T_d(F) wave function at afinite size d=1.5-1.7 fm. Table <ref> shows calculated values of the T_d and T_d(F) componentsin theVAP and VAP+αGCM wave functions at d=1.6 fm. It also shows the components in the single-base VAP wave functionΦ^Jπ(Z^ opt_J^π_1) without configuration mixing of the VAPwave functions. The 0^+_1 and 3^-_1 states contain significant T_d component as 90% and 60-70%, respectively, leading to an interpretation that they are approximatelyregarded as the T_d band members. However, there is no 4^+ state havingdominant T_d component. Since the single-base VAP wave function for the 4^+_1 statehas a distorted tetrahedral intrinsic structure, two 4^+ states obtained from K=0 and K=2 components by the J^π projection share the T_d component. These two 4^+ statescorrespond to the4^+_1 and 4^+_3 states in the VAP result.In the VAP+αGCM result, the T_d component is fragmented furtherbecause of mixing with ^12C+α components. As a result of the distortion from the T_d symmetry of the intrinsic 4αstructure and the mixing with ^12C+α components the T_d component of the 4^+_2 state is reduced to 16% in the VAP+αGCM result.For the vibration T_d(F) mode, the obtained 1^-_1 state contains significantT_d(F) component as 50-60% meaning that the 1^-_1 state can be roughly categorized into the T_d(F) band.Elliott assigned the 0^+_1, 3^-_1, and 4^+_2 states as the T_d band <cit.>, whereas Bijker and Iachello assigned the 0^+_1, 3^-_1, and 4^+_1 states as the T_d band and the 1^-_1 state as the vibrationT_d(F) band <cit.>. For the 0^+_1, 3^-_1,and 1^-_1 states,our result is approximately consistent with their assignment. However for the 4^+ state,it is indicated thatthe T_d symmetry is not stable at J^π=4^+ and its identity does not remain in 4^+ states of ^16O. As discussed previously, we assigned the 0^+_1, 3^-_1, and 4^+_2 statesas the ground band members in the VAP+αGCMbecause they containmore than 55% components of the corresponding VAP wave functions, which clearlyshow the similar tetrahedral 4α structure.Our assignment of the 4^+_2 to the ground band is consistent with that by Elliott, butthe ground band is constructedfrom tetrahedral 4α structure somewhat distorted from the ideal T_d symmetry in the present result.It should be also noted again thatsignificant state mixing occurs between 4^+_1 and 4^+_2 states.§.§ ^12C+α cluster feature Figures <ref> and <ref> show^12C(0^+_1)+α component in thepositive- and negative-parity states, respectively. The component is calculated by overlap with a ^12C(0^+_1)+α cluster wave function at a certain distance (d) in the same way as Ref. <cit.>.The ^12C(0^+_1) cluster configuration is determinedin an asymptotic region (d=8.4 fm is chosen in the present case)by diagonalization within a fixed-d model space ofΦ^Jπ K_^12 C+α(d=8.4 fm,Ω_j,Z^opt_β_C).We truncate intrinsic configurations of ^12C as β_C=0^+_1 and 0^+_2for simplicity.In Figs. <ref> and <ref>,the calculated result for positive- and negative-parity states of the VAP+αGCM is shown as functions of d.The 0^+_2, 2^+_1, and 4^+_1 states have large ^12C(0^+_1)+α component with maximum amplitude around d=5 fm, indicating that these states have the spatially developed^12C(0^+_1)+α cluster structure.The 1^-_2, 3^-_2, and 5^-_1 states alsohave large ^12C(0^+_1)+α component with maximum amplitude around d=6-7 fm and show further development of the^12C(0^+_1)+α cluster structure.It indicates that these states belong to the K^π=0^± bands constructed from the ^12C(0^+_1)+α structure, consistently with the strong in-band E2 transitions. In Table <ref>, calculated α-decay widths are compared with experimental data.We also show the theoretical values of a semi-microscopic^12C+α calculation in Ref. <cit.>. The reduced widths are evaluated from the ^12C(0^+_1)+α component at a channel radius a <cit.>.The squared dimensionless α-decay width θ^2_α evaluated fromthe experimental α-decay widths are remarkably large for the 0^+_2, 2^+_1, 4^+_1, 1^-_2, 3^-_2,5^-_1 states. The calculated θ^2_α values at a=6.0 fmreasonably agree with the experimental values and alsoare consistent with the theoretical values of Ref. <cit.>.Figures <ref> and<ref> showoccupation probability of oscillator quanta N shells in a harmonic oscillator basis expansion. Here we use the size parameterb=1/√(2ν) of the harmonic oscillator. For the ^12C+α cluster states, the occupation probability isdistributed widely in a higher shell region. In particular, the distribution is verybroad in the VAP+αGCM result because of the spatially developed ^12C+α cluster structure.The 0^+_1, 1^-_1, 3^-_1, and 4^+_1 states of the VAP result are dominated by the N=12, 13, 13, 14 shell component corresponding to the0p0h, 1p1h,1p1h,and 2p2h on the p shell, respectively. However, they also contain more than 50% higher shell components, which come from cluster correlations in the finite size 4α configurations. The higher shell mixing in these states becomes large in the VAP+αGCM result. It should be stressed that the significant higher shell mixing in the 0^+_1 state, i.e. the ground state cluster correlation enhances B(E3;0^+_1→ 3^-_1) and B(E3;0^+_1→ 4^-_1) as mentioned previously. § SUMMARY AND OUTLOOKSWe have investigated structures of the ground and excited states of ^16O with the AMD+VAP method combined with the ^12C(AMD)+α GCM. The present result reasonably reproduces the experimental data of energy spectra,E2, E3, E4, and IS1 transitions as well as α-decay properties.The formation of 4 α clusters has been confirmed from nucleon degrees of freedom in the present calculationwithout assuming existence of any clusters. They form the (distorted) tetrahedral 4α structure in the low-lying states,0^+_1, 3^-_1, 4^+, 1^-_1, and 2^-_1, and the ^12C+α cluster structures in the excited states near and above the^12C+α threshold.The 0^+_1, 3^-_1, and 4^+_2 states are assigned to the ground band constructed fromthe tetrahedral 4α structure. The tetrahedral 4α structure does not necessarily have the ideal tetrahedral configurationwith the T_d symmetry, but a somewhat distorted tetrahedral shape.Nevertheless,the 0^+_1 and 3^-_1 have significant (90% and 60%) component of theT_d-invariant 4α configuration projected onto the J^π eigen state,and therefore, they can beapproximately interpreted as the T_d band members.In 4^+ states,the T_d component is shared mainly by two 4^+ states because of the distortion of the tetrahedral shape from the T_d symmetry, and fragmented further by mixing of ^12C+α states. It indicates that the tetrahedral 4α structure may be rather fragile at J^π=4^+,and the ideal T_d member with 4^+ does not appear in ^16O. The 1^-_1 state can be roughly categorized into the vibration mode, T_d(F) band.Our assignment of the 4^+_2 to the ground band is consistent with that of Ref. <cit.>. However, the 4^+_2 state is not the ideal T_d member buthas the distorted tetrahedral 4α shape as the dominant component. It should be also noted thatsignificant state mixing occurs between 4^+_1 and 4^+_2 states. The ^12C+α cluster structure constructs the K^π=0^+ band consisting of the 0^+_2, 2^+_1, 4^+_1 and the K^π=0^- band of 1^-_1, 3^-_2, 5^-_1. These states contain the dominant ^12C(0^+_1)+α component and largeα-decay widths, which are consistent with experimental observations. The present result for theK^π=0^± ^12C(0^+_1)+α bands are consistent with those ofthe semi-microscopic and microscopic cluster model calculations<cit.>. The present assignment of the 4^+_1 state to the ^12C+α band issupported by experimental data of the strong E2 transition to the 2^+_1 and the large α-decay width.The E3 and E4 transition strengths have been discussed. The calculated B(E3;3^-→ 0^+_1) is in good agreement with the experimental data. The E3 strength is enhanced because of the tetrahedral 4α structure in the 0^+_1 and 3^-_1 states.The ground state 4α correlation plays an important role in the enhancement of the E3 strength. For the E4 strength, the present calculation reproduces well theexperimentalB(E4;4^+_1 → 0^+_1). Historically, the significant B(E4) measured by (e,e') scattering has often drownattention to cluster structure of the 4^+_1, which could be the ground band member with the T_d symmetry. In the present result of the VAP+αGCM, the 4^+_1 belongs to not the ground band butthe ^12C+α band starting from the 0^+_2 state. Although, inter-band transitions are generally weak, however, the B(E4;4^+_1 → 0^+_1) is increased bythe significant state mixing of the^12C+α and tetrahedral 4α structures between 4^+_1 and 4^+_2 states and also by slight mixing of the ^12C+α component in the 0^+_1. As a result, the calculated B(E4;4^+_1 → 0^+_1) is as large as the experimental data in spite of the different structures in the initial and final states. In the experimental measurement using (e,e') scattering <cit.>, it has been reported thatthe E4 transition strength for 4^+_2 → 0^+_1 is the same order as that for 4^+_1 → 0^+_1.It may support significant state mixing between two 4^+ states.In the traditional microscopic cluster model calculations with the Volkov interaction (a density-independent two-bodyinteraction), it has been known thatexcitation energies of the ^12C+α cluster states are highly overestimated. The excitation energies of ^12C+α cluster states are largely improved in the presentcalculation. One of the main reasons is that we used the effective interaction with the zero-range three-body term, with whichthe α-decay threshold energy is reproduced.Internal excitation and angular momentum projection of the subsystem ^12C clusteralso give significant contribution to the energy reduction ofthe ^12C+α cluster states. However, the theoretical excitation energies is still higher than the experimental data. For better reproduction of the experimental energy spectra,further improvement of the model space with more sophisticated effective interactions including the tensor force may be necessary. Moreover, the present calculation is based on the bound state approximation. Coupling with continuum states should be carefully taken into account to discussdetailed properties of resonances. 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Taniguchi,PTEP 2014, 073D02 (2014). | http://arxiv.org/abs/1705.09097v1 | {
"authors": [
"Yoshiko Kanada-En'yo"
],
"categories": [
"nucl-th"
],
"primary_category": "nucl-th",
"published": "20170525090951",
"title": "Tetrahedral $4α$ and $^{12}\\textrm{C}+α$ cluster structures in $^{16}$O"
} |
=0.0cm [email protected] de Física, Centro de Investigación y de Estudios Avanzados del IPN, Apartado Postal 14-740, 07000 Ciudad de México, Mé[email protected] de Física, Universidad Nacional de Colombia, Código Postal 11001, Bogotá, [email protected] Santiago de Cali, Facultad de Ciencias Básicas, Campus Pampalinda, Calle 5 No. 62-00, Código Postal 76001, Santiago de Cali, [email protected] de Física, Universidad de Los Andes, Código Postal 111711, Bogotá, Colombia In this work, the lepton-number-violating processes in |Δ L|=2 decays of Λ_b^0 baryon, Λ_b^0 → p π^+μ^- μ^- and Λ_b^0 →Λ_c^+ π^+μ^- μ^-, are investigated for the first time, via an intermediate on-shell Majorana neutrino N with a mass in the GeV scale. We explore the experimental sensitivity of thesedimuon channels at the LHCb and CMS experiments, in which heavy neutrino lifetimes in the accessible ranges of τ_N = [1, 100, 1000] ps are considered. For a integrated luminosity collected of 10 and 50 fb^-1 at the LHCb and 30, 300 and 3000 fb^-1 at the CMS, we found significant sensitivity on branching fractions of the order BR(Λ_b^0 → p π^+μ^- μ^-) ≲𝒪(10^-9 - 10^-8) andBR(Λ_b^0 →Λ_c^+ π^+μ^- μ^-) ≲𝒪(10^-8 - 10^-7). Exclusion regions on the parameter space (m_N,|V_μ N|^2) associated with the heavy neutrino are presented and compared with those from K^- →π^+μ^-μ^- (NA48/2) and B^- →π^+μ^-μ^- (LHCb) as well as by different search strategies such as NA3, CHARMII, NuTeV, Belle, and DELPHI. Exploring GeV-scale Majorana neutrinos in lepton-number-violating Λ_b^0 baryon decays José D. Ruiz-Álvarez December 30, 2023 ===================================================================================== § INTRODUCTION At present, the possibility of extending the Standard Model (SM) by including right-handed sterile neutrinos with masses in the GeV-scale as an explanation of the neutrino mass generation, via a low-scale seesaw model, has taken strength both from the theoretical and experimental points of view <cit.>. On the theoretical side, seesaw scenarios with neutrinos masses below the electroweak scale are technically possible, without invoking higher energy scales <cit.>. Additionally, these GeV-scale sterile neutrinos could also explain simultaneously the baryon asymmetry of the Universe via leptogenesis <cit.>. While on the experimental side, such sterile neutrinos may be produced and studied in a large variety of current and future experiments, both at the intensity and energy frontier, rendering it to a falsifiable scenario (for recent reviews, see Refs. <cit.> and references therein).An interesting search strategy for heavy Majorana neutrinos in the GeV range, it is to look for rare phenomena in which the total lepton number L is broken by two units, generally referred to as |Δ L|=2 processes. These sort of processes are forbidden in the SM and remain the best way to discern if neutrinos are Majorana fermions <cit.>. The most appealing test of such a lepton-number-violating (LNV) processes is the neutrinoless double-β (0νββ) decay <cit.>. Although the case in which the exchange of a light massive Majorana neutrino is considered as the usual interpretation (standard mechanism) <cit.>, recently, Refs. <cit.> have found that the rate for this process can also be enhanced due to a dominant contribution from heavy neutrino exchange with masses in the GeV-scale. Up to now, the 0νββ decay seems to be a rather elusive process and has not yet been observed experimentally. Currently, the best limits on their half-lives have been obtained from the nuclei ^76 Ge<cit.> and ^136 Xe<cit.>. The non-observation allow us to set strong bounds on the mixing of a heavy neutrino N with the electron (V_eN) <cit.>.The low-energy studies of rare processes in |Δ L|= 2 decays of pseudoscalar mesons and the τ lepton have been extensively studied <cit.>,as alternative LNV processes to 0νββ decay <cit.>. In these |Δ L|= 2 decays, a sterile heavy neutrino with masses around 0.1 GeV ≲ m_N ≲ few GeV can be produced on their mass-shell and its signal could be detected at different intensity frontier experiments. According to their final-state topology, they can be classified as the following: *three-body channels <cit.> * M^- → M^' +ℓ_i^- ℓ_j^-, * τ^- →ℓ_α^+ M^' - M^'' - ,*four-body channels <cit.> * M̅^0→ M^'' + M^' +ℓ_α^- ℓ_β^- ,* M^- → M^'' 0 M^' +ℓ_i^- ℓ_j^-,* τ^-→ M^' +ν_τℓ_α^- ℓ_β^-,* M^- →ℓ_i^- ℓ_j^-ℓ^' +ν_ℓ^',where M ∈{ K, D, D_s, B, B_c } represents the decaying meson, ℓ_i(j) and ℓ^(')∈{ e, μ, τ} are the leptonic flavors, and M^' and M^'' represent final hadronic states that are allowed by kinematics. The possibility of CP violation detection in Δ L=2 decays of charged mesons <cit.> and the τ lepton <cit.> have been also explored.Experimentally, some of these |Δ L|= 2 decays have been pursued for many years by different flavor facilities. No evidence has been seen so far, and upper limits on their branching fractions have been reported by the Particle Data Group (PDG) and several experiments such as NA48/2, BABAR, Belle, LHCb, and E791<cit.>. At CERN, further improvements are expected by the NA62 kaon factory <cit.> and the LHCb in Run 2 and the future upgrade Run 3 <cit.>. In addition, the forthcoming Belle II experiment aims to get ∼ 40 times more data than the those accumulated by its predecessor Belle (as well as BABAR) <cit.>. All these efforts will increase the sensitivity on |Δ L|= 2 signals by 1 or 2 orders of magnitude. For instance, the future prospect of the Belle II search for the channel B^- →π^+μ^- μ^- has been discussed in Ref. <cit.>. Furthermore, these improvements will allow the search for those signals not yet explored.Aside from the LNV processes of pseudoscalar mesons and the τ lepton, the possibility of |Δ L|= 2 decays of hyperons <cit.> and charmed baryons <cit.> has been also studied, namely, the three-body decayℬ_A^- →ℬ_B^+ℓ_i^- ℓ_j^- as is graphically represented in Fig. <ref>(a), where ℬ_A(ℬ_B) denotes an initial (final) baryon. From the experimental side, so far, the HyperCP <cit.> and E653 <cit.> Collaborations have reported limits on the branching fractions of Ξ^- → p μ^-μ^- and Λ_c^+ →Σ^- μ^+μ^+, respectively.With the large number of hyperons that are expected to be produced at the BESIII experiment, these |Δ L|= 2 hyperon decays can be also searched <cit.>. In addition to the |Δ L|= 2 three-body decays of a charged baryon [Fig. <ref>(a)], |Δ L|= 2 four-body decays of a neutral baryon ℬ_A^0 →ℬ_B^+ π^+ℓ_i^- ℓ_j^- are possible as well, as is shown in Fig. <ref>(b).In the case of the exchanged of a GeV-scale Majorana neutrino, this four-body process is generated through an s-channel in which the neutrino can be produced on-shell and dominates over the t-channel three-body one <cit.>. Keeping this in mind and since the production of the Λ_b^0 baryon is around ∼ 5% of the total b-hadrons produced at the LHC, in this work, we will study new LNV processes in the four-body |Δ L|=2 decays of Λ_b^0 baryon, Λ_b^0 → p π^+μ^- μ^- and Λ_b^0 →Λ_c^+ π^+μ^- μ^-, in the scenario provided by the production of an on-shell Majorana neutrino. Within this simplified model approach, one heavy neutrino N mixing with one flavor of SM lepton (ℓ=μ) and its interactions are completely determined by the mixing angle V_μ N. Because of the relatively high muon reconstruction system, we focus on these same-sign dimuon channels and explored their expected sensitivity at the LHCb and CMS experiments. We will show that their experimental search allow us to put bounds on the parameter space associated with the mass m_N and mixing |V_μ N|^2 of the heavy Majorana neutrino. It is worth it to mention that the present work can be easily extendible to other b-baryons such as Ω_b, Ξ_b, and Σ_b, which are expected to be produced at the LHC in a lesser number than Λ_b.This work is organized as follows. In Sec. <ref>, we study the four-body |Δ L|=2 decays of Λ_b^0 baryon. The expected experimental sensitivity for these channels at the LHCb and CMS experiments is presented in Sec. <ref>. In Sec. <ref>, based on the results of the previous sections, we estimate the constraints on the parameter space(m_N,|V_μ N|^2) of the heavy neutrino that can be achieved, in which a comparison with different search strategies is also presented. Our conclusions are presented in Sec. <ref>.§ FOUR-BODY |Δ L|= 2 DECAYS OF Λ_B BARYON In this section, we explore the four-body |Δ L| = 2 decays of the Λ_b^0 baryon Λ_b^0 →ℬ^+ π^+ℓ_i^- ℓ_j^-, which can occur via the exchange of a Majorana neutrino with a kinematically allowed mass, namely(m_ℓ_j + m_π) < m_N < (m_Λ_b - m_ℬ-m_ℓ_i), where ℬ^+ = p , Λ_c^+ denotes a final-state baryon and ℓ_i(j)=e, μ, τ. The corresponding diagram is shown in Fig. <ref>(b), with ℬ_A = Λ_b and ℬ_B =ℬ. Among the possible same-sign dilepton final states, we will focus on the dimuon channels Λ_b^0 → (p, Λ_c^+) π^+μ^- μ^- and assume that only one on-shell heavy neutrino N dominates these processes.The|Δ L| =2 decays Λ_b^0 → (p, Λ_c^+) π^+μ^- μ^- receive the effect of a heavy Majorana neutrino with a mass in the ranges, Λ_b^0→p π^+μ^- μ^- :m_N ∈ [0.25,4.57] GeV,Λ_b^0→ Λ_c^+ π^+μ^- μ^- :m_N ∈ [0.25,3.23] GeV, respectively. Within these mass ranges, the total decay width of the intermediate Majorana neutrino N(Γ_N) is much smaller than its mass, Γ_N ≪ m_N<cit.>, so the narrow width approximation is valid. This allows us to consider the Majorana neutrino as a particle that is produced on its mass shell through the semileptonic decay Λ_b^0 →ℬ^+ μ^-N, followed by the subsequent decay N →μ^-π^+. In this on-shell factorization approach, the branching fraction of Λ_b^0 →ℬ^+ π^+μ^- μ^- is then split into two subprocesses,BR( Λ_b^0 →ℬ^+π^+μ^-μ^-)=BR(Λ_b^0 →ℬ^+ μ^- N)×Γ(N →μ^-π^+) τ_N/ħ,with τ_N as the lifetime of the Majorana neutrino. The decay width of N →μ^-π^+ is given by the expression <cit.>Γ(N→ μ^-π^+) = G_F^216 π|V_ud^CKM|^2 |V_μ N|^2 f_π^2 m_N √(λ(m_N^2,m_μ^2,m_π^2)) ×[ (1- m_μ^2m_N^2)^2 - m_π^2m_N^2(1+ m_μ^2m_N^2) ],where G_F is the Fermi constant, V_ud^CKM is the up-down Cabibbo-Kobayashi-Maskawa (CKM) matrix element, and f_π is the pion decay constant. The usual kinematic Källen function is denoted by λ(x,y,z)=x^2+y^2+z^2-2(xy+xz+yz). The coupling of the heavy neutrino (sterile) N to the charged current of lepton flavor μ is characterized by the quantity V_μ N <cit.>. Both its mass m_N and V_μ N are unknown parameters that can be constrained (set) from the experimental non-observation (observation) of |Δ L| =2 processes <cit.>. The lifetime of the Majorana neutrino τ_N=ħ / Γ_N in (<ref>) can be obtained by summing over all accessible final states that can be opened at the mass m_N <cit.>. However, in further analysis (Secs. <ref> and <ref>), we will leave it as a phenomenological parameter accessible to the LHCb and CMS experiments.To obtain the branching fraction of the semileptonic subprocess Λ_b^0 →ℬ^+μ^- N, we begin from its amplitude, which is given by the expression M( Λ_b^0→ℬ^+μ^-N)= G_F√(2) V_Qb^CKM V_μ N⟨ℬ(p_ℬ) |Q̅γ_α(1-γ_5) b|Λ_b (P) ⟩ × [u̅(p_μ) γ^α (1-γ_5) v(p_N)],where V_Qb^CKM is the CKM matrix element involved, with Q = u, c for ℬ=p, Λ_c, respectively. The matrix element of the vector and axial-vector currentsassociated to the baryonic transition Λ_b →ℬ can be parametrized as <cit.>⟨ℬ(p_ℬ) |Q̅γ_α b|Λ_b(P)⟩ =u_ℬ(p_ℬ)[ γ_αf_1^V(t)+i σ_αβq^βf_2^V(t) /m_Λ_b+ q_αf_3^V(t) /m_Λ_b] u_Λ_b(P), ⟨ℬ(p_ℬ) |Q̅γ_αγ_5 b|Λ_b(P)⟩ =u_ℬ (p_ℬ)[ γ_αf_1^A(t)+i σ_αβq^βf_2^A(t) /m_Λ_b+ q_αf_3^A(t) /m_Λ_b]γ_5 u_Λ_b(P), in terms of six transition form factors (f_1^V,f_2^V,f_3^V) and (f_1^A,f_2^A,f_3^A), where q=(P-p_ℬ) is the transferred momentum and t=q^2. The spinors of ℬ and Λ_b are represented by u_ℬ and u_Λ_b, respectively. We end up with a branching ratio of Λ_b^0 →ℬ^+μ^- N given by the following expressionBR( Λ_b^0→ℬ^+ μ^- N) = G_F^2 τ_Λ_b512π^3 m_Λ_b^3 ħ |V_Qb^ CKM|^2 |V_μ N|^2∫_(m_μ +m_N)^2^Δ_-^2 dt√(λ(m_μ^2,m_N^2,t)λ(m_Λ_b^2,m_ℬ^2,t))×{16/3t^3 [f_1^V(t)]^2α_1^V(t) + 8/3 m_Λ_b^2 t^2[f_2^V(t)]^2 α_2^V(t) + 8/3 m_Λ_b^2 t[f_3^V(t)]^2 α_3^V(t) + 32/m_Λ_b t^2[ f_1^V(t)f_2^V(t)α_12^V(t)+ f_1^V(t)f_3^V(t)α_13^V(t)]+ 16/3t^3 [f_1^A(t)]^2α_1^A(t) + 8/3 m_Λ_b^2 t^2[f_2^A(t)]^2 α_2^A(t) + 8/3 m_Λ_b^2 t[f_3^A(t)]^2 α_3^A(t) +32/m_Λ_b t^2[f_1^A(t)f_2^A(t)α_12^A(t)+ f_1^A(t)f_3^A(t)α_13^A(t)] } ,where the kinematic factors areα_1^V/A(t)=m_μ^2 [ t (Σ_-^2 - 2 m_N^2Σ_+) - 2 t^2 (m_N^2 ∓ m_Λ_b m_ℬ + Δ_∓^2) + 4m_N^2 Σ_-^2 + t^3] + (t-m_N^2) [ m_N^2 (t Σ_+ -2Σ_- +t^2) - t(Δ_∓^2 - t)(Σ_±^2 +2t)] ,α_2^V/A(t)= [t (m_μ^2 + m_N^2) + (m_μ^2 - m_N^2)^2 - 2t^2 ] [t (Σ_±^2 ± 4m_Λ_b m_ℬ) - 2 (Σ_-^2 + t^2)] , α_3^V/A(t)= (Δ_±^2 - t^2) [m_μ^2(t + 2m_N^2 -m_μ^2) + m_N^2 (t- m_N^2) ] , α_12^V /A(t)= Δ_± (Δ_∓ - t) [m_μ^2(t - 2m_N^2) +m_μ^4 + m_N^4 +m_N^2 t -2t^2], α_13^V/A(t)= Δ_∓ (Δ_± - t) [ (m_μ^2 - m_N^2)^2 -t(m_μ^2 + m_N^2)],with Δ_±= m_Λ_b± m_ℬ and Σ_±= m_Λ_b^2 ± m_ℬ^2. As a cross-check, we have verified that this expression is consistent with the one obtained in Ref. <cit.>. For ensuing numerical evaluations in Sec. <ref>, we will use the theoretical predictions obtained by Lattice QCD on the form factors (f_1^V,f_2^V,f_3^V) and (f_1^A,f_2^A,f_3^A) <cit.>. Besides, we will take the following numerical inputs: |V_ud^CKM|= 0.97417, |V_cb^CKM| = 40.5 × 10^-3, |V_ub^CKM|= 4.09 × 10^-3 <cit.>, and f_π= 130.2(1.7) MeV <cit.>. The masses of the particles involved and lifetime τ_Λ_b are taken from Ref. <cit.>.We close by mentioning that there are different calculations of the Λ_b → (p, Λ_c) form factors in the literature, for instance, the covariant confined quark model <cit.>. Using this model, we have checked that one gets very similar results as the ones presented in Sec. <ref> by means of Lattice QCD <cit.>. § EXPECTED EXPERIMENTAL SENSITIVITY AT THE LHC In this section, we provide an estimation of the expected number of events at the LHC, namely, LHCb and CMS experiments, for the |Δ L|=2 channels Λ_b^0→ℬ^+π^+μ^-μ^- (with ℬ = p, Λ_c), discussed above. §.§ LHCb experiment The number of expected events in the LHCb experiment has the form N_ exp^ LHCb = σ(pp→ H_b X)_ accf(b→Λ_b)BR(Λ_b^0 →Δ L=2) ×ϵ_D^ LHCb(Λ_b^0 →Δ L=2) P_N^ LHCb ℒ^ LHCb_ int,where σ(pp→ H_b X)_ acc is the production cross-section of b-hadrons inside the LHCb geometrical acceptance; f(b→Λ_b) is the hadronization factor of a b-quark to Λ_b^0 baryons; ℒ_ int^ LHCb is the integrated luminosity; BR(Λ_b^0→Δ L=2) corresponds to the branching fraction of the given LNV process; and ϵ_D^ LHCb(Λ_b^0 →Δ L=2) is its detection efficiency of the LHCb detector involvingreconstruction, selection, trigger, particle misidentification, and detection efficiencies. Most of the the on-shell neutrinos produced in the decaysΛ_b^0→ (p,Λ_c^+)μ^-N are expected to live a long enough time to travel through the detector and decay (N →π^+μ^-) far from the interaction region. This effect is given by the P_N^ LHCb factor (acceptance factor), which accounts for the probability of the on-shell neutrino N decay products to be inside the LHCb detector acceptance <cit.>. The reconstruction efficiency will depend on this acceptance factor as well.The production cross section is well measured to be σ(pp→ H_b X)_ acc=(75.3 ± 5.4 ±13.0) μb inside the LHCb acceptance <cit.>. The hadronization factor can be related with the total hadronization factor to baryons as f(b→baryons)≃ f(b→Λ_b)(1 + 2 f(b→ B_s^0)/f(b→ B^0)), where we have used isospin symmetry described in Ref. <cit.>. Thus, the hadronization factor can be built from Ref <cit.>, wheref(b→baryons) = 0.088 ± 0.012, f(b→ B_s^0) = 0.103 ± 0.005 and f(b→ B^0) = 0.404± 0.006, where these factors are computed as an average of LEP and Tevatron measurements. This leads to f(b→Λ_b)≃ 0.053 ± 0.017. Precise computation of the detection efficiency requires fully simulated decay-specific Monte Carlo samples, reconstructed in the same manner as real data and with a simulation of the full detector.However a rough estimation can be done with detection efficiencies already reported by the LHCb experiment in the study of some Λ_b^0 decays with the same or similar final-state particle content as our LNV modes, such as Ref. <cit.>. Here, the study of Λ_b^0 →[cc̅]pK− modes ( [cc̅] stands for Jψ, χ_c1, and χ_c2 charmonia states which decay into a pair of muons tracks) is performed, and the measured yields and efficiency corrected yields are given, and therefore ϵ_D^ LHCb per mode can be extracted. From that information, it is extracted that ϵ( Λ_b^0 →J/ψ(→μ^+μ^-)pK−)≃ 0.0246±0.0001, where itmust be mentioned that tight selection criteria are used, to maximize signal over background, given the large suppression of the decays under study, something similar to the |Δ L| = 2 decays of interest. On the other hand, in Ref. <cit.>,it is determined the following ratio of efficiencies ϵ( Λ_b^0 →μ^+μ^-pπ−)/ϵ( Λ_b^0 →J/ψ(→μ^+μ^-)pπ−) =0.49± 0.02, which are channels with identical topologies to Λ_b→ pπ^+μ^-μ^-, therefore we can safely state ϵ_D^ LHCb(Λ_b^0→ pπ^+μ^-μ^-)≃ 0.0121± 0.0005. As, Λ_b^0 →Λ_c^+(→ pK^-π^+)π^+μ^-μ^- involves two additional charged tracks, we can multiply above expression for 90% for each additional track, keeping same uncertainty in a conservative scenario, to obtain ϵ_D^ LHCb(Λ_b^0→Λ_c^+π^+μ^-μ^-)≃ 0.0098±0.0005. Finally, in Ref. <cit.>, reconstruction efficiencies for hypothetical long-lived particles inside the LHCb acceptance are given. Here we can observe that a maximum variation of about 25% is measured in the efficiencies of particles living in the [5 - 100] ps range, with masses up to 200 GeV/c^2. Thus, to account for this effect, we will just add a 25% relative uncertainty to our efficiency prediction, obtaining finally ϵ_D^ LHCb(Λ_b^0→ pπ^+μ^-μ^-) P_N^ LHCb ≃0.0121± 0.0030,ϵ_D^ LHCb(Λ_b^0→Λ_c^+π^+μ^-μ^-)P_N^ LHCb ≃0.0098± 0.0025 . The combination of all inputs to the number of expected events leads to a relative uncertainty of 45% in both LNV modes, where to compute N(Λ_b^0 →Λ_c^+(→ pK^-π^+)π^+μ^-μ^-) we have usedBR(Λ_c^+ → pK^-π^+)=(6.35±0.33)%.Assuming above assumptions on the efficiency and cross section, in Figs. <ref>[top] and <ref>[bottom] we plot the number of expected events to be observed in the LHCb experiment as a function of the branching fraction for Λ_b^0→ pπ^+μ^-μ^- and Λ_b^0→Λ_c^+π^+μ^-μ^-, respectively. The red and magenta regions correspond to a luminosity of ℒ_ int^ LHCb = 10 and 50 fb^-1 for LHC Run2 and LHC Run3, respectively. In Table <ref> we present the number of expected events for some selected values of branching ratios. We can see that values of branching fractions of the order 10^-9 - 10^-8 for Λ_b^0→ pπ^+μ^-μ^- and 10^-8 - 10^-7 for Λ_b^0→Λ_c^+π^+μ^-μ^- might be within the experimental reach of the LHCb. §.§ CMS experiment For the CMS experiment, the number of expected events is given by the expressionN_ exp^ CMS = σ(pp→Λ_b^0 X) BR(Λ_b^0 →Δ L=2)×ϵ_D^ CMS(Λ_b^0 →Δ L=2) P_N^ CMSℒ_ int^ CMS,where ℒ_ int^ CMS is the integrated luminosity, σ(pp→Λ_b^0 X) is the production cross-section of Λ_b^0 baryons inside the CMS geometrical acceptance proton-proton collisions, P_N^ CMS is the acceptance factor at the CMS, ϵ_D^ CMS(Λ_b^0 →Δ L=2) is the CMS experiment efficiency that involves the detection and trigger efficiencies and the geometrical acceptance to detect our signal, and BR(Λ_b^0 →Δ L=2) is its respective branching ratio. To calculate the efficiency of the CMS experiment to accept our signal we have to take into account the track reconstruction efficiency for charged pions (due to the lack of the particle identification, CMS assumes that all charged tracks are pions) as well as the muon reconstruction efficiency for the kinematics of the signal. The produced particles from Λ_b^0 decay have relatively low p_T. We consider that the pions and muons from our signal have mainly a p_T<20 GeV. From <cit.> CMS reconstruction efficiency of charged tracks in the tracker, for the p_T spectrum of interest, the reconstruction efficiency of pions from our signal is of 90% at 7 TeV proton-proton collisions. We assume that this efficiency remains mainly unchanged at 13 TeV. At 8 TeV the muon reconstruction efficiency for a p_T>3 GeV and p_T<20 GeV has been measured to be around 90% <cit.>. We also assume this efficiency remains unchanged at 13 TeV. Additionally, precise computation of the detection efficiency of our signal events requires fully simulated decay specific Monte Carlo samples, reconstructed using the same techniques as in real data and with a full simulation of the detector. However, a rough estimation can be done with detection efficiency already reported by the CMS experiment in the study of the Λ_b^0 baryon cross section <cit.>, which is 0.73% (with an uncertainty of approximately 10%). For the decay channel Λ_b^0 → pπ^+μ^-μ^- we have the same final-state particle, and thus we assume the same reconstruction efficiency, i.e., ϵ_D^ CMS(Λ_b^0 → pπ^+μ^-μ^-) = (0.73 ± 0.07)%. For the channel Λ_b^0 →Λ_c^+π^+μ^-μ^-, there are two other tracks (considering the chain decay Λ_c^+→ pK^-π^+). For this reason, the efficiency detection will be reduced, and taking into account the efficiency of these additional tracks, we will assume it to be ϵ_D^ CMS(Λ_b^0 →Λ_c^+π^+μ^-μ^-) = (0.59 ± 0.06)%. This is not an optimistic case, since the CMS Collaboration is making an important effort to improve the reconstruction capabilities in Run 2. We have considered a minimum and maximum neutrino lifetime of τ_N= 1 ps and 1000 ps, respectively, where the detector has sensitivity. Considering that the neutrino travels at nearly the speed of light and taking into account that the neutrino comes from the Λ_b^0 decay, the decay length of the neutrino is L_N= 0.3 cm (30 cm) for τ_N= 1 ps (1000 ps) lifetime.We consider that the decay length of Λ_b^0 is 0.4 cm because its lifetime is 1.466 ps <cit.>. According to Ref. <cit.>, the reconstruction efficiency for tracks originated at a distance of 30 cm from the collision point is 55% and for 1 cm is 100%, where we can observe a maximum variation of about 18%. Then, to consider this effect, we will add an 18% relative uncertainty to our efficiency prediction. Therefore, we estimate an acceptance factor0.55 ≤ P_N ≤ 1 for1 ps ≤τ_N ≤ 1000 ps. Of course, this is an estimation from studies performed by the CMS experiment at 7 TeV, and a precise estimation will require knowledge of the production and decay vertex, which can be adequately included during the data analysis. Finally, the efficiencies will be ϵ_D^ CMS (Λ_b^0 → pπ^+μ^-μ^-) P_N^ CMS ≃0.073 ± 0.015,ϵ_D^ CMS (Λ_b^0 →Λ_c^+π^+μ^-μ^-) P_N^ CMS ≃0.059 ± 0.013 .The cross section times the branching fraction σ(Λ_b)× BR(Λ_b→ J/ψΛ) [with p_T(Λ_b) > 10 GeV and | y(Λ_b)|<2.0] at 7 TeV measured by the CMS experiment is 1.16± 0.06± 0.12 nb <cit.>. It is not possible to directly infer the BR(Λ_b→ J/ψΛ) because there is not a published measurement of f(b →Λ_b). However, we will use the value estimated in the previous section f(b→Λ_b)≃ 0.053 ± 0.017. Now, the PDG reports BR(Λ_b→ J/ψΛ) × f(b →Λ_b) = (5.8 ± 0.8) × 10^-5. Using the previous numbers, we can find a cross section value of σ(pp→Λ_b X)=(1.06 ± 0.39)μb at 7 TeV . Extrapolating this value to 13 TeV assuming that the cross sections grows as the energy collision σ(pp→Λ_b X)=(1.97 ± 0.72)μb.Considering the values found for the cross section and efficiencies, Fig. <ref> shows the number of expected events to be observed in the CMS experiment for ℒ_ int^ CMS= 30, 300, and 3000 fb^-1 as a function of the branching ratio of Λ_b^0 into our LNV channels, where to compute N(Λ_b^0 →Λ_c^+(→ pK^-π^+)π^+μ^-μ^-) we have usedBR(Λ_c^+ → pK^-π^+)=(6.35±0.33)%. For integrated luminosities of 30 and 300 fb^-1, in Table <ref>, we present the number of expected events for some selected values of branching ratios. It is easy to see that for 3000 fb^-1 the number of events will increase 1 order of magnitude, since these scale in the same way as the luminosity.We found significant sensitivity at the CMS on branching fractions of the order 10^-9 - 10^-8 for Λ_b^0→ pπ^+μ^-μ^- and10^-8 - 10^-7 for Λ_b^0→Λ_c^+π^+μ^-μ^-. In the analysis of the next section, we will take these values of branching fractions as the most conservative ones and accessible to the LHCb and CMS experiments. § CONSTRAINTS ON THE (M_N,|V_Μ N|^2) PLANE Experimental limits from the search of |Δ L| =2 processes can bereinterpreted as constraints on the parameter space of a heavy sterile neutrino (m_N,|V_μ N|^2), namely, the squared mixing element |V_μ N|^2 as a function of the massm_N <cit.>. Based on the analysis presented in the previous section, here we explore the constraints on the (m_N,|V_μ N|^2) plane that can be achieved from the experimental searches on Λ_b^0 →(p, Λ_c^+) π^+μ^- μ^- at the LHC. From Eq. (<ref>), it is straightforward to obtain the relation|V_μ N|^2 = [ħBR(Λ_b^0 →ℬ^+ π^+μ^- μ^-)/ BR(Λ_b^0 →ℬ^+ μ^- N) ×Γ(N →μ^-π^+) τ_N]^1/2whereBR(Λ_b^0 →ℬ^+ μ^- N)=BR(Λ_b^0 →ℬ^+ μ^- N) / |V_μ N|^2 , Γ(N →μ^-π^+)= Γ(N →μ^-π^+)/ |V_μ N|^2,are the normalized branching ratio BR(Λ_b^0 →ℬ^+ μ^- N) [Eq. (<ref>)] and decay width Γ(N →μ^-π^+) [Eq. (<ref>)] to the neutrino mixing |V_μ N|^2. As was already discussed in Sec. <ref> and following the analysis of NA48/2 <cit.> and the LHCb <cit.>, we will consider heavy neutrino lifetimes of τ_N = [1, 100, 1000] ps as benchmark points in our analysis. This will allow us to extract limits on |V_μ N|^2 without any additional assumption on the relative size of the mixing matrix elements. To illustrate the constraints that can be achieved from the experimental searches on Λ_b^0 →p π^+μ^- μ^-, in Figs. <ref>(a) and <ref>(b), we show the exclusions regions on |V_μ N|^2 as a function of m_N obtained by taking an expected sensitivity on the branching fractions of the order BR(Λ_b^0 →pπ^+μ^- μ^-) < 10^-8 and < 10^-9, respectively.In both cases, the gray, blue, light-gray regions represents the constraints obtained for heavy neutrino lifetimes of τ_N = [1, 100, 1000] ps, respectively. For the purpose of comparison, we also plotted the available exclusion limits obtained from searches on |Δ L|=2 channels: K^- →π^+μ^-μ^- (NA48/2) <cit.> and B^-→π^+μ^-μ^- (LHCb) <cit.>. The limit from the K^- →π^+μ^-μ^- channel is taken for τ_N = 1000 ps <cit.>. While for the B^-→π^+μ^-μ^- channel, we compare with the revised limit <cit.> from the LHCb analysis <cit.>. We can see in Figs. <ref>(a) and <ref>(b) that the most restrictive constraint is given by K^- →π^+ μ^-μ^- which can reach |V_μ N|^2∼𝒪(10^-5) but only for a very narrow range [0.25, 0.38] GeV of Majorana neutrino masses. For m_N > 0.38 GeV, the four-body channel Λ_b^0 →p π^+μ^- μ^- (CKM suppressed) would be able to extend the region of |V_μ N|^2 covered by the channel B^- →π^+μ^-μ^- (also CKM suppressed). As for the searches on Λ_b^0 →Λ_c^+ π^+μ^- μ^-, with a similar expected sensitivity at the LHC of BR(Λ_b^0 →Λ_c^+ π^+μ^- μ^-) < 10^-8 and < 10^-9, we show, respectively, the exclusion curves on the (m_N,|V_μ N|^2) plane in Figs. <ref>(a) and <ref>(b). Again, in both cases, the gray, blue, and light-gray regions present the constraints obtained for heavy neutrino lifetimes of τ_N = [1, 100, 1000] ps, respectively. It is important to remark that, due to the CKM mixing elements involved, the Λ_b^0 →Λ_c^+ π^+μ^- μ^- channel is a CKM-allowed process, and therefore it would be able to exclude regions of |V_μ N|^2 that are weaker than K^- →π^+μ^-μ^- and stronger than B^- →π^+μ^-μ^-.In addition, it is important to comment that for the GeV-scale of sterile neutrino masses relevant to this work, m_N ∈ [0.25,5.0] GeV, different search strategies have been used to get constraints on the mixing element |V_μ N| (for a recent review on the theoretical and experimental status see Refs. <cit.> and references therein). The lack of experimental evidence of searches of peaks in the muon spectrum of leptonic K^± decays (PS191, E949) and searches through specific visible channels of heavy neutrino decays produced in beam dump experiments (such as NA3, CHARM, and NuTeV, among others) allows us to put constraints on |V_μ N|^2 ∼𝒪(10^-8 - 10^-6)for masses of thesterile neutrino ranging from 0.2 to 2.0 GeV <cit.>. It is expected that the recently proposed high-intensity beam dump experiment SHiP <cit.> can significantly improve those bounds <cit.>. Moreover, in the mass range [0.5,5.0] GeV searches of heavy neutrinos have been performed by Belle using the inclusive decay mode B → X ℓ N followed by N →ℓπ (with ℓ = e, μ) <cit.>, and by DELPHI using the possible production of heavy neutrinos in the Z-boson decay Z →ν N<cit.>. For masses 5.0 GeV <m_N < m_W, the possibility of a heavy neutrino produced in W and Higgs boson decays has been studied as well; see, for instance Ref. <cit.>.In Fig. <ref>, we show the exclusion bounds on the (m_N,|V_μ N|^2) plane coming from Belle <cit.>, DELPHI <cit.>, NA3 <cit.>, CHARMII <cit.>, and NuTeV <cit.> experiments, in the mass range [0.5,5.0] GeV. In comparison, the constraints obtained from the searches on Λ_b^0 → (p,Λ_c^+) π^+μ^- μ^-are represented by the gray and blue regions, for a branching fraction of BR < 10^-8 and BR < 10^-9, respectively. In both cases, a lifetime τ_N = 1000 ps have been taken as a representative value. We observe that our Δ L=2 channels proposal would complement these bounds in the mass region around m_N ≃ 2.0 - 3.0 GeV.§ CONCLUSIONS In the present work, we have studied new LNV processes in the four-body |Δ L|=2 decays of the Λ_b^0 baryon, Λ_b^0 → p π^+μ^- μ^- and Λ_b^0 →Λ_c^+ π^+μ^- μ^-. We have investigated the production of a GeV-scale Majorana neutrino through these |Δ L|=2 decays, particularly, neutrino masses m_N ∈ [0.25,4.57] GeV and m_N ∈ [0.25,3.23] GeV. Because of the relatively high muon reconstruction system, we have paid attention on these same-sign dimuon channels and explored their experimental sensitivity at the LHCb and CMS. We considered heavy neutrino lifetimes of τ_N = [1, 100, 1000] ps, where the probability for the on-shell neutrino N decay products to be inside the detector (acceptance factor P_N) has been taken into account in our analysis. According to this analysis, it is found as conservative values, that for a integrated luminosity collected of 10 and 50 fb^-1 at the LHCb and 30, 300, and 3000 fb^-1 at the CMS; one would expect sensitivities on the branching fractions of the order BR(Λ_b^0 → p π^+μ^- μ^-) ≲𝒪(10^-9 - 10^-8) andBR(Λ_b^0 →Λ_c^+ π^+μ^- μ^-) ≲𝒪(10^-8 - 10^-7), respectively. With such sensitivities, we extracted constraints on the parameter space (m_N,|V_μ N|^2) that might be obtained from their experimental search for neutrino lifetimes of τ_N = [1, 100, 1000] ps. Depending on the τ_N value, these channels would be able to exclude regions of |V_μ N|^2that are weaker than K^- →π^+μ^-μ^- (NA48/2)and stronger than B^- →π^+μ^-μ^- (LHCb). In addition, we observed that our Δ L=2 channels proposal would complement the bounds given by different search strategies (such as NA3, CHARMII, NuTeV, Belle, and DELPHI), in the mass region around m_N ≃ 2.0 - 3.0 GeV. 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D88, 113001 (2013). | http://arxiv.org/abs/1705.10606v3 | {
"authors": [
"Jhovanny Mejia-Guisao",
"Diego Milanes",
"Nestor Quintero",
"Jose D. Ruiz-Alvarez"
],
"categories": [
"hep-ph"
],
"primary_category": "hep-ph",
"published": "20170526224621",
"title": "Exploring GeV-scale Majorana neutrinos in lepton-number-violating $Λ_b^0$ baryon decays"
} |
http://arxiv.org/abs/1705.09227v2 | {
"authors": [
"Paul M. Alsing",
"Edwin E. Hach III"
],
"categories": [
"quant-ph"
],
"primary_category": "quant-ph",
"published": "20170525152835",
"title": "Photon pair generation in a lossy microring resonator. I. Theory"
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|
http://arxiv.org/abs/1705.09118v3 | {
"authors": [
"Keisuke Ohashi",
"Toshiaki Fujimori",
"Muneto Nitta"
],
"categories": [
"cond-mat.quant-gas",
"hep-th"
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"primary_category": "cond-mat.quant-gas",
"published": "20170525101529",
"title": "Conformal symmetry of trapped Bose-Einstein condensates and massive Nambu-Goldstone modes"
} |
|
printfolios=true [ICFP'17]ACM SIGPLAN International Conference on Functional ProgrammingSeptember 04–06, 2017Oxford, UK PACMPL 2017 978-x-xxxx-xxxx-x/YY/MM 10.1145/nnnnnnn.nnnnnnn 1 ACM-Reference-Formatacmauthoryearcalc ifundefinedifeditingifundefinediflong=+ 6cm=+ 0cm=+ 6cm=+ 3cm ifundefinedlhs2tex.lhs2tex.sty.read namedeflhs2tex.lhs2tex.sty.read#1 OT1cmtex OT1cmtexmn <5><6><7><8>cmtex8<9>cmtex9<10><10.95><12><14.4><17.28><20.74><24.88>cmtex10 OT1cmtexmit <-> ssub * cmtt/m/itOT1cmttbxn <5><6><7><8>cmtt8<9>cmbtt9<10><10.95><12><14.4><17.28><20.74><24.88>cmbtt10 OT1cmtexbxn <-> ssub * cmtt/bx/n ifundefinedmathindent polycode.fmtoldplainhscode=normalcrarrayhscode=normalcrtexthscode framedhscode ==@|p-2-2-2pt|@ -1.5ex==normalcr.5ex= inlinehscode ##1##2 ##1##2##3##3 myhscode forall.fmt= lambda.fmt = Experience Report – Extended Version Experience Report University of Pennsylvania3330 Walnut [email protected] Implementing multi-player networked games by broadcasting the player's input and letting each client calculate the game state – a scheme known as lock-step simulation– is an established technique. However, ensuring that every client in this scheme obtains a consistent state is infamously hard and in general requires great discipline from the game programmer. The thesis of this report is that in the realm of functional programming – in particular with Haskell's purity and static pointers – this hard problem becomes almost trivially easy.We support this thesis by implementing lock-step simulation under very adverse conditions. We extended the educational programming environment CodeWorld, which is used to teach math and programming to middle school students, with the ability to create and run interactive, networked multi-user games. Despite providing a very abstract and high-level interface, and without requiring any discipline from the programmer, we can provide consistent lock-step simulation with client prediction. Lock-step simulation is child's play Chris Smith December 30, 2023 ====================================Networked multi-user games must tackle the challenge of ensuring that all participating players on a network with potentially significant latency still see the same game state. In some circumstances, an appealing choice is lock-step simulation. In this scheme, which dates back to the age of Doom, the state of the game itself is never transmitted over the network. Instead, the clients exchange information about their player's interactions – as abstract game moves or just the actual user input events – and each client independently calculates the state of the game.Of course, this only works as intended if all clients end up with the same state. The technique is fraught with danger if the programmer is not very careful and disciplined about managing that state. <cit.>, who implemented the network code for the real time strategy games Age of Empires 1 & 2,report:As much as we check-summed the world, the objects, the pathfinding, targeting and every other system – it seemed that there was always one more thing that slipped just under the radar. […] Part of the difficulty was conceptual – programmers were not used to having to write code that used the same number of calls to random within the simulation.More drastic words were voiced by <cit.>, also a video game software engineer:One of the most vile bugs in the universe is the desync bug. They're mean sons of bitches. The grand assumption to the entire engine architecture is all players being fully synchronous. What happens if they aren't? What if the simulations diverge? Chaos. Anger. Suffering. [16]r0.4< g r a p h i c s > The Snake gameThe pitfalls facing a programmer implementing lock-step simulation include reading the system clock, querying the random number generator, other I/O, uninitialized memory, and local or hidden statefulness. In short: side effects! What if we chose a programming language without such side-effects? Would these problems disappear? Intuitively, we expect that pure functional programming makes lock-step simulation easy.This experience report corroborates our expectation. We have implemented lock-step simulation in Haskell under very adverse conditions. The authors of the quotes above are professional programmers working on notable games. They can be expected to maintain a certain level of programming discipline, and to tolerate additional complexity. Our implementation is part of CodeWorld[<https://code.world/haskell>], an educational, web-based programming environment used to teach mathematics and coding to students as early as middle school. These children, who are just learning to code, can write and run arbitrary game logic, using a simple API, without adhering to any additional requirements or coding discipline. Nevertheless, we still guarantee consistent lock-step simulation and avoid the dreaded desync bug.The main contributions of this experience report are:*With a bold disregard for pesky implementation detail, we design a natural extension to CodeWorld's existing interfaces that can describe multi-user interactive programs in as straightforward, simple and functional a manner as possible (<ref>).*We identify a complication – unwanted capture of free variables – which can thwart consistency of such a program.We solve it using either using the module system (<ref>) or the Haskell language extension static pointers (<ref>).*We explain how to implement this interface. Despite its abstractness, we present an eventually consistent implementation that works for arbitrary client code, and includes client prediction to react immediately to local input while still reconciling delayed input from other users (<ref>).*We share lessons learned in stress-testing the system (<ref>). Testing was successful, but we identified an inconsistency in floating point transcendental functions. Replacing these with deterministic approximations recovers the consistency that we rely upon (<ref>).*Haskell’s promise of purity is muddied by underspecification of floating point transcendental functions. Replacing these with deterministic approximations recovers the consistency that we rely upon (<ref>).*We show that, even with no knowledge of the structure of the program's state, our approach still allows us to smooth out artifacts that arise due to network latency (<ref>).*Overall, we show that pure functional programming makes lock-step simulation easy. § CODEWORLD In this section, we give a brief overview of how students interact with the CodeWorld environment, the programming interfaces that are provided by CodeWorld and how student programs are executed. Many of the figures illustrating this paper are created by students.These and more can be found in the CodeWorld gallery at <https://code.world/gallery.html>.To ease deployment, students need only a web browser to use CodeWorld.They write their code with an integrated editor inside the browser. Programs are written in Haskell, which the CodeWorld server compiles to JavaScript using GHCJS <cit.> and sends that back to browser to execute in a canvas beside the editor. These programs are always graphical: students create static pictures, then animations, and finally interactive games and other activities. §.§ Two flavors of Haskell [16]r0.4< g r a p h i c s > Haunted House, by Lisha (8th grade)The standard Haskell language is not an ideal vessel for the children in CodeWorld's target audience. Therefore, CodeWorld by default provides a specially tailored educational environment. In this mode, a custom prelude is used to help students avoid common obstacles. Graphics primitives are available without an import, to create appealing visual programs. Functions of multiple arguments are not curried but rather take their arguments in a tuple, both to improve error messages and match mathematical notation that students are already learning. Finally, a singletype (isomorphic to ) is provided to avoid the need for type classes, and thelanguage extension makes literals monomorphic. Compiler error messages are post-processed to make them more intelligible to the students. Nevertheless, the code students write is still Haskell, and is accepted by GHC.However, at <https://code.world/haskell> instead of <https://code.world/>, one finds a standard Haskell environment, with full access to the standard library. In this paper we focus on the latter variant. §.§ API design principles [14]r0.4 trim=.2.2.2.2,scale=1.666,clip< g r a p h i c s > SmileyAn important principle of CodeWorld is to provide students with the simplest possible abstraction for a given task.This allows them to concentrate on the ideas they want to express and think clearly about the meaning of their code, and hides as many low-level details as possible. The first and simplest task that students face is to produce a static drawing.This is done with the abstract data type , with a simple compositional API (<ref>) which was heavily inspired by the Gloss library <cit.>.Complex pictures are built by combining and transforming simple geometric objects.The entry point used for this has the very simple type B@>l<@ E@>l<@ [B]drawingOf::Picture→.IO ()[E]This function takes care of the details of displaying the student's picture on the screen, redrawing upon window size changes and so on.So all it takes for a student to get the computer to smile like in <ref> is to write B@>l<@ 11@>l<@ E@>l<@ [B] import CodeWorld[E] [B]smiley=[11] [11]translated (-4) 4 (solidCircle 2)&translated 4 4 (solidCircle 2)&[E] [11]thickArc 2 (-pi) 0 6&colored yellow (solidCircle 10)[E] [B]main=drawingOf smiley[E] [15]r0.4< g r a p h i c s > Yo Grandma, by Sophia (6th grade)As a next step, the students can create animations and simulations to make their pictures move, before eventually making their programs react to user input in interactions. The game in <ref> is a typical interaction, where the player saves flying Grandma from various obstacles by attaching balloons or parachutes to her wheelchair.These are created by calling the following interface: B@>l<@ 16@>c<@ 16E@l@ 20@>l<@ E@>l<@ [B]interactionOf[16] [16]::[16E] [20]world[E] [16]→.[16E] [20](Double→.world→.world)[E] [16]→.[16E] [20](Event→.world→.world)[E] [16]→.[16E] [20](world→.Picture)[E] [16]→.[16E] [20]IO ()[E]In a typical call B@>l<@ E@>l<@ [B]main=interactionOf start step handle draw[E]the student passes four arguments, namely:*an initial state, ,*a time step function, , which calculates an updated state as time passes,*an event handler function, , which calculates an updated state when the user interacts with the program and*a visualization function, , to depict the current state as a . Thetype, shown in <ref>, is a simple algebraic data type that describes the press or release of a key or mouse button, or a movement of the mouse pointer.The type of the state, , is chosen by the user and consists of the domain-specific data needed by the program. Thetype is completely unconstrained, and this will be an important factor influencing our design. It need not even be serializable, nor comparable for equality. In particular, the state may contain first-class functions and infinite lazy data structures. One way that students commonly make use of this capability is by defining infinite lazy lists of anticipated future events, based on a random number source fetched before the simulation begins. § AN INTERFACE FOR MULTI-PLAYER GAMES We would like students to extend their programming to networked multi-user programs, so that they can invite their friends to join over the internet and collaborate together on a drawing, fight each other in a fierce duel of Snake, or interact in any other way the student designs and implements. In this section, we turn our attention to choosing an API for such a task. §.§ Wishful thinking Let us apply “API design by wishful thinking", and ask: What is the most convenient abstract model of a multi-player game we can hope for, independent of implementation concerns or constraints?[16]r0.4< g r a p h i c s > Dot Grab, by Adrian (7th grade) As experienced programmers, our thoughts might drift to network protocols or message passing between independent program instances, each with its own local state. Our students, though, care about none of this, and ideally we would not burden them with it. In fact, motivated students have already implemented games to be played with classmates, using different keys on the same device. An example is shown in <ref>, where the red player uses the keys WASD and the blue player the keys , in a race to consume more dots.Their games, which they have already designed, are described in terms of one shared global state.Why should the programming model change drastically simply because of one detail – that the code will now run on multiple nodes communicating over a network?We conclude, then, that an interactive multi-user program is a generalization of an interactive single-user program, and the centerpiece of the API is still a single, global state, which is mutually acted upon by all players. Basing the API on , we make only minimal changes to adapt to the new environment:*A new first parameter specifies the number of players.*The parametersandremain as they are.*Theparameter, though, ought to know which user pressed a certain button or moved their mouse, so it receives the player number (a simple ) as an additional parameter.*Different players may also see different views of the state, so thefunction also receives the player number for which it should render the screen – but it is free to ignore that parameter, of course.All together, we arrive at the following “ideal” interface that we call collaborations, which allows students to build networked multi-player games and other activities: B@>l<@ 18@>c<@ 18E@l@ 22@>l<@ E@>l<@ [B]collaborationOf[18] [18]::[18E] [22]Int[E] [18]→.[18E] [22]world[E] [18]→.[18E] [22](Double→.world→.world)[E] [18]→.[18E] [22](Int→.Event→.world→.world)[E] [18]→.[18E] [22](Int→.world→.Picture)[E] [18]→.[18E] [22]IO ()[E] [16]r0.4 < g r a p h i c s > Two players' mouse movementsA small example will clarify how this interface is used. The following code traces the mouse movements of two players using colored, fading circles, and <Ref> shows this program in action. The green player is a bot that simply mirrors the red player's movements.B@>l<@ 9@>l<@ 12@>l<@ 14@>l<@ 35@>l<@ 51@>l<@ E@>l<@ [B] import CodeWorld[E] [B] type World=[1.5mu (Color,Double,Double,Double)1.5mu][E] [B]step::Double→.World→.World[E] [B]step dt dots=[1.5mu (c,exp (-dt)*r,x,y)| (c,r,x,y)←dots,r≥0.11.5mu][E] [B]handle::Int→.Event→.World→.World[E] [B]handle [9] [9]0 [12] [12](MouseMovement (x,y)) [35] [35]dots=(red,[51] [51]1,x,y):dots[E] [B]handle [9] [9]1 [12] [12](MouseMovement (x,y)) [35] [35]dots=(green,[51] [51]1,x,y):dots[E] [B]handle [9] [9] [12] [12] [35] [35]dots=dots[E] [B]draw::Int→.World→.Picture[E] [B]drawdots[14] [14]=mconcat [1.5mu translated x y (colored c (solidCircle r))| (c,r,x,y)←dots1.5mu][E] [B]main::IO ()[E] [B]main=collaborationOf 2 [1.5mu 1.5mu] step handle draw[E] A collaboration begins with a lobby, featuring buttons to create or join a game. Upon creating a new game, the player is given a four-letter code to be shared with friends. Those friends may enter the four-letter code to join the game. Once enough players have joined, the game begins. §.§ Solving random problems with the module system Likebefore it, the parameters ofprovide enough information to completely determine the behavior of the program from the sequence of time steps and UI events that occur. Unlike , however, a collaboration involves more than one use of theAPI, as the function is executed by each participating player. To ensure that there is a single, well-defined behavior, it is essential that all players runwith the same arguments. Obviously, we need to ensure that all clients run the same program, and the CodeWorld server does so. But even with the same code, the arguments tocan differ from client to client: B@>l<@ 3@>l<@ E@>l<@ [B]main= do[E] [B]3[3] [3]r←randomRIO (0,1)[E] [B]3[3] [3]collaborationOf numPlayers start step (handle r) draw[E]The event handling function now depends on I/O – specifically, the choice of a random number – and it is very unlikely that all clients happen to pick the same random number. Despite sharing the same code, the clients will disagree about the correct behavior of the system.The problem is not the use of random numbers per se, but rather the unconstrained flow of client-specific state resulting from any I/O into theAPI via free variables in its parameters. Since most of the parameters tohave function types, we cannot just compare them to establish consistency at runtime.We solve this problem in two ways: one in the educational environment, and the other in the standard Haskell environment.In the former, we have tight control over the set of library functions available to the student. No packages are exposed except for a custom standard library with a heavily customizedmodule, and this library simply does not provide any functions to compose IO operations, such as as the monadic bind operators (, ). This also rules out the use of Haskell's -notation, which under the regime ofrequires an operator called () to be in scope. A valid Haskell program requires a top-level function , and since the only available way to obtain anis through our API entry points (, , and so on), we know that all CodeWorld collaborations are of essentially the formIn particular, no I/O can be executed prior to the collaboration, and hence no client-dependent behavior is possible. §.§ Solving random problems syntactically This solution is not suitable for the standard Haskell environment, where we do not want to restrict the user's access to the standard library. We can still prevent the user from using the results of client-specific I/O in arguments to .To accomplish this, we creatively use the work of <cit.>, who sought to bring Erlang-like distributed computing to Haskell. They had to exchange functions over the network, which is possible by passing code references, as long as no potentially unserializable values are captured from the environment.To guarantee that, they introduced a Haskell language extension, static pointers, which introduces:*a new type constructor , which wraps values of type ,*a new syntactic construct , such that for any expressionof type , the expressionhas type , but is only valid ifdoes not contain any locally bound free variables,*a pure function , to unwrap the static pointer, and*a pure functionwhich produces a key that – within one program – uniquely identifies a static pointer.The requirement thatvalues cannot have locally bound free variables turns out to be exactly what we need to prevent programs from smuggling client-specific state obtained with I/O actions into collaborations. We therefore further refine the API to require its arguments to be static pointers: B@>l<@ 18@>c<@ 18E@l@ 22@>l<@ E@>l<@ [B]collaborationOf[18] [18]::[18E] [22]Int[E] [18]→.[18E] [22]StaticPtr world[E] [18]→.[18E] [22]StaticPtr (Double→.world→.world)[E] [18]→.[18E] [22]StaticPtr (Int→.Event→.world→.world)[E] [18]→.[18E] [22]StaticPtr (Int→.world→.Picture)[E] [18]→.[18E] [22]IO ()[E]The mouse tracing program in <Ref> must now change its definition ofto B@>l<@ E@>l<@ [B]main=collaborationOf 2 ( static [1.5mu 1.5mu]) ( static step) ( static handle) ( static draw).[E]On the other hand, writingto smuggle in a randomly drawn number , as in the example above, will fail at compile time.Requiring thekeyword here admittedly muddies the clarity of the API a bit. We believe that the target audience of CodeWorld's standard Haskell mode can handle this. Beginners working within the educational mode need not deal with this slight complication. A somewhat more clever attempt, though, still causes problems: B@>l<@ 3@>l<@ E@>l<@ [B]main= do[E] [B]3[3] [3]coinFlip←randomIO[E] [B]3[3] [3] let step= if coinFlipthenstatic step1elsestatic step2[E] [B]3[3] [3]collaborationOf 2 ( static [1.5mu 1.5mu]) step ( static handle) ( static draw)[E]This program is accepted by the compiler because the arguments toare indeedvalues of the right types, yet it raises the same questions when clients disagree on the choice of step function. While we cannot prevent this case at compile time, we can at least detect it at runtime. Static pointers can be serialized using the function . Before a game starts, the participating clients compare the keys of their arguments to check that they match. This is a subtly different use of static pointers from the original intent of sending functions over a network in a message-passing protocol. We need not actually receive the original values on the remote end of our connections, but instead use the serialized keys only to check for consistency.With this check in place – short of using unsafe features such as – we are confident that every client is indeed running the same functions. However, this forces our games to be entirely deterministic. This is a problem, since many games involve an element of chance! To restore the possibility of random behavior, we supply a random number source to use in building the initial state, with a consistent seed in all clients. The type of theparameter is now .(This is not entirely new: CodeWorld's educational environment has never exported a random number generator, and its simulations and interactions have always been initialized with an infinite list of random numbers.) This completes our derivation of , which in its final form is B@>l<@ 18@>c<@ 18E@l@ 22@>l<@ E@>l<@ [B]collaborationOf[18] [18]::[18E] [22]Int[E] [18]→.[18E] [22]StaticPtr (StdGen→.world)[E] [18]→.[18E] [22]StaticPtr (Double→.world→.world)[E] [18]→.[18E] [22]StaticPtr (Int→.Event→.world→.world)[E] [18]→.[18E] [22]StaticPtr (Int→.world→.Picture)[E] [18]→.[18E] [22]IO ()[E] § FROM WISHFUL THINKING TO RUNNING CODE How can we implement this interface? It turns out that our implementation options are severely narrowed down by the following requirements: *We need to handle any code using the API. Given the educational setting of CodeWorld, we cannot require any particular discipline.*The players need to see an eventually consistent state. They may have different ideas about the state of the world, but only until everybody receives information about everybody’s interactions.*The effects of a player's own interactions are immediately visible to that player. Even a “local” interaction, such as selecting a piece in a game of Chess, will have to represented in the game state, and any latency here would make the user interface sluggish. The first requirement in particular implies that the game state is completely opaque to us. This already rules out the usual client-server architecture, where only the central server manages the game state and the clients send abstract moves (e.g., “white moves the knight to e8") and render the game state that they receive from the server. We have neither insight into what constitutes an abstract move, nor how to serialize and transmit the game state.We could avoid this problem by sending the raw UIinstead of an abstract move to the server, and letting the server respond to each client with theto show. This “dumb terminal” approach however would run afoul of our third requirement, as every user interaction would be delayed by the time it takes messages to travel to the server and back.The requirement of immediate responsiveness implies that every client needs to manage its own copy of the game state, and being abstract in the game state implies that there is nothing else but the UI events that the clients can transmit to synchronize the state. In other words, lock-step simulation is the only way for us. This approach assumes the integrity of client code. Since all clients track the entire game state, malicious players could trick CodeWorld into running a modified version of the program which, among other things, could then reveal hidden parts of the game state. Given the educational goals of CodeWorld, we are willing to trade this security for a cleaner API.§.§ Types and messages We seek, then, to implement the API by exchanging UI events between clients. For the purposes of this paper, it does not matter how events are transmitted from client to client. The CodeWorld implementation uses a very simple relay server that broadcasts messages from one client to the others via WebSockets (a full-duplex server-client protocol for web applications), but peer-to-peer communication using WebRTC (a peer-to-peer protocol for web applications) or other methods would work equally well, as long as they deliver events reliably and in order.Every such message obviously needs to contain the actualand the player number. In addition, it must contain a timestamp, so that each client applies the event at the same time despite differences in network latency. Otherwise – assuming a time-sensitive game with a non-trivialfunction – the various clients would obtain different views of the world. Timestamps arevalues, measured in seconds since the start of the game.B@>l<@ 17@>c<@ 17E@l@ 20@>l<@ E@>l<@ [B] type Timestamp[17] [17]=[17E] [20]Double[E] [B] type Player[17] [17]=[17E] [20]Int[E] [B] type Message[17] [17]=[17E] [20](Timestamp,Player,Event)[E]§.§ Resettable state Having fixed the message type still leaves open the question of what to do with these messages, which is non-trivial due to the network latency.Assume that 23.5 seconds into a real-time strategy game, I send my knights to attack the other player. My client sends the corresponding messageto the other player. The message arrives, say, 100ms later. As mentioned before, the other player cannot simply let my knights set out a bit later. What else?The classical solution <cit.> is to not act on local events immediately, but add a delay of, say, 200ms. The message would be , and assuming it reaches all other players in time, all are able to apply the event at precisely the same moment. This solution works well if the UI can somehow respond to the user's actions immediately, e.g. by letting the knight audibly confirm the command, so hide this delay from the user.The luxury of such a separation is not available to us – according to the third requirement, each client must immediately apply its own events – and the message really has to have the timestamp 23.500. This leaves the other player, when it receives the message 100ms later, with no choice but to roll back the game state to time 23.500, apply my event, and replay the following 100ms. While rollback and replay are hard to implement in imperative programming paradigms, where every piece of data can have local mutable state, they are easy in Haskell, where we know that the value of typereally holds all relevant bits of the program's state.One way of allowing such recalculation is to simply not store the state at all, and re-calculate it every time we draw the game screen. The function to do so would expect the game specification, the current time and the list of messages that we have seen so far, including the locally generated ones, and would calculate the game state. Its type signature would thus be B@>l<@ E@>l<@ [B]currentState::Game world⇒Timestamp→.[1.5mu Message1.5mu]→.world[E]where the hypothetical type classcaptures the user-defined game logic; we introduce it here to avoid obscuring the following code listings by passing it explicitly around as an argument: B@>l<@ 3@>l<@ 11@>c<@ 11E@l@ 15@>l<@ E@>l<@ [B] class Game worldwhere[E] [B]3[3] [3]start[11] [11]::[11E] [15]world[E] [B]3[3] [3]step[11] [11]::[11E] [15]Double→.world→.world[E] [B]3[3] [3]handle[11] [11]::[11E] [15]Player→.Event→.world→.world[E] Assume, for a short while, that there was nofunction, i.e. the game state changes only when there is an actual event. Then the timestamps are only required to put the events into the right order and to disregard events which are not yet to be applied (which can happen if the player’s game time started at slightly different points in time): B@>l<@ 3@>l<@ 10@>l<@ E@>l<@ [B]currentState::Game world⇒Timestamp→.[1.5mu Message1.5mu]→.world[E] [B]currentState now messages=applyEvents to95 apply start[E] [B]3[3] [3] where to95 apply=takeWhile ( (t, , )→.t≤now) (sortMessages messages)[E] [B]sortMessages::[1.5mu Messages1.5mu]→.[1.5mu Messages1.5mu][E] [B]sortMessages=sortOn ( (t,p, )→.(t,p)) messages[E] [B]applyEvents::Game world⇒ [1.5mu Message1.5mu]→.world→.world[E] [B]applyEvents messages w=foldl apply w messages[E] [B]3[3] [3] where [10] [10]apply w ( ,p,e)=handle p e w[E] Eventually, every client receives the same list of messages, up to the interleaving of events from different players. After a stable sort by timestamp and player, the lists of events will be identical, so all clients will calculate the same game state. §.§ A few more steps This is nice and simple, but ignores thefunction, which models the evolution of the state as time passes. Clearly, we have to callbefore each event, and again at the end. In order to calculate the time passed since the last event, we also have to keep track of which timestamp a snapshot of the game state corresponds to:B@>l<@ 3@>l<@ 10@>l<@ 22@>l<@ E@>l<@ [B]currentState::Game world⇒Timestamp→.[1.5mu Message1.5mu]→.world[E] [B]currentState now messages=step (now-t) world[E] [B]3[3] [3] where [10] [10]to95 apply[22] [22]=takeWhile ( (t, , )→.t≤now) (sortMessages messages)[E] [10](t,world)[22] [22]=applyEvents to95 apply (0,start)[E] [B]applyEvents::Game world⇒ [1.5mu Message1.5mu]→.(Timestamp,world)→.(Timestamp,world)[E] [B]applyEvents messages ts=foldl apply ts messages[E] [B]3[3] [3] where [10] [10]apply (t0,world) (t1,p,e)=(t1,handle p e (step (t1-t0) world)))[E] Unfortunately, students would not be quite happy with this implementation. Thefunction is commonly used to calculate a single step in a physics simulation, which requires that it is called often enough to achieve a decent simulation frequency.For instance, when simulating a projectile, a common technique is to adjust the position linearly along the velocity vector, and the velocity linearly according to forces like gravity or drag. The result is a stepwise-linear approximation, the precision of which depends on the sampling frequency. Another common technique is to do collision detection only once per time step, and again the result depends on the frequency of steps.It is important, then, that thefunction is called at a reasonably high frequency.We could leave students to resolve this themselves, by dividing time steps into multiple finer steps, if necessary, in theirimplementation.However, imposing that burden would violate our first requirement: not requiring any discipline from the user. Therefore, we have to ensure that thefunction is called often enough, even if there is no user event for a while.In simulations and interactions, the implemented behavior is to evaluate the step function as quickly as possible between animation frames. Thus, simulations running on faster computers may take smaller steps and be more accurate. The need for eventual consistency precludes this strategy here. Instead, the desired step length foris defined globally and set to one-sixteenth of a second: B@>l<@ E@>l<@ [B]gameRate::Double[E] [B]gameRate=1/16[E] We can obtain the desired resolution by wrapping the student's step function in one that iterateson time steps larger than the desired rate:B@>l<@ 20@>c<@ 20E@l@ 23@>l<@ 38@>l<@ E@>l<@ [B]gameStep::Game world⇒Double→.world→.world[E] [B]gameStep dt world[20] [20]|[20E] [23]dt≤0[38] [38]=world[E] [20]|[20E] [23]dt>gameRate[38] [38]=gameStep (dt-gameRate) (step gameRate world)[E] [20]|[20E] [23]otherwise[38] [38]=step dt world[E]Replacingwithin the implementation ofandabove yields a correct solution.To see this code in action, we construct the following program: As the time passes, a column grows on the screen, from bottom to top. Initially, it is gray. When a player presses a number key, the column begins to grow in a different color. Additionally, wheneveris called, this current height of the column is marked with a black line.Because the program output is one-dimensional, we can use the horizontal dimension to show in <ref> how the players' displays evolves over time. The dashed arrows indicate the transfer of each packet to the other player, which is not instant. When a message from the other player arrives, the state is updated to reflect this change. Because this game essentially records its history, these delayed updates result in a “flicker" as the client updates the state. In many cases the effect will be less noticeable than it is here. We can see that the algorithm achieved eventual consistency, as the right edge of the drawing looks identical for both clients. §.§ Limiting time travel In the course of a game, quite a large number of events occur. As time goes by, the cost of calculating the current state from scratch grows without bound, and will eventually become too large to be completed between each frame, and animations will stop being smooth. Clearly, some of that computation is quite pointless to repeat.Our message transport guarantees that messages from each client are delivered in order, so that when we receive a message, we know that we have seen all messages from the sender up to that timestamp. If we call this the client's commit time, then we know that no new events will be received before the earliest commit time of any client, which we call the commit horizon. We can now precompute the game state up to the commit horizon, forget all older state and events, and use this as the basis for future state recalculations.In the following we will explain the data structure and associated operations that CodeWorlduses to keep track of the committed state, the pending events and each player's commit time. The main data type is B@>l<@ 23@>c<@ 23E@l@ 26@>l<@ 37@>c<@ 37E@l@ 41@>l<@ E@>l<@ [B] data Log world=Log [23] [23]{1.5mu [23E] [26]committed[37] [37]::[37E] [41](Timestamp,world),[E] [26]events[37] [37]::[37E] [41][1.5mu Message1.5mu],[E] [26]latest[37] [37]::[37E] [41][1.5mu (Player,Timestamp)1.5mu]1.5mu}[E] Initially, there are no events, and everything is at timestamp zero: B@>l<@ E@>l<@ [B]initLog::Game world⇒ [1.5mu Player1.5mu]→.Log world[E] [B]initLog ps=Log (0,start) [1.5mu 1.5mu] [1.5mu (p,0)|p←ps1.5mu][E]When an event comes in, the message is added tovia the publicfunction. B@>l<@ 3@>l<@ 10@>l<@ E@>l<@ [B]addEvent::Game world⇒Message→.Log world→.Log world[E] [B]addEvent (t,p,e) log=recordActivity t p (log {1.5mu events=events'1.5mu})[E] [B]3[3] [3] where [10] [10]events'=sortMessages (events log++[1.5mu (t,p,e)1.5mu])[E]Then, the client's commit time inis updated. B@>l<@ 3@>l<@ 10@>l<@ 26@>c<@ 26E@l@ 29@>l<@ 44@>l<@ E@>l<@ [B]recordActivity::Game world⇒Timestamp→.Player→.Log world→.Log world[E] [B]recordActivity t p log[26] [26]|[26E] [29]t<t95 old[44] [44]=error Messages out of order[E] [26]|[26E] [29]otherwise[44] [44]=advanceCommitted (log {1.5mu latest=latest'1.5mu})[E] [B]3[3] [3] where [10] [10]latest'=(p,t):delete (p,t95 old) (latest log)[E] [10]Just t95 old=lookup p (latest log)[E]This might have moved the commit horizon, and if some of the messages from the listare from before the commit horizon, we can integrate them into thestate. B@>l<@ 3@>l<@ 10@>l<@ 29@>c<@ 29E@l@ 32@>l<@ 43@>l<@ E@>l<@ [B]advanceCommitted::Game world⇒Log world→.Log world[E] [B]advanceCommitted log=log [29] [29]{1.5mu [29E] [32]events[43] [43]=to95 keep,[E] [32]committed[43] [43]=applyEvents to95 commit (committed log)1.5mu}[E] [B]3[3] [3] where [10] [10](to95 commit,to95 keep)=span ( (t, , )→.t<commitHorizon log) (events log)[E] [B]commitHorizon::Log world→.Timestamp[E] [B]commitHorizon log=minimum [1.5mu t| (p,t)←latest log1.5mu][E]The final public function is used to query the current state of the game. Starting from the committed state, it applies the pending events. B@>l<@ 3@>l<@ 10@>l<@ E@>l<@ [B]currentState::Game world⇒Timestamp→.Log world→.world[E] [B]currentState now log|now<commitHorizon log=error Cannot look into the past[E] [B]currentState now log=gameStep (now-t) world[E] [B]3[3] [3] where [10] [10]past95 events=takeWhile ( (t, , )→.t≤now) (events log)[E] [10](t,world)=applyEvents past95 events (committed log)[E] This algorithm, printed in <ref> in its entirety, relies on these assumptions:*The list of players provided tois correct.*For each player, events are added in order, with monotonically increasing timestamps.*The state is never queried at a time that lies before . The first assumption is ensured by the CodeWorld framework. The second is ensured by using a monotonic time source to create the timestamps, and by using an order-preserving communication channel. The third follows from the fact that every client's own timestamps are always in that player’s past, and therefore the argument tois later than the commit horizon.If one of the players were to stop interacting with the program, that client would not send any messages. In this case, no events can be committed and the list of events to be processed bywould again grow without bound. To avoid this, each client sends empty messages (“pings") whenever the user has not produced input for a certain amount of time. When such a ping is received, thefunction advances thefield without adding a new event: B@>l<@ E@>l<@ [B]addPing::Game world⇒ (Timestamp,Player)→.Log world→.Log world[E] [B]addPing (t,p) log=recordActivity t p log[E] This way, the number of events in thefield is bound bymax input rate× (max network delay + max time between events or pings) × (number of players-1)which is independent of how long the game has been running.As long as no players disconnect. Not sure how best to phrase that. Maybe not important. But one reviewer noticed. Worth a sentence? Assuming a bounded input event rate and network delay, this bounds the size of thefield. More tweaks are possible. In the CodeWorld implementation, we also cache the current state, so that querying the current state again, when no new events were received, is much cheaper. When an input event from another player comes in, we discard this cached value and recalculate it based on the committed state and the stored events.The main property of the code in <ref> is: No matter the interleaving of events from the various players, the result ofis the same. To increase our confidence that this property holds we used the QuickCheck library to randomly generate pairs of lists of events with monotonically increasing timestamps, considered all possible interleavings and checked that the resultingdata structure is identical. § EXPERIENCES AND DISCUSSION[11]r0.4 < g r a p h i c s > The tank gameThe interface from <ref> allows the creation of multi-user applications with great ease, and with the algorithms in <ref>, CodeWorld can provide a smooth user experience. The reader may wonder, though, how well this works in practice, and what the drawbacks are for this approach. §.§ Early experience For a first practical evaluation of the system, the second author organized a stress test, involving four colleagues, a selection of games with different styles, and small prizes for winners.During the event, participants play-tested the games, hoping to uncover any bugs or unexpected quirks of the format.The games involved, which can be played at <https://code.world/gallery-icfp17.html>, include:*The Dot Grab game (<ref>), which was originally written by a student as a single-computer interaction. Since the API for games is a straightforward extension of the one for interactions, it was trivial to make this game networked.*The game “Snake" (<ref>), where a player has to move across the playing field while avoiding the other player's trails and the walls.*A tank game (<ref>) where each player steers a tank using the keyboard, aims using the mouse and fires bullets that explode after a certain time. Here the game evolves over time and manages a larger number of moving parts – tanks, bullets, and explosions.Manual testing showed that the system is nicely responsive and that the artifacts due to network latency are noticeable, but not irritating. The system handled the more complex tank game well. A separate test using a high latency satellite connection remained playable, but with more pronounced latency-related artifacts, as expected.We plan to introduce the API to students in the Spring semester of 2017.Let’s erase this or replace it with something more timeless.§ DISCUSSIONThe interface from <ref> allows the creation of multi-user applications with great ease, and with the algorithms in <ref>, CodeWorld can provide a smooth user experience. But surely there are drawbacks and open problems worth discussing. §.§ Floating point calculation A dominant concern in the implementation in <ref> was to guarantee eventual consistency of all clients, so that game states would always converge over time. We achieve that requirement, on the assumption that the code passed toconsists of pure functions. This result relies on a strong notion of pure function, though, which requires that outputs are predictable even between instances of the code running on different machines, operating systems, and runtime environments.In this sense, even functions in Haskell may not always be pure!A notable source of nondeterminism in Haskell is underspecified floating point operations. Thetype in Haskell is implementation-defined, and “should cover IEEE double-precision" <cit.>. Our interest is limited to the Haskell-to-JavaScript compiler GHCJS <cit.>, which inherits the floating point operation semantics from JavaScript. The ECMA standard <cit.> specifies a JavaScript number to be a “double-precision 64-bit binary format IEEE 754-2008 value" – which is luckily already a quite specific specification.We are optimistic that the basic arithmetic operations are deterministic,and this optimism is supported by anecdotal reports from a game developer with Gas Powered Games <cit.> We have never had a problem with the IEEE standard across any PC CPU, AMD and Intel, with this approach. None of our […] customers have had problems with their machines either, and we are talking over 1 million customers here. We would have heard if there was a problem with the FPU not having the same results as replays or multi-player mode wouldn't work at all.encouraged by reports from professional game developers <cit.>.However, transcendental functions (, , , , etc.) are not completely specified by IEEE-754, and different browser/system combinations are allowed to yield slightly different results here.We tested this with a double pendulum simulation, which makes heavy use ofandin every simulation step. The double pendulum is a well-known example of a chaotic system, and we expect it to quickly magnify any divergence in state. Indeed, after running the program on two different browsers (Firefox and Chrome, on the same Linux machine) for several minutes, the simulations take different paths, confirming the worries about these functions.If, however, we use a custom implementation of – based on a quadratic curve approximation – the simulation runs consistently.We tested this variant on multiple JavaScript engines (Chrome, Firefox, and Microsoft Edge), on different operating systems (Windows, Linux, Android, and ChromeOS) and on different CPUs (Intel and ARM),We tested this variant on multiple JavaScript engines, OSs and CPUs,and did not uncover any more consistency issues.The tests confirm again that, apart from inconsistent implementations of transcendental functions, basic floating point operations are reliably deterministic in practice.We can deploy a fix to transcendental functions in two ways. In CodeWorld's educational mode, where we have implemented a custom standard library, it is easy to just substitute new implementations of these functions. In the plain Haskell variant, however, we would like to allow the programmer to make use of existing libraries, which may use standard floating point functions. To achieve this, we can instead replace these operations at the JavaScript level, ensuring that even third-party Haskell libraries are deterministic. In the future, we also plan to automate checks for synchronization problems like this. We cannot directly compare program states in our implementation, since they are of arbitrary type.However, we can compare the generated pictures – or a hash thereof – to achieve essentially the same effect.§.§ Interpolating the effects of delayed messagesAnother trick in the game programming toolbox is interpolation to smooth out artifacts that result from corrections to the game state. These artifacts can be clearly seen in <ref>: The moment the message 2 reaches the first player, the top segment of the growing column abruptly changes from green to red.Similarly, in a game like the tank-fighting game in <ref>, an opponent can appear to teleport to a new location.Similarly, in a first-person shooter, an opponent can appear to teleport to a new location.In this situation, many games would instead interpolate the position smoothly over a fraction of a second. This can introduce new anomalies of its own,such as characters passing through walls, or tanks moving sideways,but in most cases, it is hoped the result will appear more realistic than the alternative. By providing an API that is completely abstract in the game state, it seems that we have shut the door on implementing this trick. We lack the ability to look inside the state and adjust positions. Surprisingly, though, a form of interpolation is possible. All that is needed is a sort of change of coordinates. While we cannot interpolate in space, we can interpolate in time!It seems that our API is too abstract to allow us to implement this trick, as we cannot look inside the state to adjust positions. But although we cannot interpolate in space, we can interpolate in time!When a delayed event arrives, we initially treat it as if its timestamp is “now" and then slide it backward in time over a short interpolation period until it reaches its actual time.Usually, thefunction is approximately continuous, and as a result, moving an event backwards in time gives a smooth interpolation in the state as well. This can be seen in <ref>: After the message 2 arrives at Player 1, the column smoothly changes its color from green to red, from the tip downwards, until the correct state is reached.Like all interpolation, though, anomalies can still happen.This scheme introduces abrupt artifacts as we slide a delayed message past another event with a non-commuting effect. In <ref> the second player smoothly integrates the delayed 3 message, and the top of the column changes color from blue to yellow. But the moment this event is pushed before the local event 4, the column abruptly changes its color back to blue.This is an elegant trick to recover the ability to do interpolation.However, it is not clear if interpolation is always the best experience, and a jerky, abrupt update may be preferred for certain games.§.§ Irreversible updates In some cases, the visual artifacts due to delayed messages, whether smooth or jerky, pose a serious problem. Consider, for example, a card game in which both players click to draw cards from the same deck.Suppose player 1 clicks to draw a card first, but the message from player 1 to player 2 arrives after player 2 clicks as well.For a brief moment before the message is received, player 2 sees the top card, even though it ultimately ends up in the first player's hand! This is an example of a case where eventual consistency in the game state is not good enough.This problem is hard to avoid, given our constraints and the third requirement of responding immediately to local events.It can be mitigated by the game programmer, by adding a short delay before major events such as those that reveal secrets. The delay can sometimes be creatively hidden by animations or effects. This trick dodges the problem as long as network latency is shorter than this delay, but it provides no guarantee. A complete solution to this problem must involve the programmer in a way that is undesirable in our setting, since only the programmer understands which state changes represent a significant enough event to postpone.§.§ Lock-step simulation and CRDTs Our approach to lock-step simulation may remind some readers of conflict-free replicated data types (CRDTs), introduced by <cit.> as a lightweight approach to providing strong consistency guarantees in distributed systems, even in the face of network failure, partition or out-of-order event delivery. These data types come in two forms: “convergent" replicated data types (CvRDTs) are based on transmitting state directly, while “commutative" replicated data types (CmRDTs), are based on transmitting operations that act on that state. Despite the similarity, our game state does not form a CmRDT, as these require that update operations on the game state are commutative.This limits the types of data that can used in such an approach and is inconsistent with our first requirement of supporting arbitrary game state.We find, however, that typetype defined in <ref> forms a CmRDT. Theevents from different players commute, as both just add the event to the set. The theory of CRDTs hence provides another argument that the resulting game state is eventually consistent (in fact, strongly so). § CONCLUSIONS By implementing lock-step simulation with client prediction generically in the educational programming environment CodeWorld, we have demonstrated once more that that pure functional programming excels at abstraction and modularity. In addition, this work will directly support the education of our next generation of programmers. The first author has been supported by the GS100000001National Science Foundationhttp://dx.doi.org/10.13039/100000001 under Grant No. GS100000001CCF-1319880 and Grant No. GS10000000114-519. We thank Stephanie Weirich, Zach Kost-Smith, Sterling Stein and Justin Hsu for helpful comments on a draft of this paper, as well as the reviewers for their comments.§ QUOTES (Many found via <https://randomascii.wordpress.com/2013/07/16/floating-point-determinism/>.)<https://blog.forrestthewoods.com/synchronous-rts-engines-and-a-tale-of-desyncs-9d8c3e48b2be>:*If each player is independently updating the game state does that mean the game simulation must be fully deterministic? It sure does. Isn't that hard? Yep. *One of the most vile bugs in the universe is the desync bug. They're mean sons of bitches. The grand assumption to the entire engine architecture is all players being fully synchronous. What happens if they aren't? What if the simulations diverge? Chaos. Anger. Suffering.<http://www.gamasutra.com/view/feature/131503/1500_archers_on_a_288_network_.php?page=1>:*As much as we check-summed the world, the objects, the pathfinding, targeting and every other system – it seemed that there was always one more thing that slipped just under the radar. […]Part of the difficulty was conceptual – programmers were not used to having to write code that used the same number of calls to random within the simulation (yes, the random numbers were seeded and synchronized as well). <http://gafferongames.com/networking-for-game-programmers/floating-point-determinism/>:*I work at Gas Powered Games and i can tell you first hand that floating point math is deterministic.*We have never had a problem with the IEEE standard across any PC cpu AMD and Intel with this approach. None of our SupCom or Demigod customers have had problems with their machines either, and we are talking over 1 million customers here (supcom1 + expansion pack). We would have heard if there was a problem with the fpu not having the same results as replays or multi-player mode wouldn't work at all.*For all of the instructions that are IEEE operations (*,+,-,/,sqrt, compares, regardless of whether they are SSE or x87), they will produce the same results across platforms with the same control settings (same precision control and rounding modes, flush to zero, etc.) and inputs. This is true for both 32-bit and 64-bit processors… On the x87 side, the transcendental instructions like, fsin, fcos, etc. could produce slightly different answers across implementations. They are specified with a relative error that is guaranteed, but not bit-for-bit accuracy. <http://www.box2d.org/forum/viewtopic.php?f=4 t=175#p1092>:*During development, we discovered that AMD and Intel processors produced slightly different results for transcendental functions (sin, cos, tan, and their inverses), so we had to wrap them in non-optimized function calls to force the compiler to leave them at single-precision. That was enough to make AMD and Intel processors consistent, but it was definitely a learning experience.Also interesting: <http://drewblaisdell.com/writing/game-networking-techniques-explained-with-pong/><http://gafferongames.com/networking-for-game-programmers/what-every-programmer-needs-to-know-about-game-networking/>:*it’s exceptionally difficult to ensure that a game is completely deterministicAlso hasa long list of resources about floating-point determinancy at: <http://gafferongames.com/networking-for-game-programmers/what-every-programmer-needs-to-know-about-game-networking/#comment-1176> | http://arxiv.org/abs/1705.09704v1 | {
"authors": [
"Joachim Breitner",
"Chris Smith"
],
"categories": [
"cs.PL"
],
"primary_category": "cs.PL",
"published": "20170526201100",
"title": "Lock-step simulation is child's play"
} |
A well-balanced meshless tsunami propagationand inundation modelRüdiger Brecht^†^, Alexander Bihlo^†, Scott MacLachlan^†and Jörn Behrens^^† Department of Mathematics and Statistics, Memorial University of Newfoundland,St. John's (NL) A1C 5S7, Canada ^ Department of Mathematics, Universität Hamburg, Bundesstraße 55, Hamburg 20146, Germany and CEN – Center for Earth Systems Research and Sustainability, Universität Hamburg,Grindelberg 5, Hamburg 20144, Germany [email protected], [email protected], [email protected], [email protected] present a novel meshless tsunami propagation and inundation model. We discretize the nonlinear shallow-water equations using a well-balanced scheme relying on radial basis function based finite differences. The inundation model relies on radial basis function generated extrapolation from the wet points closest to the wet–dry interface into the dry region. Numerical results against standard one- and two-dimensional benchmarks are presented.§ INTRODUCTION The shallow-water equations are a simplified, albeit important, model in geophysical fluid mechanics, playing a central role in both atmospheric and ocean sciences. In ocean dynamics, the shallow-water equations govern the evolution of long waves and, thus, are a particularly suitable model for tsunami propagation <cit.>. An important characterization of a tsunami event is the involvement of vastly different scales, with wave propagation happening over thousands of kilometers in the open ocean and inundation taking place on coastlines with diverse features on the scale of just a few meters. This presence of vastly different scales has justified the extensive use of unstructured meshes and adaptive mesh refinement for the numerical solution of the shallow-water equations <cit.>.Current numerical methodologies for the solution of the shallow-water equations for tsunami modeling include finite-difference methods <cit.>, finite-volume methods <cit.>, and finite-element and discontinuous Galerkin methods <cit.>. The application of so-called meshless methods to the problem of tsunami modeling has received considerably less attention. One primary appeal of general meshless methods is that they do not rely on a predefined topologically connected mesh but, rather, operate on an (in principle) arbitrary collection of nodes where the numerical solution of the model problem is sought. For this reason, meshless methods are, by design, suitable for problems that benefit from variable spatial resolution. Several meshless methods have been proposed already for the shallow-water equations, such as those based on smoothed particle hydrodynamics <cit.> or radial basis functions <cit.>. To the best of our knowledge, the shallow-water equations with variable bottom topography have not been considered extensively within the general meshless methodology, as it is quite challenging to preserve the inherent balance between the bottom topography source term and the flux term in the momentum equations. For a review on these difficulties and possible solutions, we refer to <cit.>.The purpose of this paper is to develop a meshless tsunami model based on radial basis functions generated finite differences (RBF-FD). The RBF-FD method was first introduced in <cit.> and has since seen an extensive development both theoretically and with regards to different fields of applications, in particular in the geosciences <cit.>. This method has methodology similar to classical finite differences, yet it can be used on both arbitrary nodal layouts and general smooth embedded manifolds <cit.>, such as on the sphere <cit.>. This makes RBF-FD a natural method for both far- and near-field simulations of tsunamis. Moreover, we demonstrate in this paper that RBF-based extrapolation is also a suitable method for the simulation of tsunami inundation.The further organization of the paper is the following. In Section <ref>, we present the shallow-water equations and discuss the well-balanced meshless RBF-FD methodology employed for discretizing them. Section <ref> is devoted to the description of the RBF-based inundation algorithm. In Section <ref>, we present the results of various standard benchmark tests for the one- and two-dimensional shallow-water equations aiming at assessing the quality of the newly proposed method. The conclusions of the paper as well as a discussion of necessary further developments are found in Section <ref>.§ WELL-BALANCED MESHLESS DISCRETIZATION OF THE SHALLOW-WATER EQUATIONS In this section, we introduce the shallow-water equations and describe a well-balanced meshless method for their discretization. §.§ The shallow-water equations We consider the following generalized 2-dimensional conservation lawρ_t+𝐅_x+𝐆_y=𝐒.In order to define shallow-water equations, we specify (following <cit.>): ρ=(h,hu,hv)^ T is the transport vector of total mass and horizontal momentum, 𝐅=(hu,hu^2+gh^2/2,huv)^ T and 𝐆=(hv,huv,hv^2+gh^2/2)^ T are the associated flux vectors, and 𝐒=(0,-ghb_x,-ghb_y)^ T is the source term. The height of a constant density water column is denoted by h=h(t,x,y), the two-dimensional, vertically averaged fluid velocity is denoted by (u,v)^ T=(u(t,x,y),v(t,x,y))^ T, the sea bottom topography is b=b(x,y), and g is the gravitational constant. For the sake of brevity, we use subscripts to denote the partial derivatives with respect to t, x and y, i.e. h_t=∂ h/∂ t, etc.We point out that the form of the shallow-water equations used here does not include any bottom friction, although such friction terms can be introducedwith the aim of obtaining more accurate inundation results <cit.>. Here, we present mostly canonical benchmarks, which are typically tested without the use of bottom friction and, so, we also exclude this in our model. The inclusion of bottom friction as well as the application to more elaborate test cases will be the subject of future investigations. §.§ RBF-FD discretization We discretize (<ref>) using the RBF-FD method. Radial basis function based discretizations have seen a rapid development over the past 20 years, both for the global formulation (which is akin to the pseudospectral method <cit.>) and for the local RBF-FD formulation, see e.g. <cit.> for recent reviews. The RBF methodology has also already been used for solving the shallow-water equations, both in planar geometry and on the surface of the sphere <cit.>. The discretization of the partial derivative operators in (<ref>) within the framework of the RBF method proceeds as follows. Consider a set of N nodes, 𝐱_i∈ℝ^2, covering the spatial domain Ω⊆ℝ^2 such that no two points coincide. Given the values of the field functions f∈{h,u,v} at those nodes, i.e. f_j=f(𝐱_j), we approximate the action of a linear differential operator ℒ on f at 𝐱_i byℒ f|_𝐱=𝐱_i≈∑_j=1^Nw^ℒ_ijf_j.In other words, the action of ℒ on f at 𝐱_i is approximated as a weighted linear sum of the values of f at the N nodes 𝐱_j. Within the context of RBF methods, the weights w^ℒ_ij are found by enforcing (<ref>) to be exact when evaluated for radial basis functions ϕ_k(𝐱)=ϕ(||𝐱-𝐱_k||) centered at 𝐱_k, i.e. ℒϕ_k(𝐱_i)=∑_j=1^Nw^ℒ_ijϕ_k(𝐱_j), k=1,…, N,which constitutes a linear system for w_ij^ℒ, 1≤ j ≤ N,at each node 𝐱_i. In the RBF-FD method, one assumes that only points close to the node 𝐱_i contribute to the approximation of the derivative at 𝐱_i and, thus, most of the N weights w^ℒ_ij vanish. In practice, this is done by determining the n nearest neighbors of 𝐱_i, where typically n≪ N. Solving the resulting (small) linear system (<ref>) at each node 𝐱_i allows one to compose the (sparse) differentiation matrix W^ℒ=(w_ij^ℒ), such that the action of the linear differential operator ℒ on the field functions f at all nodal points 𝐱=(𝐱_1,…,𝐱_N)^ T can be approximated as ℒ𝐟≈ W^ℒ𝐟, where 𝐟=(f(𝐱_1),…,f(𝐱_N))^ T. In the following, we have ℒ∈{∂/∂ x,∂/∂ y}. Note that it is also customary to include certain low-degree polynomials while solving (<ref>), as with a pure RBF basis it is not possible to obtain the correct derivatives of constants, linear functions, etc. For further details, see e.g. <cit.>.Several RBFs are typically used, with the multiquadric, ϕ(r)=√(1+(ϵ r)^2), and the Gaussian RBF, ϕ(r)=exp(-(ϵ r)^2) being amongst the most popular choices. The parameter ϵ is called the shape parameter as it governs the flatness of the RBF. In the following we will work with the Gaussian RBF, since for hyperbolic PDEs the RBF-FD method typically requires the use of hyperdiffusion for stabilization which is most easily accomplished using the Gaussian RBF <cit.>. When implementing a numerical scheme for the shallow-water equations for ocean modeling, the presence of a nonflat sea bottom topography generally presents a challenge for both meshbased and meshless numerical schemes. More specifically, preserving the so-called lake at rest solution is a nontrivial, but crucial endeavor, since the violation of the lake at rest solution typically leads to the stimulation of spurious numerical waves that can render the correct simulation of the actual physical waves extremely challenging. There has been a considerable body of literature devoted to the construction of so-called well-balanced numerical schemes for the shallow-water equations, which are schemes that can preserve the lake at rest solution numerically, see <cit.> for some examples for such well-balanced schemes. Most recently, in <cit.>, a unifying strategy was proposed for developing general well-balanced meshbased and meshless schemes which is in particular suitable for the RBF-FD methodology. For the sake of completeness of the present exposition, we briefly review the key idea of <cit.> here for the case of the one-dimensional form of the shallow-water equations,h_t+(hu)_x=0, (hu)_t+(hu^2+1/2gh^2)_x=-ghb_x.The two-dimensional case is treated analogously by enforcing the well-balanced condition given below for both the x- and y-derivatives.As u=0 in the lake at rest solution, any well-balanced discretization has to preserve the identity1/2∂_xh^2=-h∂_xb,numerically in the case that h+b=c, for c=. It is found in <cit.> that this is in general only possible if one discretizes the balance equation (<ref>), at each point 𝐱_i, so that1/2(D^ f_xh^2)_i=-h_i(D^ s_x(c-h))_i,for any mesh function, h, and constant, c, where D_x^ f and D_x^ s are the discrete approximations to the spatial first derivative operators in the flux and source terms of the shallow-water equations, respectively, and h_i=∑_j=1^nm_ijh_j is a specific, consistent average (meaning that ∑_j=1^nm_ij=1) of the total water height h over the stencil of 𝐱_i.As noted in <cit.>, the above equality implies that(D^ s_xc)_i=0,1/2(D^ f_xh^2)_i=h_i(D^ s_xh)_i,must hold at all nodes 𝐱_i. The first condition requires the consistent (exact) derivative of constants by the derivative operator D^ s_x (which in our case of RBF-FD based derivatives requires the inclusion of zero-degree polynomials in the basis), and the second condition can be satisfied provided we interpret D^ f_xh^2 as a bilinear form, i.e. (1/2D_x^ fh^2)_i=1/2𝐡^ TW^ f_i𝐡.This procedure defines a third order differentiation tensor, 𝐖^ f, with the matrix W^ f_i being its ith slice. For the approximation of the source derivative, D_x^ s, we write(D^ s_xh)_i=(𝐰^ s_i)^ T𝐡,where 𝐰^ s_i=(W_ij^ s)_1⩽ j⩽ n is the ith row of the associated differentiation matrix W^ s. In a similar manner, we can write 𝐦_i = (m_ij)_1⩽ j⩽ n for the ith row of the averaging matrix M. The second condition of (<ref>) then naturally translates to1/2𝐡^ TW^ f_i𝐡=(𝐦^ T_i𝐡)((𝐰^ s_i)^ T𝐡),for all i, which (taking W^ f_i to be symmetric) requires thatW^ f_i=𝐦_i(𝐰^ s_i)^ T + 𝐰^ s_i𝐦_i^ T,holds at all nodes 𝐱_i. Practically speaking, one is thus free to choose the averaging matrix M and the derivative matrix W_x^ s corresponding to the derivative approximation used in the source term and then Eq. (<ref>) prescribes how to choose the weights in the flux derivative D_x^ fh^2, given through the derivative tensor 𝐖^ f, so as to obtain a well-balanced scheme for the shallow-water equations. For a more in-depth discussion and results regarding the consistency of the resulting discrete derivative operators, consult <cit.>.We use the RBF-FD method for discretizing the two-dimensional shallow-water equations (<ref>), invoking condition (<ref>) to guarantee that the resulting meshless scheme will be well-balanced. For the time-stepping, the second-order explicit midpoint scheme is used. It is pointed out in <cit.> that the application of the RBF-FD method to purely convective PDEs is prone to numerical instability as the eigenvalues of the associated derivative matrices tend to scatter to the complex right-half plane. As a remedy, the inclusion of hyperdiffusion was proposed, which we have included in our discretization as well. In other words, instead of solving the shallow-water equations (<ref>), we solve ρ_t+𝐅_x+𝐆_y=𝐒̃, where 𝐒̃ is a modified source term of the form 𝐒̃=𝐒+𝐃, where𝐃=(-1)^ℓ+1νΔ^ℓ(0,hu,hv)^ T,ℓ⩾1with ν being the diffusion parameter and Δ=∂^2/∂ x^2+∂^2/∂ y^2 being the two-dimensional Laplacian operator. If the underlying RBF is the Gaussian RBF, then Δ^ℓϕ(r)=ϵ^2ℓp_ℓ(r)ϕ(r), with p_ℓ(r) being computable through the recurrence relationp_0(r)=1, p_1(r)=4(ϵ r)^2-4, p_ℓ+1(r)=4((ϵ r)^2-2ℓ-1)p_ℓ(r)-16ℓ^2p_ℓ-1(r),ℓ⩾1.The polynomials p_ℓ(r) are related to the Laguerre orthogonal polynomials, see <cit.> for further details. Below, we use ℓ=2, unless otherwise noted.To summarize, we present below the algorithm used at every time step. The initialization phase of the algorithm uses the RBF-FD methodology to define the averaging matrix, M, source derivative, W_i^ s, as well as the discrete approximation of the hyperdiffusion operator, 𝐃.Additionally, we precompute b^ d_x and b^ d_y, the RBF-FD derivatives of the bottom topography function, b(x,y).In the algorithm below, we use D_x^ d and D_y^ d to denote the discrete derivative operators of the well-balanced scheme, with the discrete hyperdiffusion operator, D_Δ^2^ d, computed using (<ref>).At each node, i, we store unknowns h_i, (hu)_i, and (hv)_i; the products and ratios of vectors h, hu, and hv given in the algorithm below are computed componentwise.We note that the use of the inundation algorithm presented below ensures that no division by zero is ever performed.In what follows, at each time step, we make use of the natural partitioning of the set of all nodes, X = {𝐱_i}_i=1^N, into sets of wet and dry points,X_ wet={𝐱_i∈ X : h_i> δ}, X_ dry=X\ X_ wet,where δ∈ℝ is a user-defined (small) positive parameter.We note that this partitioning is updated after every stage of the time-step, but we supress superscripts or subscripts denoting the time-step and stage to simplify notation. For every timestep: Compute first stage of explicit midpoint rule: Compute hu^2=h·(hu/h)^2, hv^2=h·(hv/h)^2, hvu=h·hv/h·hu/h andh̅ = Mh, then set (𝔥_x)_i=(h̅)_i· (D_x^ d h)_i and (𝔥_y)_i=(h̅)_i· (D_y^ d h)_i for all nodes 𝐱_i ∈ X_ wet.h = h - Δ t/2( D_x^ d(hu)+D_y^ d(hv))hu = hu-Δ t/2(D_x^ d(hu^2)+ g 𝔥_x + D_y^ d(hvu) + gh̅ b^ d_x -ηD_Δ^2^ d(hu) ) hv = hv-Δ t/2( D_y^ d(hv^2)+ g 𝔥_y + D_x^ d(hvu) + gh̅ b^ d_y -ηD_Δ^2^ d(hv) ) Apply boundary conditions. Apply inundation algorithm. Compute second stage of explicit midpoint method: Compute hu^2=h·(hu/h)^2, hv^2=h·(hv/h)^2, hvu=h·hv/h·hu/h and h̃ = Mh̃, then set (𝔥_x)_i=(h̃)_i· (D_x^ dh)_i and (𝔥_y)_i=(h̃)_i· (D_y^ dh)_i for all nodes 𝐱_i ∈ X_ wet.h ← h -Δ t (D_x^ d(hu)+ D_y^ d(hv)) hu ← hu-Δ t ( D_x^ d(hu^2)+ g 𝔥_x + D_y^ d(hvu) + gh̃ b^ d_x -ηD_Δ^2^ d(hu))hv ← hv-Δ t ( D_y^ d(hv^2)+ g 𝔥_y +D_x^ d(hvu) + g h̃ b^ d_y -ηD_Δ^2^ d(hv)) Apply boundary conditions. Apply inundation algorithm. We note that, since the scheme is explicit, this allows a very efficient implementation of the time-stepping using only sparse matrix multiplications with M and W_i^ s, along with componentwise vector operations.Furthermore, the fine-scale parallelism of these operations allows a natural avenue for parallelization, on both classical compute clusters and modern manycore and accelerated architectures.§ MESHLESS INUNDATION ALGORITHM Of particular importance in a tsunami model is the treatment of the wet–dry interface when the incoming wave hits the coastline. Several algorithms have been proposed to deal with this moving boundary condition in numerical models that solve the governing equations in the strong form (for associated results for the governing equations in the weak form, consult, e.g. <cit.>). In <cit.>, the authors use a finite-difference model with variable grid spacing near the boundary that ensures existence of a shoreline boundary point on the surface of the beach at all times. Similarly, in <cit.>, based on the earlier work of <cit.> for the shallow-water equations, a finite-difference model for the nonlinear Boussinesq equations with fixed grid spacing was used that handles the moving boundary by employing (one-dimensional) linear extrapolation of the wave run-up from the last wet points to the first dry points on the beach. This idea was re-considered in <cit.> where true two-dimensional bilinear extrapolation was used to compute the run-up in a two-dimensional finite-difference model for the Boussinesq equations. Since our numerical scheme is based on the RBF methodology using the strong form of the shallow-water equations, it is natural to use RBFs also for the inundation model. This is, in particular, justified as (multiquadric) RBFs were originally proposed by Hardy for two-dimensional scattered data interpolation <cit.> and later found by Franke to be the most accurate of all the 29 methods tested in <cit.>. Of further interest are the studies carried out in <cit.> where the authors investigated the Runge phenomenon in the context of RBF interpolation. It has been found that the Runge phenomenon can be controlled by a suitable choice of the shape parameter in the RBF (with spatially varying shape parameters most favorable) and non-equally chosen data points, typically much better than the Runge phenomenon can be controlled in standard polynomial interpolation.In light of the favorable performance of RBF-based interpolation schemes, we propose to use the extrapolation technique of <cit.> but instead of using polynomial-based extrapolation, we use RBF-based extrapolation.We again make use of the partitioning of the set of all nodes, X, into wet and dry points, as X = X_ wet∪ X_ dry, with X_ wet={𝐱_i∈ X : h_i> δ}. Since the shallow-water equations are not defined at the dry points (where h_i is too small or negative), we need to extrapolate the wet values to the dry points to have a numerical solution defined at all points X. The advantage of this procedure is that then the same derivative approximation can be invoked at all points, even those close to the wet–dry interface, where some of the nearest neighbors on which the RBF derivatives are defined will be dry points.For each dry point immediately neighboring a wet point we find its n_ e nearest wet points through which we define an RBF interpolant. Once the RBF interpolant based on the wet points is defined, we use it to extrapolate the field functions u, v and h+b to the dry point under consideration.We found experimentally that the multiquadric RBF basis augmented with first degree polynomials (i.e. constants and the monomimals x and y) is most suitable for the extrapolation procedure. Note that in order to preserve the lake at rest solution in the presence of dry points it is necessary to extrapolate h+b, not h alone, to the dry points. The extrapolation is done by the following algorithm: for each 𝐱^∈ X_ immediately neighboring a wet point find {𝐱_j_1^, …, 𝐱_j_nn^} nearest neighbours of 𝐱_k^ in X_ extrapolate h(𝐱^), hu(𝐱^), hv(𝐱^): * for f∈{h, hu, hv} solve the linear system for w [ϕ(r_1,1) … ϕ(r_1,nn) 1(𝐱_j_1^)^T; ⋮ ⋱ ⋮ ⋮; ϕ(r_nn,1) …ϕ(r_nn,nn) 1 (𝐱_j_nn^)^T; 1 … 1 0 0;𝐱_j_1^ … 𝐱_j_nn^ 0 0 ][w_1;⋮; w_nn; w_nn+1; 𝐰_nn+2 ] = [f(𝐱^_j_1);⋮; f(𝐱^_j_nn);0;0 ]withr_i, l=𝐱^_j_i-𝐱^_j_l Set f(𝐱^)=∑_i=1^nn w_i ϕ(r_i) + w_nn+1+𝐰_nn+2·𝐱^ Here, we note that the coefficient 𝐰_nn+2 has the same dimension as 𝐱^dry.§ NUMERICAL SIMULATIONS Having described the meshless discretization of the shallow-water equations (<ref>) and the associated inundation model, we now proceed to present the results of several classical benchmark tests for both the one-dimensional and two-dimensional form of the shallow-water equations. §.§ One-dimensional benchmarks We repeat here some of the one-dimensional tests carried out in <cit.>. In all experiments we compute the RBF-FD approximation D_x^ s, obtained from the procedure outlined in Section <ref>. The RBF used is the Gaussian RBF with shape parameter ϵ=0.1/Δ x, where Δ x is the (uniform) nodal spacing. Eq. (<ref>) is solved based on the three nearest neighbors of each nodal point (which includes each node itself). The averaging matrix M needed for (<ref>) is obtained from a normalized Gaussian filter over the three nearest neighbors of each node. With this, each time step of the explicit midpoint scheme can be implemented with 𝒪(1) work per spatial mesh point. §.§.§ Lake at rest solution It was shown in <cit.> that the numerical scheme based on a discretization that respects (<ref>)–(<ref>) is well-balanced. Since there the shallow-water equations were considered without an inundation model, we carry out a test case where part of the domain is initially dry, so as to check that the discretization remains well-balanced in the presence of wet–dry interfaces.Specifically, we consider the domain Ω=[0,1], with the smooth bottom topographyb_ s={[ aexp(-0.5/(r_m^2-r^2))exp(-0.5/r_m^2) r<r_m; 0 ].,where we set r=|x-0.5|, a=1.2 and r_m=0.4. We use h=max(0,1-b(x)) and u=0 as initial conditions. A total of N=50 (regularly spaced) grid points were used, the time step is Δ t=0.002 and the final integration time is t=20. The extrapolation tolerance is δ=0.0025 with the three nearest neighbors being used for the extrapolation. The shape parameter for the RBF extrapolation is ϵ_ e=1. Since realistic bathymetry is usually not smooth, in a second experiment, we add some noise on top of the smooth bottom topography b_s. In particular, we consider a noisy bathymetry of the formb_ n=b_ s+∑_j=1^3a_jsin(16jπ x+p_j),where a=(0.1,0.2,0.3) and p=(1.6,3.2,0.5). Reflecting boundary conditions were used for both experiments. The results of the two numerical experiments are depicted in Figures <ref> and <ref>, which demonstrate that the extrapolating boundary conditions preserve the well-balanced properties of the meshless RBF-FD discretization. §.§.§ Oscillatory flow in a parabolic basin We consider the oscillatory flow in a parabolic basin, which is described by the following exact solution first reported in <cit.>. The domain for this problem is Ω=[-5000,5000] with parabolic bottom topography b=h_0(x/a)^2, where a=3000 and h_0=10. The initial conditions are chosen so that the exact solution to the shallow-water equations for this benchmark ish_ a(t,x)=h_0-B^2/4g(1+cos2ω t)-Bx/2a√(8h_0/g)cos(ω t),u_ a(t,x)=Baω/√(2h_0g)sinω twhere ω=√(2gh_0)/a and B=5. In the first experiment we use N=200 equally spaced nodes and integrate the one-dimensional shallow-water equations up to t=3000 using the time step Δ t=1. The extrapolation parameter is set to δ=0.01. Since the water surface never reaches the boundaries of the domain, no specific boundary conditions have to be imposed. Snapshots of the numerical solution and the exact solution at times t=0, t=1000, t=2000 and t=3000 are depicted in Figure <ref>. The associated time series of the l_∞ relative height and mass errors are shown in Figure <ref>.Note that although our numerical scheme does not preserve mass by construction, the relative change in the total mass over time is small, bounded and oscillatory only. That is, no spurious trend in the mass is introduced by the numerical scheme and the inundation model.To numerically verify the convergence of the numerical solution, we next carry out a sequence of numerical integrations using N∈{200,400,800,1600} equally spaced points. The time steps are halved each time the number of nodes is doubled. The final integration time is again t=3000. In Table <ref>, we record the maximum l_∞ errors (relative height error, relative mass error, absolute momentum error), occurred over the integration period. The table shows also the experimental convergence rate e_c/e_f, where e_c is the error of the coarse and e_f of the fine grid.Similar data is shown in Table <ref>, for the l_2 norm; we note that both measures of the error show similar overall convergence, but the l_2 norm shows steadier convergence rates than the maximum norm data in Table <ref>.§.§.§ Tsunami run-up on a sloping beach The run-up of waves on a sloping beach is a classical test case for numerical schemes for the shallow-water equations within the area of tsunami modeling. As in the case of the parabolic bowl, there exists an analytical solution for this test case, which was first found in the seminal paper <cit.>.Here, we follow <cit.> and consider the computational domain Ω=[-500,50 000], with the bottom topography being defined as b=5000-0.1x. Since we will consider the solution at times t=160, t=175 and t=220, simple reflecting boundary conditions can be employed at the right boundary since the reflected waves cannot travel to the sloping beach in that time. We choose Δ x=20 with a time step of Δ t=0.025. The extrapolation parameter was set to δ=0.1. The initial condition, numerical solutions and exact solutions at the sampling times are displayed in Figure <ref>, which demonstrates that the inundation process is correctly approximated by the numerical model.§.§ Two-dimensional benchmarks We consider three two-dimensional benchmarks, again the lake at rest, but on a non-uniform mesh, flow around a conical island, and the Monai Valley experiment. Again, the well-balanced RBF-FD discretization described in Section <ref> is used. The RBF-FD discretization again uses the Gaussian RBF with shape parameter ϵ=0.1/√(Δ x^2+Δ y^2), where Δ x and Δ y are the spacings in x- and y-direction. A total of nine nearest neighbors are used for the derivative approximation (which again includes each node as the center of the stencil itself), unless otherwise indicated. A two-dimensional normalized Gaussian filter over these nine nearest neighbors is used to construct the averaging matrix, M.§.§.§ Two-dimensional lake at rest We repeat the smooth lake at rest test case in two dimensions but, in addition, we use a non-uniform mesh. We consider the domain Ω = [0,1]^2 covered by n=2500 nodes. To demonstrate that our scheme is meshless, we start from a uniform 50× 50 mesh of the unit square and add (0.1Δ x 𝒩(0,1), 0.1Δ y 𝒩 (0,1)) as a disturbance to the coordinates of each grid point, where Δ x = Δ y = 0.02 and 𝒩(0,1) is a normally distributed random variable with zero mean and variance one. We use reflecting boundary conditions. The bottom topography is given byb(x,y)=b_s((x-0.5,y-0.5)_2)where b_s is defined in Equation (<ref>). Again we use h=max(0,1-b(x, y)) and u=0 as initial conditions. The time step is Δ t=0.0015 and the final integration time is t=20. The extrapolation parameter is set to δ=0.05. In this case, we use 25 neighbours for the derivative approximation, to ensure sufficiently accurate approximations on the non-uniform mesh. The initial surface elevation and node distribution can be seen in Figure <ref>. Figure <ref> shows the errors in the computed solution over time.§.§.§ Flow around a conical island The flow around a conical island is another classical benchmark test for tsunami models, following the experimental study described in <cit.>; see also <cit.>. The experiment is an idealization of the 1992 Flores Island tsunami run-up on Babi Island.The setup of the experiment is a 25m long and 30m wide basin with a flat ground. The border of the basin absorbs the incoming wave, therefore there are no reflections.The origin of this conical island is located at x=12.96m and y=13.8m. In Figure <ref>, the setup, shape and properties of the island can be seen. The initial water depth is h_0=0.32m. The y-axis is parallel to the wavemaker, which generates solitary waves.We will compare the data of 4 of the 27 gauges originally used in the experiment, to measure the surface wave elevations. The position of the four gauges are indicated in Figure <ref>.We choose Δ x=0.125, Δ y=0.14 and Δ t=0.02. Then we simulate case A, withheight-to-depth ratio H=0.04, and case C, with H=0.18. The recorded water heights at gauges 6, 9, 16 and 22 can be seen in Figure<ref>. Figure <ref> shows snapshots of the propagating wave for both cases. The maximal horizontal run-up agrees well with the measured data of the experiment, which can be seen in Figure <ref>. §.§.§ Monai Valley experiment The Monai Valley experiment <cit.> is a model of the 1993 Okushiri tsunami,which caused an extreme 32 meter run-up in the Monai Valley on Okushiri island. The purpose of this experiment is to recreate the run-up in a 1/400 model of the relevant part of Okushiri island, see also <cit.> for further details.We discretize the domain Ω=[0,5.488]×[0,3.402] with a regular mesh with step sizes Δ x=Δ y=0.014. Reflective boundary conditions are employed everywhere except at x=0, where the incident wave is prescribed up to t=22.5. For t>22.5 we use open boundary conditions at x=0. Water levels are recorded at the three points (4.521,1.196), (4.521,1.696) and (4.521,2.196), which correspond to gauges 1, 2 and 3 of the experimental setup at which points measurements of the water height are provided. We integrate the shallow-water equations until t=25, which is long enough to record the maximum run-up which occurs at approximately t=20 at the reference locations. The time step in the simulation was set to Δ t=0.01. All experimental data as well as the incident wave profile were obtained from <cit.>. The evolution is illustrated by four snapshots in Figure <ref>. The recorded water heights at the gauges is in good accordance with the experimentally recorded values, see Figure <ref>. § CONCLUSIONS In this paper, we have proposed a novel numerical procedure for solving the shallow-water equations, which is suitable for tsunami modeling. In particular, both the well-balanced numerical scheme and the inundation algorithm are based on radial basis functions. This makes the model truly meshless and, hence, capable of employing variable resolution as well as operating on arbitrary coastal regions, without the need to use an underlying orthogonal mesh. First benchmark tests demonstrate the competitiveness of the proposed methodology both in the one- and two-dimensional setting.A main defining characteristic of tsunamis is that they occur on a multitude of spatial and temporal scales, which can pose considerable challenges to numerical solvers for the shallow-water equations. An advantage of the RBF methodology is that is can be easily adopted to both arbitrary geometries (e.g. the plane and the sphere) and arbitrary nodal layouts, both features important for the far- and near-field modeling of tsunamis. While, in the present work, we have concentrated on the near coast propagation of tsunamis as well as coastal inundation, in future work we will combine the methodology proposed here with a global scale tsunami propagation model. Suitable candidates for numerical models have already been proposed in <cit.>, which we will adopt to be able to handle arbitrary sea bottom topographies.With the exception of the two-dimensional lake at rest solutions, all other benchmark tests were carried out on a regular, orthogonal mesh. This was done in order to facilitate comparison with other numerical models for the same benchmark problems that are usually done on an orthogonal mesh as well. A full demonstration of the meshless capabilities of the proposed shallow-water discretization, as well as more complicated real-world tsunami simulations, is a subject for future work.One potential problem of the RBF-FD methodology as used in the present paper is that the resulting schemes are not exactly mass conserving. While we have numerically verified that the error in mass conservation is purely oscillatory (and showing no spurious growth or decay), in order for the meshless methodology to be competitive with standard finite volume or discontinuous Galerkin methods (which both typically preserve mass) it will be essential to develop a meshless methodology that is conservative when applied to hyperbolic conservation laws. We will consider this issue in future investigations.§ ACKNOWLEDGMENTS This research was undertaken, in part, thanks to funding from the Canada Research Chairs program and the NSERC Discovery Grant program.J.B. acknowledges support by the Cluster of Excellence CliSAP (EXC177), Universität Hamburg, funded by the German Science Foundation (DFG). Further support by Volkswagen Foundation (Project ASCETE, AZ 88 470) is acknowledged. The authors thank Stefan Vater for valuable advice on setting up and interpreting test cases. 10 urlstylemona17a Tsunami Runup onto a Complex Three-dimensional Beach; Monai Valley, <http://nctr.pmel.noaa.gov/benchmark/Laboratory/Laboratory_MonaiValley/>, accessed: 2017-02-18.audu04a Audusse E., Bouchut F., Bristeau M.O., Klein R. and Perthame B., A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows, SIAM J. Sci. 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Elem. 28 (2004), 967–973. | http://arxiv.org/abs/1705.09831v1 | {
"authors": [
"Rüdiger Brecht",
"Alexander Bihlo",
"Scott MacLachlan",
"Jörn Behrens"
],
"categories": [
"physics.ao-ph",
"physics.comp-ph",
"physics.flu-dyn",
"physics.geo-ph"
],
"primary_category": "physics.ao-ph",
"published": "20170527153101",
"title": "A well-balanced meshless tsunami propagation and inundation model"
} |
[ [ December 30, 2023 =====================fancy shortarticle§ INTRODUCTION Linear canonical transform (LCT), first introduced in <cit.>, is a generalization of the fractional Fourier transform (FRFT). It unifies a variety of transforms such as Fourier transform (FT),FRFT and Fresnel transform. It has four parameters with three degrees of freedom, and thus more important and useful in optics <cit.> and many signal processing applications including filter design, radarsystemanalysis, signal synthesis, phasereconstruction, time-frequency analysis, patternrecognition, encryption and modulation <cit.>. To extend the 1D LCT to two dimensions (x,y), an easy and straightforward approach is performing two independent 1D LCTs in the two transverse directions x and y, respectively. Since the two-dimensional (2D) kernel can be separated, this 2D transform is called 2D separable LCT (2D SLCT) <cit.>. The 2D SLCT can produce affine transformations in the (x,ω_x) and (y,ω_y) planes, where ω_x and ω_y are the spatial-frequency coordinates with respect to x and y. Since the 1D LCT has three degrees of freedom, the 2D SLCT has six degrees of freedom.A further generalization of the 2D SLCT is the 2D nonseparable LCT (2D NsLCT) <cit.>, named after its nonseparable 2D kernel. The 2D NsLCT provides four more (i.e. ten) degrees of freedom to represent all transformations not onlyin(x,ω_x) and (y,ω_y) planes but also in (x,y), (x,ω_y), (ω_x,y)and (ω_x,ω_y) planes. The 1D/2D FT, FRFT and Fresnel transform, as well as the 1D LCT, 2D SLCT and gyrator transform <cit.> are all its special cases. All the applications of these special cases in optics,signal processing and digital image processing can be extended and become more flexible by the 2D NsLCT. For example, in <cit.>, the authors show that the noise with nonseparable term cannot be removed clearly by the 2D SLCT filter but can be by the 2D NsLCT filter. In optical system analysis, the 2D NsLCT is more effective when analyzing systems containing quadratic phase components misaligned in both x and y axes <cit.>. Consider an affine transformation in the space-spatial-frequency plane: [u;v; ω _u; ω _v ] =[ A B; C D ] [x;y; ω _x; ω _y ] = [ a_11 a_12 b_11 b_12; a_21 a_22 b_21 b_22; c_11 c_12 d_11 d_12; c_21 c_22 d_21 d_22 ][x;y; ω _x; ω _y ].where (ω_x,ω_y) and (ω_u,ω_v) are the spatial-frequency coordinates with respect to (x,y) and (u,v), respectively. The 4×4transformation matrix in (<ref>) is called ABCD matrix and also denoted by ( A, B; C, D) in this paper.Assume z=[x,y]^T and z'=[u,v]^T.The 2D NsLCT that can result in the space-spatial-frequency transformation in (<ref>) is given by <cit.>G( z') =O_^( A, B; C, D){g( z)}=1/2π√( -(B))∫exp[ j/2( z'^TDB^ - 1z'... . / - 2z^TB^ - 1z' + z^TB^ - 1Az)]g( z)d z, where g( z)=g(x,y) and G( z')=G(u,v) are the 2D input and output signals, respectively. The ABCD matrix ( A,B;C,D) for a valid 2D NsLCT should satisfyAB^T = BA^T, CD^T = DC^T,AD^T-BC^T= I,or equivalentlyA^TC = C^TA, B^TD = D^TB,A^TD-C^TB= I.Either (<ref>) or (<ref>) leads to six linear equations, i.e. six constraints. Although there are a total of 16 parameters inA, B, C and D, the number of degrees of freedom is 10 due to the six constraints. It is obvious that the definition in (<ref>) is valid only when ( B)≠0, i.e. B is invertible. When B= 0, the 2D NsLCT reduces into a 2D affine transformation multiplied by a 2D chirp function:G( z') = √( (D))exp( j/2z'^TCD^Tz')g( D^Tz'). When ( B)=0 but B≠0, the definition of 2D NsLCT is subdivided into several different cases. One can refer to <cit.> for a detailed details.The digital implementation of the 1D LCT has been widely studied in many papers such as <cit.>. The digital implementation of the 2D SLCT can be easily realized by performing any of these 1D implementation techniques twice, one in the x direction and one in the y direction. However, there are less works regarding the digital implementation of the 2D NsLCT <cit.>. Zhao et al.'s works <cit.> mainly focus on the sampling of the 2D NsLCT to ensure unitary property. Koç et al.'s work <cit.> and Ding et al.'s work <cit.> focus on the development of digital implementation algorithms to improve complexity and accuracy.To develop the 2D NsDLCT, the simplest method is to discretize the 2D NsLCT in (<ref>) by sampling and summation:G[p,q] = O_^( A, B; C, D){g[m,n]}=Δ_xΔ_y/2π√( -(B))∑_m ∑_n exp[ j/2.. /·(r'^TDB^ - 1r' - 2r^TB^ - 1r' + r^TB^ - 1Ar) ]g[m,n],where r=[mΔ_x,nΔ_y]^T and r'=[pΔ_u,qΔ_v]^T are the input and output sampling points, respectively. This 2D NsDLCT is named direct method in this paper. The direct method is very inefficient. In order to reduce computational complexity, the 2D NsLCT is decomposed into several simpler 2D operations, and it follows that one can develop a low-complexity 2D NsDLCT by connecting several low-complexity 2D discrete operations. Decomposition methods have been widely used in the digital implementation of 1D/2D FRFT, 1D LCT, gyrator transform, etc. In <cit.>, the authors decomposed the 2D NsLCT into one 2D chirp multiplication (2D CM), one 2D FRFT and two 2D affine transformations. However, 2D affine transformations will introduce interpolation error in the discrete case. According, in <cit.>, another decomposition that involves two 2D CMs, one 2D FT and only one 2D affine transformation is proposed.In this paper, the accuracy is further improved by decomposing the 2D NsLCT into two 2D CMs and two 2D chirp convolutions (2D CCs), called CM-CC-CM-CC decomposition. More precisely, only 2D CMs, 2D FTs and 2D inverse FTs (2D IFTs) are used. No 2D affine transformations are involved. Based on this decomposition, two types of 2D NsDLCT are proposed. Compared to each other, one has higher accuracy while the other one has lower complexity. All the proposed 2D NsDLCTs have lower complexity and higher accuracy than the previous works <cit.>. Plus, the proposed methods have lower error in additivity property, which is a useful property in applications such as filtering and encryption/decryption. Besides, another decomposition called CM-CC-CM-CC decomposition is introduced such that the proposed methods have perfect reversibility property. That is, the input discrete signal/image can be perfectly recovered from the output of the proposed 2D NsDLCTs.§ BASIC 2D DISCRETE OPERATIONSThe 2D nonseparable discrete LCTs (NsDLCTs) proposed in this paper will be compared with Koç's method <cit.> andDing's method <cit.>. In this section, some basic 2D discrete operations used in these 2D NsDLCTs are introduced. The computational complexity of these operations is also analyzed. Denote G(u,v) as the output of a 2D continuous operation with g(x,y) being the input.Given N× N sampled input signal g[m,n]Δ =g(mΔ_x,nΔ_y), the corresponding 2D discrete operation is designed to generate N× N output G[p,q] that can approximate G(pΔ_u,qΔ_v). §.§ 2D Discrete Chirp Multiplication (Discrete CM)Consider that the ABCD matrix is given by [ A B; C D ] = [ I 0; C I ], C=[ c_11 c_12; c_12 c_22 ]is symmetric, i.e.C= C^T, because of the constraints in (<ref>) or (<ref>).Thus, this ABCD matrix has only three degrees of freedom. Since B= 0, the definition in (<ref>) is used, and the 2D NsLCT becomesthe multiplication with a 2D chirp function, called 2D chirp multiplication (CM) for short:G(u,v)=O_^( I, 0; C,I){g(x,y)}=e^j/2( c_11u^2 + 2c_12uv + c_22v^2)g(u,v). Sampling (<ref>) with sampling intervals Δ_u=Δ_x and Δ_v=Δ_y, the 2D discrete CM, denoted by O_^ C, is given by G[p,q] =O_^ C{g[m,n]}Δ =e^j/2( c_11p^2Δ_u^2 + 2c_12pqΔ_uΔ_v + c_22q^2Δ_v^2 )g[p,q].Supposing the exponential kernel function in (<ref>) can be precomputed and stored in memory,only one N× N pointwise product that involves N^2 complex multiplications is required. §.§ 2D Discrete Fourier Transform (DFT) and inverse DFT (IDFT)In (<ref>), if ABCD matrix is given by[ A B; C D ] = [0I; -I0 ][0 -I;I0 ],the 2D NsLCT reduces to the 2D Fourier transform (FT) or the 2D inverse FT (IFT) multiplied by constant phase 1/j: G(u,v)= 1/j2π∫_ - ∞^∞∫_ - ∞^∞e ^ ∓ jux ∓ jvyg(x,y)dxdy.If Δ_uΔ_x=Δ_vΔ_y=2π/N, the discrete version of (<ref>),denoted by F_x,y or F^-1_x,y, are simply the 2D DFT or 2D IDFT multiplied by some constant:G[p,q] = F_x,y{g[m,n]}Δ = Δ _xΔ _y/j2π∑_m^∑_n^e^ - j2π/N(pm + qn)g[m,n],G[p,q] = F_x,y^-1{g[m,n]}Δ =Δ _xΔ _y/j2π∑_m^∑_n^e^ + j2π/N(pm + qn)g[m,n]. The 2D DFT and IDFT can be implemented by 2D FFT with N^2/2log_2N^2 complex multiplications. Zero-padding the input signal to size N'× N',where N'>N, canreduce the output sampling intervals to Δ_u=2π/N'Δ_x and Δ_v=2π/N'Δ_y, but the cost is higher computational complexity. §.§ 2D Discrete Chirp Convolution (Discrete CC)Suppose the ABCD matrix is of the following form[ A B; C D ] = [ I B; 0 I ],B=[ b_11 b_12; b_12 b_22 ]. B is symmetric because of the constraints in (<ref>) or (<ref>) and thus has only three degrees of freedom.In (<ref>), ( I,B; 0,I) leads to G(u,v)= O_^( I,B; 0,I){g(x,y)}=1/2π√( -(B))·∫_ - ∞^∞∫_ - ∞^∞e^j/2 (B)[ b_22(u - x)^2 - 2b_12(u - x)(v - y) + b_11(v - y)^2]g(x,y)dxdy,which is a 2D convolution with a 2D chirp function and called 2D chirp convolution (CC) for short. In the discrete case, directly calculating the sampled version of (<ref>) by summationleads to computational complexity up to O(N^4). Fortunately, the ABCD matrix in (<ref>) can be decomposed as[ I B; 0 I ] = [ 0 - I; I 0 ]_[ I 0; - B I ]_[ 0 I; - I 0 ]_.Taking the benefit of the additivity property of 2D NsLCT, the above equality implies that the 2D CC can bealternatively implemented by one 2D IFT, one 2D CM and one 2D FT, i.e.O_^( I,B; 0,I)=O_^( 0, - I; I,0)O_^( I,0;- B,I)O_^( 0,I;- I,0).Therefore, the 2D discrete CC with chirp matrix B, denoted byO_^ B, is defined as a cascade of one 2D IDFT, one 2D discrete CM with chirp matrix - B, and one 2D DFT:O_^ BΔ =F^-1_x,yO_^- BF_x,y.Two 2D FFTs and one pointwise product are used andtotally requireN^2log_2N^2+N^2 complex multiplications. §.§ 2D Discrete Fractional Fourier Transform (DFRFT)The 2D FRFT with two parameters α and β is a specialcase of the 2D NsLCT. If A, B, C and D are given byA = D = [ cosα0;0 cosβ ],B =- C = [ sinα0;0 sinβ ],the 2D NsLCTreduces tothe 2D FRFT <cit.> with some constant phase difference:G(u,v)=O_^( A,B;- B, A){g(x,y)}=1/2π√( - sinαsinβ)∫_ - ∞^∞∫_ - ∞^∞K_α(u,x)K_β(v,y)g(x,y)dxdy,whereK_α and K_β are 1D FRFT kernels <cit.> with fractioanl angles α and β, respectively:K_α(u,x) = exp( jα/2u^2 - jα ux + jα/2x^2).Obviously, the 2D FRFTis separable andcan be implemented by two 1D FRFTs in two transverse directions, x and y. There are a variety of implementation algorithms for 1D FRFT, and a review of some of them is given in <cit.>. Here, we introduce the algorithm used in Koç's method <cit.>. IfΔ_uΔ_x=2π|sinα|/N, the sampled version of the kernel in (<ref>) is given byK_α[p,m] = exp( jα/2p^2Δ_u^2 ∓ j 2π/N pm + jα/2m^2Δ_x^2 ).For the minus-plus sign ∓ in the above kernel,minus is usedwhen sinα>0 while plus is used when sinα<0. (<ref>) shows that the 1D DFRFT can be implemented by two discrete CMs and one DFT/IDFT.Once the 1D DFRFT is developed, the 2D DFRFT can be commutated by two separate 1D DFRFTs in two transverse directions, m and n:G[p,q] =F^ α,β_x,y{g[m,n]}Δ =Δ_xΔ_y/2π√( - sinαsinβ)∑_m^∑_n^K_α[p,m]K_β[q,n]g[m,n].According to (<ref>), if Δ_uΔ_x=2π|sinα|/N, Δ_vΔ_y=2π|sinβ|/N, sinα>0 and sinβ>0, the 2D DFRFT can be implemented by two 2D discrete CMs and one 2D DFT:F^ α,β_x,y=j/√( - sinαsinβ) O_^ HF_x,yO_^ H,where chirp matrix H is given byH= A B^-1= B^-1 A= [ α 0; 0 β ]. One 2D FFT and two pointwise products are used in F^ α,β_x,y and thustotally involveN^2/2log_2N^2+2N^2 complex multiplications.§.§ 2D Discrete Affine TransformationWhen B and C are both 0, one has A=( D^T)^-1 since AD^T= I:[ A B; C D ] = [ ( D^T)^-1 0; 0 D ],D=[ d_11 d_12; d_21 d_22 ].From(<ref>),the above ABCD matrix leads toG(u,v) =O_^(( D^T)^-1,0; 0, D){g(x,y)}=√( (D)) g(d_11u + d_21v,d_12u + d_22v),which is a 2D affine transformation.Sampling (<ref>) with Δ_u and Δ_v yieldsG(pΔ_u,qΔ_v)=√( (D)) g(d_11pΔ_u + d_21qΔ_v,d_12pΔ_u + d_22qΔ_v).However, G(pΔ_u,qΔ_v)is often not available when there are a limited number of input samples. Accordingly, 2D interpolation is necessary.Here, we introduce the bilinear interpolation method that isused inDing's method <cit.>. With the discrete input g[m,n]=g(mΔ_x,nΔ_y), the 2D discrete affine transformation, denoted by O_^ D, based on bilinear interpolation is given by G[p,q]=O_^ D{g[m,n]}Δ =√( (D)){(1 - k)(1 - l)· g[p_2,q_2] + (1 - k)l· g[p_2,q_2 + 1]..+ k(1 - l) · g[p_2 + 1,q_2] + kl · g[p_2 + 1,q_2 + 1]},wherep_1=d_11pΔ_u + d_21qΔ_v/Δ_x, q_1=d_12pΔ_u + d_22qΔ_v/Δ_y,p_2=⌊p_1⌋, q_2=⌊q_1⌋,k=p_1-p_2, l=q_1-q_2.The symbol ⌊ ⌋ denotes floor function.In (<ref>), if (1-k)(1-l), (1-k)l, k(1-l) and kl arepre-computed, each output sample G[p,q]requires 8 real multiplications, or equivalently 2 complex multiplications. Therefore, the total number of required complex multiplications is 2N^2.Smaller Δ_x and Δ_y can decrease the approximation error between G[p,q] and G(pΔ_u,qΔ_v), but the additional upsamplingpreprocess requires more computation time. § REVIEW OF PREVIOUS WORKSMost digital implementation methods of the 1D FRFT can be extended to the 1D LCT. However, the 2D NsLCT is much more complicated. Most digital implementation methods of the 2D FRFT is not suitable for the 2D NsLCT or need extensive modification. In this section, the implementation algorithms of Koç's 2D NsDLCT <cit.> andDing's 2D NsDLCT <cit.> are introduced . These two methods are based on the additivity property of 2D NsLCT and decompositions of ABCD matrix. Again, assume the discrete input and output are both of size N× N. §.§ Koç's 2D NsDLCT <cit.>In <cit.>, Koç, Ozaktas and Hesselink proposed a fast algorithm to compute the 2D NsLCT. It is based on the Iwasawa decomposition <cit.> that decomposes the ABCD matrix into[ A B; C D ] = [ I 0; G I ][S0;0 S^ - 1 ][ X Y; - Y X ],whereS = (AA^T + BB^T)^1/2,G = (CA^T + DB^T)S^ - 2, X = S^ - 1A Y = S^ - 1B.In (<ref>), the first and second matrices correspond to 2D CM and 2D affine transformation, respectively. The last matrix can befurther decomposed as follows:[ X Y; - Y X ] = [ R_ϕ 0; 0 R_ϕ ][ E F; - F E ][ R_θ 0; 0 R_θ ],where the first and third matrices correspond to 2D affine transformations (more precisely, geometric rotations) while the second matrix corresponds to 2D FRFT:R_θ = [ cosθ sinθ; - sinθ cosθ ], R_ϕ = [ cosϕ sinϕ; - sinϕ cosϕ ], E = [ cosα0;0 cosβ ], F = [ sinα0;0 sinβ ].The fractional angles α and β can be obtained fromexp[ j(α+ β )]= ( X + jY), cos (α- β )=X +Y.And then the values of rotation angles θ and ϕ can be determined by solving the following two equations:exp[ j( θ+ ϕ+ (α+ β )/2)]= X_11 + X_22 - Y_12 + Y_21 + j( X_12 - X_21 + Y_11 + Y_22)/2cos[ (α- β )/2]exp[ j( θ- ϕ+ (α+ β )/2)] = j(- X_11 + X_22 + Y_12 + Y_21) + X_12 + X_21 + Y_11 - Y_22/2sin[ (α- β )/2].Substituting(<ref>) into (<ref>) leads to[ A B; C D ] = [ I 0; G I ]_[SR_ϕ 0; 0 S^ - 1R_ϕ ]_[ E F; - F E ]_[ R_θ 0; 0 R_θ ]_. Based on this decomposition, Koç et al.'s developed a 2D NsDLCT, denoted by O_^( A,B; C, D), consisting of one 2D discrete CMs, two 2D discrete affine transformation and one 2D DFRFT:O_^( A,B; C, D) Δ = O_^ GO_^S^ - 1R_ϕF^ α,β_x,yO_^R_θ=O_^ GO_^S^ - 1R_ϕ O_^ HF_x,yO_^ HO_^R_θ.These basic discrete operations used above have been defined in (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), and H is given byH= E F^-1= F^-1 E=[ α 0; 0 β ]. If the input of the 2D DFT F_x,y is zero-padded to N'× N' where N'>N, the output of the 2D DFT will have smaller sampling intervals. Then, as mentioned inSec. <ref>.<ref>, the interpolation in the 2D discrete affine transformation O_^S^ - 1R_ϕ will have higher accuracy. But the cost is higher computational complexity. If two-times upsampling (N'=2N) is employed,the number of complex multiplications used in Koç's method becomesN^2+2N^2+N'^2+N'^2/2log_2N'^2+N^2+2N^2=2N^2log_2N^2+14N^2.The six terms in (<ref>) (from left to right) are the numbers of complex multiplications used in the six discrete operations in (<ref>) (from left to right), respectively.§.§ Ding's 2D NsDLCT <cit.> Since 2D discrete affine transformations will introduce interpolation error, Ding, Pei and Liu proposed a more accurate 2D NsDLCT in which only one 2D discrete affine transformation is used. In their work, the ABCD matrix is decomposed into another form:[ A B; C D ]=[ I 0; DB^ - 1 I ]_[B0;0 (B^T)^ - 1 ]_[ 0 I; - I 0 ]_[ I 0; B^ - 1A I ]_.Based on (<ref>), Ding et al.'s developed another type of 2D NsDLCT, denoted by O_^( A,B; C, D), which connects a 2D discrete CM, a 2D discrete affine transformation, a 2D DFT and another 2D discrete CM in series: O_^( A,B; C, D) = O_^DB^ - 1O_^(B^T)^ - 1F_x,yO_^B^ - 1A.These basic discrete operations aredefined in (<ref>), (<ref>) and (<ref>).Again, supposetwo-times upsampling is performed inF_x,y for higher accuracy in the 2D discrete affine transformation O_^(B^T)^ - 1. Assumetwo-times upsampling (N'=2N) is employed. The number of complex multiplications required by Ding's method is given byN^2 + 2N^2 + N'^2/2log _2N'^2 + N^2 = 2N^2log _2N^2+8N^2.Comparing (<ref>) and (<ref>), Ding's method has lower computational complexity thanKoç's method. § PROPOSED 2D CM-CC-CM-CC NSDLCTSSince the 2D discrete affine transformation will introduce interpolation error, in this paper, some 2D NsDLCTs are developed with no 2D discrete affine transformation involved. In <cit.>, Koç et al. have introduced a varietyof decompositions for 1D LCT, where the parameter matrix is decomposed into three, four or five matrices. The CM-CC-CM decomposition <cit.> is the only one without using scaling operation.Therefore, to avoid affine transformations in two dimensions, a possible method is decomposing the ABCD matrix into CM matrices ( I,0; C,I) and CC matrices ( I,B; 0,I). Accordingly, all the proposed methods are composed of only the 2D discrete CMs and 2D discrete CCs.More precisely, only the 2D discrete CMs, 2D DFTs and 2D IDFTs are used. These basic discrete operations have been defined in (<ref>), (<ref>), (<ref>) and (<ref>). §.§ 2D NsDLCT Based on CM-CC-CM Decomposition When B= B^T and B≠0 First, suppose the 2D NsLCT can also be decomposed into CM-CC-CM. Then, it implies that the ABCD matrix can be expressed as the following form:[ A B; C D ] = [I0; (D- I)B^ - 1I ]_[ I B; 0 I ]_[I0; B^ - 1(A- I)I ]_.Obviously, B must be invertible. And since all the matrix in (<ref>) should satisfy the constraints in (<ref>) or (<ref>),the necessary and sufficient condition is that B is symmetric. Therefore, ifB= B^T B≠0,the 2D NsDLCTs based on the CM-CC-CM decomposition is given byO_^( A,B; C, D) = O_^(D- I)B^ - 1 O_^ BO_^B^ - 1(A- I)=O_^(D- I)B^ - 1 F_x,y^-1O_^- BF_x,yO_^B^ - 1(A- I). In this method, two 2D FFTs and three pointwise products are utilizedand totally require N^2log_2N^2+3N^2 complex multiplications.The 2D FRFT and gyrator transform <cit.> are the special cases of the 2D NsLCT withB= B^T and B≠0, and thus can be digitally implemented by (<ref>).§.§ 2D NsDLCT Based on CM-CC-CM-CC Decomposition When B≠ B^T or B=0 The CM-CC-CM decomposition in (<ref>) is valid only when B≠ B^T and B=0. Consider the more general case that ABCD matrix is arbitrary (but the constraints in (<ref>) or (<ref>) should be satisfied) and at most has ten degrees of freedom. The CM matrix ( I,0; C,I) and CC matrix ( I,B; 0,I) have only three degrees of freedom.Four CM/CC matrices are required todescribe the ABCD matrix. Accordingly, the CM-CC-CM-CC decomposition that has twelve degrees of freedom is considered.First, decompose the ABCD matrix into[ A B; C D ] = [A B';C D' ][ I H; 0 I ] =[A A H+B';C C H+D' ],where H= H^T, B'= B- A H and D'= D- C H. If B' is symmetric and invertible, according to (<ref>), the ABCD matrix can be further decomposed as follows:[ A B; C D ] = [I0; (D'- I)B'^ - 1I ]_[I B';0I ]_[ I 0; B'^ - 1(A- I) I ]_[ I H; 0 I ]_.This decomposition is valid even if B≠ B^T or B=0. The 2D NsDLCT based on the above CM-CC-CM-CC decomposition is given byO_^( A,B; C, D)= O_^(D'- I)B'^ - 1 O_^ B'O_^B'^ - 1(A- I)O_^ H.Since the CM-CC-CM-CC decomposition has two more degrees of freedom, there are infinite number ofpossible decomposition results (i.e. infinite choices of B',D'and H).There are two approaches to determine B',D'and H. Firstly, ifA is invertible, once B' is determined, H can be obtained from H= A^-1( B- B') and then D' from D'= D- C H. A valid B' for the CM-CC-CM-CC decomposition should satisfy the following three conditions:B'= B'^T, B'≠0AB'^T = B'A^T;that is,B'=[ b'_11 b'_12; b'_12 b'_22 ], b'_11b'_22-b'^2_12≠0, a_11b'_12+a_12b'_22=a_21b'_11+a_22b'_12.Since ( A,B; C, D) and ( A,B'; C, D') need to satisfy the constraints in (<ref>), it is required thatAB^T = BA^T and AB'^T = B'A^T. Thus, one has A( B-B')^T = ( B-B')A^T, and it follows that H= H^T:H=A^-1( B-B')=( B-B')^T( A^-1)^T = H^T.Secondly, if A is non-invertible, one has to determine Hfirst, and then obtain B' and D' from B'= B- A H and D'= D- C H, respectively. A valid H should lead toH= H^T, B'= B'^T B'≠0.AB'^T = B'A^T doesn't need to be considered because it is true when H= H^T is true:AB'^T=AB^T- A H^T A^T = BA^T- A H A^T = B'A^T. Since there are infinite number of solutions of H to (<ref>) (or B' to (<ref>)),the problem is what the best choice of H(or B') is. In the following, two types of 2D NsDLCT are proposed based on two types of H. One is for high accuracy while the other one is for low complexity. §.§.§ 2D High-Accuracy NsDLCT (HA-NsDLCT)It has been shown in (<ref>) that the 2D NsLCT will produce an affine transformation in the space-spatial-frequency plane. The 2D CM with chirp matrix C= C^T=(c_11,c_12;c_12,c_22) will produce shearing in spatial-frequency domain:ω _u = (c_11x + c_12y) + ω _x, ω _v = (c_12x + c_22y) + ω _y.Consider the simple case that the input signal occupies -S/2<x,y<S/2 in space domain and -S/2<ω_x,ω_y<S/2 in spatial-frequency domain, i.e. has space-spatial-bandwidth product S^4. After the shearing in (<ref>), the space-spatial-bandwidth product becomes γ( C)· S^4, where the ratio function γ(·) is defined as γ( C)=(|c_11| + |c_12| + 1)(|c_12| + |c_22|+1).From (<ref>), the 2D CCwith chirp matrix B= B^T=(b_11,b_12;b_12,b_22) will lead to shearing in space domain:u= x + (b_11ω _x + b_12ω _y),v= y + (b_12ω _x + b_22ω _y).The space-spatial-bandwidth product S^4 through the 2D CC becomes γ( B)· S^4. Too large space-spatial-bandwidth product will yield serious overlapping and aliasing effects. In order to minimize the increase of space-spatial-bandwidth product caused by the two CMs and two CCs in (<ref>), H is determined bythe following optimization problem:min_ H γ( (D' - I)B^'^ - 1) ·γ( B^') ·γ( B^'^ - 1(A - I)) ·γ( H) H= H^T, B'= B'^T B'≠0,where B'= B- A H and D'= D- C H. The resulting 2D NsDLCTis called 2D high-accuracy NsDLCT (HA-NsDLCT) and given byO_^( A,B; C, D)= O_^(D'- I)B'^ - 1 O_^ B'O_^B'^ - 1(A- I)O_^ H=O_^(D'- I)B'^ - 1 F_x,y^-1O_^- B'F_x,yO_^B'^ - 1(A- I)F_x,y^-1O_^- HF_x,y.Four 2D FFTs and four pointwise products are used and totally require 4·1/2N^2log_2N^2+4· N^2=2N^2log_2N^2+4N^2complex multiplications. The computational complexity of solving the optimization problem in (<ref>) is negligible when N is large enough. From (<ref>), (<ref>) and (<ref>), we can find out thatthe 2D HA-NsDLCT has lower complexity than Koç's method <cit.> and Ding's method<cit.>.§.§.§ 2D Low-Complexity NsDLCT (LC-NsDLCT)To achieve lower computational complexity, a symmetric H of either of the following two forms is considered:H=[ h_11 h_12; h_12 h_22 ]=[ h 0; 0 0 ][ 0 0; 0 h ].The above assumption is based on the fact that the CM-CC-CM-CC decomposition in (<ref>) has two more degrees of freedom than the ABCD matrix. Therefore, the number of variables in H can be reduced to one for lower complexity. The 2D CC with H in (<ref>) will reduce into1D CC with chirp rate h operating in the x direction or y direction. We call this type of 2D NsDLCT as 2D low-complexity NsDLCT (LC-NsDLCT). A valid H should satisfy the constraints in (<ref>).The matrix B'= B- A H needs to be symmetric. Thus, h in (<ref>) is given byh=(b_21-b_12)/a_21, H=(h,0;0,0),h=(b_12-b_21)/a_12, H=(0,0;0,h).If both (<ref>) and (<ref>) are available, we can use the criterion in (<ref>) to choose the one havinghigher accuracy.There are two cases that the 2D LC-NsDLCT is not suitable. Whena_12=a_21=0, both (<ref>) and (<ref>) are invalid. In this case, another type of decomposition is used: ( A,B; C,D)=(- B,A;- D,C)( 0, - I; I,0), where (- B,A;- D,C) can be decomposed into CM-CC-CM when A is symmetric. The corresponding 2D NsDLCT has lower complexity than the 2D LC-NsDLCT.Another case is that neither of the two forms of H in (<ref>) can yield invertible B'. For example, when B= 0, (<ref>) and (<ref>) yield H= 0, and it follows that B'= B- A H= 0 is invertible.Accordingly, if the assumption in (<ref>) leads to B'=0, the 2D HA-NsDLCT in Sec. <ref>.<ref> is utilized instead.Denote F_x and F_x^-1 as 1D DFT and 1D IDFT applied in thex direction, respectively. When H=(h,0;0,0),the 2D LC-NsDLCT is given byO_^( A,B; C, D)= O_^(D'- I)B'^ - 1 O_^ B'O_^B'^ - 1(A- I)O_^ H=O_^(D'- I)B'^ - 1 F_x,y^-1O_^- B'F_x,yO_^B'^ - 1(A- I)F_x^-1O_^- HF_x,where the last 2D discrete CM O_^ H reduces into 1D discrete CM in the x direction. When H=(0,0;0,h), the F_x and F_x^-1 in the above equation arereplaced byF_y and F_y^-1, respectively. The 2D DFT/IDFT is equivalent to performing two 1D DFTs/IDFTs inx and y directions.It implies that F_x and F_x^-1 (or F_y and F_y^-1) have half the computational complexity of F_x,y and F_x,y^-1, respectively. Accordingly, the total number of complex multiplications used in the 2D LC-NsDLCT is2·1/2N^2log_2N^2+4· N^2+2·1/2·1/2N^2log_2N^2=3/2N^2log_2N^2+4N^2.Compared with (<ref>), the 2D LC-NsDLCT has computational complexity 1/2N^2log_2N^2 lower than the 2D HA-NsDLCT. The complexity of the 2D FFT is also 1/2N^2log_2N^2. Since the 2D HA-NsDLCT involves four 2D FFTs and four pointwise products, we can say that the 2D LC-NsDLCT has complexity equivalent to three 2D FFTs and four pointwise products. Although the 2D LC-NsDLCT features low complexity, later we will show that it also has higher accuracy than Koç's method <cit.> and Ding's method<cit.>.§.§ 2D NsDLCT Based on CC-CM-CC-CM Decomposition When B≠ B^T or B=0If we want to perfectly reconstruct g[m,n] from G[p,q], the inverse transforms of the 2D HA-NsDLCT in (<ref>) and 2D HA-NsDLCT in (<ref>) should be of the following form:O_^- HO_^- B'^ - 1( A- I)O_^- B'O_^-( D'- I) B'^ - 1.Accordingly, another type of decomposition, called CC-CM-CC-CM decomposition, is introduced. First, decompose the ABCD matrix, say ( A_1,B_1; C_1, D_1),into[ A_1 B_1; C_1 D_1 ]=[ I H_1; 0 I ][ A'_1 B'_1;C_1D_1 ]=[ A'_1+ H_1 C_1 B'_1+ H_1 D_1; C_1 D_1 ],where H_1= H_1^T, B'_1= B_1- H_1 D_1 and A'_1= A_1- H_1 C_1. If B'_1 is symmetric and invertible, according to(<ref>), the ABCD matrix can be further decomposed as follows:[ A_1 B_1; C_1 D_1 ] =[ I H_1; 0 I ]_[ I 0; ( D_1- I) B_1'^ - 1 I ]_[I B_1';0I ]_×[ I 0; B_1'^ - 1( A'_1- I) I ]_. Based on the CC-CM-CC-CM decomposition above, the 2D NsDLCT can be designed asO_^( A_1,B_1; C_1, D_1)= O_^ H_1O_^( D_1- I) B_1'^ - 1O_^ B'_1O_^ B_1'^ - 1( A_1'- I).Again, there are infinite number of possible decompositions. So next we will prove that when ( A_1,B_1; C_1, D_1)=( A,B; C, D)^-1, (<ref>) will become (<ref>) if H_1=- H.Since theABCD matrix is given by[ A_1 B_1; C_1 D_1 ]=[ A B; C D ]^-1=[D^T -B^T; -C^TA^T ],one hasB'_1 = B_1- H_1 D_1=- B^T- H_1 A^T,A'_1 = A_1- H_1 C_1= D^T+ H_1 C^T. Recall H= H^T, B'= B- A H and D'= D- C H used in the CM-CC-CM-CC decomposition in (<ref>). If H_1=- H=- H^T, (<ref>) becomesB'_1 =- B^T+ H^T A^T=- B'^T,A'_1 = D^T- H^T C^T= D'^T.The chirp matrices used in the four 2D CMs and CCs in (<ref>) must be symmatric, and thus (<ref>) can be written asO_^( A_1,B_1; C_1, D_1)= O_^ H_1^TO_^( B_1'^T)^ - 1( D_1^T- I)O_^ B_1'^TO_^( A_1'^T- I)( B_1'^T)^ - 1.From (<ref>), (<ref>) and H_1=- H=- H^T, the above eqution can be rewritten as O_^( A,B; C, D)^-1= O_^- HO_^- B'^-1(A- I)O_^- B'O_^-( D'- I) B'^-1,which is the same as (<ref>) and thus can be used as the inverse transform of the 2D NsDLCT based on CM-CC-CM-CC decomposition.Similarly, one can develop 2D HA-NsDLCT and 2D LC-NsDLCT based on the CC-CM-CC-CM decomposition. With H_1determined by approaches similar to (<ref>), (<ref>) and (<ref>), one has O_^( A_1,B_1; C_1, D_1) = F_x,y^-1O_^- H_1F_x,yO_^(D_1- I)B_1'^ - 1 F_x,y^-1O_^- B'_1F_x,yO_^B_1'^ - 1(A_1'- I),O_^( A_1,B_1; C_1, D_1) = F_x^-1O_^- H_1F_xO_^(D_1- I) B_1'^ - 1 F_x,y^-1O_^- B'_1F_x,yO_^B_1'^ - 1(A_1'- I).Note that F_x and F_x^-1 in the 2D LC-NsDLCT arereplaced by F_y and F_y^-1, respectively, if H_1 is of the form (0,0;0,h_1). §.§ 2D HA-NsDLCT and 2D LC-NsDLCT With Perfect Reversibility PropertyLike the continuous 2D NsLCT, the reversibility property for 2D NsDLCT is defined asO_^( A,B; C, D)^-1 O_^( A,B; C, D){g[m,n]}=g[m,n].(<ref>) and (<ref>) show the 2D NsDLCT based on "CC-CM-CC-CM decomposition" can be used as the inverse transform of the 2D NsDLCT based on "CM-CC-CM-CC decomposition", and in fact vise versa. In order to let O_^( A,B; C, D)^-1 and O_^( A,B; C, D) use different types of decompositions, we make the following assumption:O_^( A,B; C, D)={[ O_^(D'- I)B'^ - 1O_^ B'O_^B'^ - 1(A- I)O_^ H,( B)>0; O_^ HO_^( D- I) B'^ - 1O_^ B'O_^ B'^ - 1( A'- I),( B)<0 ].,where (·) denotes matrix trace. For example,given some ( A,B; C, D) where ( B)<0, the forward transform O_^( A,B; C, D) will use the second type of (<ref>).Since ( A,B; C, D)^-1=( D^T, - B^T;- C^T, A^T) leads to (- B^T)>0, the inverse transformO_^( A,B; C, D)^-1 will use the first type. Then, perfect reconstruction is achieved.Replacing (<ref>) by (<ref>) and (<ref>) leads to the reversible 2D HA-NsDLCT. Replacing (<ref>) by (<ref>) and (<ref>) leads to the reversible 2D LC-NsDLCT.§ COMPARISONS BETWEEN PROPOSED 2D NSDLCTS AND PREVIOUS WORKSIn this section, the proposed 2D HA-NsDLCT and 2D LC-NsDLCT will be compared with Koç's method <cit.> andDing's 2D method <cit.> in computational complexity, accuracy, additivity property and reversibility property. The computational complexity has been analyzed in (<ref>), (<ref>), (<ref>) and (<ref>), and is summarized in TABLE <ref>.§.§ AccuracyAssume G(u,v) is the 2D NsLCT ofinput signal g(x,y).Given g[m,n] Δ = g(mΔ_x,nΔ_y) as the discrete input, a highly accurate 2D NsDLCT should have discrete output approximating the sampled 2D NsLCT, i.e. G(pΔ_u,qΔ_v), with very small error. Accordingly, the accuracy is measured by the normalized mean-square error (NMSE)defined as_ = ∑_p^∑_q^|G[p,q]-G(pΔ_u,qΔ_v) |^2 /∑_p^∑_q^|G(pΔ_u,qΔ_v) |^2 ,where G[p,q] is the output ofKoç's, Ding'sor the proposed 2D NsDLCT:G[p,q]= O_^( A,B; C, D){g[m,n]}.In the following simulations, 2D Hermite Gaussians (HGs) are used as the input. The 1D HG of order k is defined asHG_k(x) = ( 1/2^k k!√(π))^1/2e^ - x^2/2H_k(x),where H_k(x) is the kth-order physicists' Hermite polynomial. The 2D HG of order (k,l) is a separable function defined asHG_k,l(x,y) = HG_k(x) HG_l(y).Two signals composed of 2D HGs are used:g_1(x,y) =HG_1,2(x,y)+HG_3,1(x,y),g_2(x,y) =HG_2,18(x,y)+HG_14,11(x,y).Fig. <ref>(a) shows the 100×100 sampled g_1(x,y), i.e. g_1[m,n], with sampling intervals Δ_x=Δ_y=0.25, while Fig. <ref>(b) shows 165×165 g_2[m,n] with Δ_x=Δ_y=0.2. We can find out that g_1[m,n] has energy more concentrated than g_2[m,n].In order to analyze the accuracy by (<ref>), we need to derive the 2D NsLCTs of g_1(x,y) and g_2(x,y), denoted by G_1(u,v) and G_2(u,v), respectively. However, there are no closed-form expressions for G_1(u,v) and G_2(u,v). Therefore, we use the direct method in (<ref>) and discrete input with very large size and very small sampling intervals (i.e. 1024×1024g_1[m,n] and g_2[m,n] with Δ_x=Δ_y=0.078) to approximate G_1(pΔ_u,qΔ_v) and G_2(pΔ_u,qΔ_v).Fig. <ref>(c) depicts the 100×100 approximate G_1(pΔ_u,qΔ_v) with Δ_u=Δ_v=0.25 and ABCD matrix given by[ A_1 B_1; C_1 D_1 ]= [ 01.1217 -0.7754 -0.3765; -1.0934 -1.88261.10051.3878;0.1697 -1.4013 -0.53521.2447; -0.2014 -0.5209 -0.59160.3141 ].Fig. <ref>(d) shows the 165×165 approximate G_2(pΔ_u,qΔ_v) with Δ_u=Δ_v=0.2 and ABCD matrix given by[ A_2 B_2; C_2 D_2 ]= [0.3042 -0.23061.7626 -0.5090; -0.2641 -0.7314 -1.2221 -1.2080; -0.47650.4020 -0.1935 -0.0623;0.33220.96710.70810.5295 ]. When the input is g_1[m,n], the NMSEs of Koç's method, Ding's method, the 2D HA-NsDLCT and 2D LC-NsDLCT are 8.7×10^-4, 1.0×10^-4, 1.7×10^-6 and1.7×10^-6, respectively. The proposedmethods have higher accuracy. However, when the input is g_2[m,n], the 2D HA-NsDLCT (NMSE 1.1×10^-3) and 2D LC-NsDLCT (10^-2) are better thanKoç's method (1.6×10^-2) but somewhat worse than Ding's method (10^-3). This is becausethe output occupies larger space, as shown in Fig. <ref>(d). There would be some aliasing/overlapping effect around the boundary. To solve this problem, the discrete input is first zero-padded to larger size, say (N+Δ N)×(N+Δ N).Fig. <ref> and Fig. <ref> show the accuracy versus 0≤Δ N≤50 when the input is g_1[m,n] and g_2[m,n], respectively.We can find out that the proposedmethods significantly outperform Koç's and Ding's methods when Δ N is large enough. Besides, these simulations verify the feature "high accuracy" of the 2D HA-NsDLCT. The 2D HA-NsDLCT has higher accuracy than the 2D LC-NsDLCT. §.§ Additivity PropertyWith the additivity property, someapplications of 2D NsDLCT would have lower computational complexity. For example, consider that input g[m,n] getting through twofilters operating in two different 2D NsLCT domains, sayg'[m,n]= O_^( A_1,B_1; C_1, D_1)^-1{ O_^( A_3,B_3; C_3, D_3)^-1{H_3[p_3,q_3] /.. ..· O_^( A_3,B_3; C_3, D_2){H_1[p_1,q_1] O_^( A_1,B_1; C_1, D_1){g[m,n]}}}}, where H_1[p,q] and H_3[p,q] are filters. If the additivity property is satisfied, only three 2D NsDLCTs are required:g'[m,n]= O_^( A_1,B_1; C_1, D_1)^-1( A_3,B_3; C_3, D_3)^-1{H_3[p_3,q_3] /..· O_^( A_3,B_3; C_3, D_2){H_1[p_1,q_1] O_^( A_1,B_1; C_1, D_1){g[m,n]}}}, The existing 2D NsDLCTs don't satisfy the additivity property, and neither do the proposed methods. However, we will show that the proposed 2D NsDLCTs have "approximate" additivity.If a 2D NsDLCT has perfect additivity, it should satisfyO_^( A_3,B_3; C_3, D_3) O_^( A_1,B_1; C_1, D_1) =O_^( A_3,B_3; C_3, D_3)( A_1,B_1; C_1, D_1).Therefore, a 2D NsDLCT is referred to as being approximately additive if the difference between the left side and right side of (<ref>) is small enough. The error of additivity is measured by the NMSE defined below: _ = ∑_p^∑_q^|G'[p,q]-G[p,q] |^2 /∑_p^∑_q^|G[p,q] |^2 ,where G[p,q] and G'[p,q] are given byG[p,q] = O_^( A_3,B_3; C_3, D_3){ O_^( A_1,B_1; C_1, D_1){g[m,n]}},G'[p,q] = O_^( A_3,B_3; C_3, D_3)( A_1,B_1; C_1, D_1){g[m,n]},respectively.In the following,two simulations are presented. The first one is using (100+Δ N)×(100+Δ N) zero-padded g_1[m,n] as the input with ( A_1,B_1; C_1, D_1) given in (<ref>) and ( A_3,B_3; C_3, D_3) given by[ A_3 B_3; C_3 D_3 ]= [ -0.4742 -0.87002.4284 -2.6166;4.12051.80382.6786 -7.5360; -4.3025 -0.6572 -7.2020 12.8085;3.86712.5257 -0.6080 -2.8661 ].Fig. <ref> shows the outputs of (<ref>) and (<ref>) when the original g_1[m,n] (without zero-padding shown in Fig. <ref>(a)) and 2D HA-NsDLCT are used. Comparing the additivity by (<ref>), the 2DHA-NsDLCT and 2D LC-NsDLCT have NMSE 3.6×10^-5 lower than the Koç's method (7.2×10^-3) and Ding's method (3.9×10^-3). If g_1[m,n] is zero-padded to larger size, lower and more satisfactory NMSE can be achieved, as shown in Fig. <ref>. This example shows that the proposed methods have approximate additivity. In the second example, (165+Δ N)×(165+Δ N) zero-padded g_2[m,n] gets through 2D NsDLCT with ( A_2,B_2; C_2, D_2) given in (<ref>) followed by 2D NsDLCT with ( A_4,B_4; C_4, D_4) shown below:[ A_4 B_4; C_4 D_4 ]= [0.75970.24181.40551.5125;0.93050.18062.3170 -0.7412; -0.0147 -0.50680.5030 -0.7006;0.49430.50591.87260.1543 ].Fig. <ref> depicts the outputs of (<ref>) and (<ref>) when the original g_2[m,n] (without zero-padding shown in Fig. <ref>(b)) and 2D LC-NsDLCT are used. These's some difference between these two outputs, especially around the boundary. Analyzing the additivity by (<ref>), the NMSEs of the 2D HA-NsDLCT and 2D LC-NsDLCT are 0.052 and 0.059, respectively, somewhat higher than the Koç's method (0.037) and Ding's method (0.012). However, Fig. <ref> shows that the 2D HA-NsDLCT and 2D LC-NsDLCT have lower NMSEs than Koç's and Ding's methods and have approximate additivity when g_2[m,n] is zero-padded to larger size. The 2D HA-NsDLCT has approximate additivity with lower error than the 2D LC-NsDLCT because of its higher accuracy feature. §.§ Reversibility PropertyThe reversibility property is a special case of the additivity property. A 2D NsDLCT with reversibility property should satisfyO_^( A,B; C, D)^-1{ O_^( A,B; C, D){g[m,n]}} = g[m,n].Even without the additivity property, the reversibility property may be satisfied. The reversibility is analyzed by the NMSE given by_= ∑_m^∑_n^| O_^( A,B; C, D)^-1{ O_^( A,B; C, D){g[m,n]}}- g[m,n] |^2 /∑_m^∑_n^| g[m,n] |^2 .When the input is zero-padded g_1[m,n] and ( A,B; C, D)=( A_1,B_1; C_1, D_1), the NMSEs of the reversibilityare depicted in Fig. <ref>, while the NMSEs for zero-padded g_2[m,n] and ( A_2,B_2; C_2, D_2) are shown in Fig. <ref>. These examples verify that the proposed 2D NsDLCTs have perfect reversibility evenwithout performing zero-padding (i.e. Δ N=0). At last, we use 128×128 Lena image as an example to present image reconstruction by each method. The ABCD matrix ( A_2,B_2; C_2, D_2) in (<ref>) and Δ_x=Δ_y=Δ_u=Δ_v=0.22 is adopted. The PSNRs of thereconstructed images are shown in Fig. <ref>. Here, the zero-padding process is not employed. The proposedmethods have perfect reconstruction with PSNR about 279 dB, while Koç's method andDing'smethod have higher reconstruction errors, 11.1 dB and 19.6 dB, respectively.Ding'smethod would have perfect reversibility by using the sampling theorem and unitary discretization of 2D NsLCT in <cit.>. However, there will be some restrictions on the locations of the output sampling points or the value of B in the ABCD matrix. One can refer to <cit.> for more details and derivations of the restrictions. Therefore, Ding'smethod remains irreversible in most cases. § OPTICAL APPLICATIONSThe LCT can be used to model any lossless first-order optical system, also known as ABCD system <cit.>, such as light propagation through free space, gradient index (GRIN) medium, elliptic GRIN medium, thin lens, spherical lens, cylindrical lens, tilted cylinder lens, prisms, Fourier transformer or a combination of several different spaces, media and lenses. Thus, one can use the 2D NsDLCTs to simulate and analyze many 2D optical systems. §.§ Light Propagation Through Fourier Transformer and Elliptic GRIN Medium If the refractive index distribution n(x,y) of an elliptic GRIN medium satisfiesn^2(x,y) = n_0^2[ 1 - n_1/n_0(x + py)^2 - n_2/n_0(qx + y)^2],where n_0,n_1,n_2 are the GRIN medium parameters, the result of the light propagation in the medium can be expressed as <cit.>G_1(u,v)= e^ - j2πn_0L/λ O_^( A_1,B_1; C_1,D_1){g(x,y)},where L and λ are the propagation length and the wavelength, respective. The ABCD matrix is given by[ A_1 B_1; C_1 D_1 ] = [ (S^T)^-10;0S^T ][ E F; F^ - 1(E^2 - I) F ][S0;0 S^-1 ],whereS=[ 1 p; q 1 ], E=[ cosα0;0 cosβ ], F=[ √(n_0/n_1)sinα/k0;0 √(n_0/n_2)sinβ/k ],and k=2π/λ, α=2L/π√(n_1/n_0) and β=2L/π√(n_2/n_0).In this section, we consider an optical system that consists of a Fourier transformer and an elliptic GRIN medium; that is, the corresponding ABCD matrix is given by[ A B; C D ]= [ A_1 B_1; C_1 D_1 ][0I; -I0 ],where ( A_1,B_1; C_1,D_1) is defined in (<ref>).A more accurate 2D NsDLCT is always a better choice to simulate the optical system. AssumeL=10^4 , λ=532 , p=0.6, q=0.2,n_0=1.5, n_1=5×10^-8 ^-2, n_2=2×10^-8 ^-2,and the input is a S-shaped function used in <cit.>. The 257×257 sampled S-shaped function is depicted in Fig. <ref>(a), while the proposed 2D HA-NsDLCT without zero-padding the input is depicted in Fig. <ref>(b). The NMSE 0.038 is lower thanKoç's method 0.24 and Ding's method 0.073.§.§ Self-Imaging Phenomena in OpticsFor an optical system and its corresponding LCT, one can use the eigenfunction of the LCT to analyze the self-imaging phenomena of the optical system. For 1D LCT, the eigenfunctions for all cases have been discussed in <cit.>, and further simplifiedinto more compact closed forms without integral in <cit.>. For the 2D NsLCT, the eigenfunctions for all cases of ABCD matriceshave been proposed in <cit.>. However, so far there is no general form for all cases of eigenfunctions, and the eigenfunctions in many cases are still expressed in integral form. For example, consider an optical system corresponding to the followingABCD matrix[ A B; C D ]= [ -0.1516 -0.0982 -1.5946 -0.1626; -0.09730.46410.0577 -0.9005;0.63870.09850.2636 -0.0564; -0.10390.9866 -0.15990.1940 ].In this case, the Method A in <cit.> is used and results in the eigenfunctions of the following form:Ψ (x,y)= e^ - j0.392x^2∫_ - ∞^∞e^j(x - τ )^2/1.3499ψ _1( - 0.9368τ- 0.3601y)·ψ _2(0.4388τ- 0.9027y)dτ ,where these parameters are obtained from a complicated procedure. One can refer to <cit.> for more details. The functions ψ_1 and ψ_2 in (<ref>) are the eigenfunctions of two different 1D LCTs with parameters(0.1709,-0.8242;1.2515 , -0.1843) and (0.5189,-0.8528;1.0116, 0.2646), respectively, in this case.A different method instead of Method A would be used for a different ABCD matrix. And in the discrete case, the samples of the eigenfunctions (<ref>) won't be orthogonal anymore.A general method that is suitable for all ABCD matrices and can generate a set of discrete orthogonal eigenfunctions is to calculate the eigenvectors of the 2D NsDLCT. Given some N× N discrete input, reshape the input into an N^2×1 column vector, say g. From (<ref>), the proposed 2D HA-NsDLCT can be expressed as the following matrix formG = L·g = O_CM^(D' - I)B'^ - 1F^†O_CM^B'FO_CM^B^'^ - 1(A - I)F^†O_CM^HF g.F is an N^2× N^2 matrix obtained from the Kronecker product of two N× N DFT matrices. F and F^† correspond to the 2D DFT and 2D IDFT, respectively. The four O_CM are N^2× N^2 diagonal matrices performing 2D CMs. Similarity, one can derive the matrix form for the 2D LC-NsDLCT. Because the 2D HA-NsDLCT has perfect reversibility, the N^2× N^2 matrix L in (<ref>) is unitary. And one can obtain a set of orthogonal eigenvectors from L, which can approximate the continuous eigenfunctions of the 2D NsLCT. For example, when N=51, three eigenvectors of L with ABCD matrix given in (<ref>) are depicted in Fig. <ref>. The errors between the eigenvectors and the samples of continuous eigenfunctions are all below 10^-4. § CONCLUSIONS AND FUTURE WORKTo reduce computational complexity, the 2D NsLCT is decomposed into several simpler 2D operations. Then, one can develop a low-complexity 2D NsDLCT by connecting the discrete versions of these 2D operations. In this paper, a new decomposition called CM-CC-CM-CC decomposition is proposed. Based on this decomposition, two types of 2D NsDLCTs are proposed, consisting of tow 2D discrete CMs and two 2D discrete CCs. More precisely, only 2D discrete CMs, 2D DFTs and 2D IDFTs are used. The proposed 2D NsDLCTs outperform the previous works in computational complexity, accuracy and additivity property. 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Huang, Fast discrete linear canonical transform based on cm-cc-cm decomposition and fft, IEEE Transactions on Signal Processing, accepted for publication(2015). | http://arxiv.org/abs/1707.03688v1 | {
"authors": [
"Soo-Chang Pei",
"Shih-Gu Huang"
],
"categories": [
"cs.CV",
"physics.optics"
],
"primary_category": "cs.CV",
"published": "20170526164554",
"title": "Two-dimensional nonseparable discrete linear canonical transform based on CM-CC-CM-CC decomposition"
} |
accepted for publication in Phys. Rev. [email protected] Department of Chemistry, Yale University, New Haven, CT 06520, USADepartment of Chemical Physics, University of Science & Technology of China, Hefei, Anhui 230026, ChinaDepartment of Chemistry and Chemical Biology, University of New Mexico, Albuquerque, NM 87131, [email protected] Department of Chemistry, Yale University, New Haven, CT 06520, USAElectronic friction and the ensuing nonadiabatic energy loss play an important role in chemical reaction dynamics at metal surfaces. Using molecular dynamics with electronic friction evaluated on-the-fly from Density Functional Theory, we find strong mode dependence and a dominance of nonadiabatic energy loss along the bond stretch coordinate for scattering and dissociative chemisorption of H_2 on the Ag(111) surface. Exemplary trajectories with varying initial conditions indicate that this mode-specificity translates into modulated energy loss during a dissociative chemisorption event. Despite minornonadiabatic energy loss of about 5%, the directionality of friction forces induces dynamical steering that affects individual reaction outcomes, specifically for low-incidence energies and vibrationally excited molecules. Mode-specific friction induces enhanced loss of rovibrational rather than translational energy and will be most visible in its effect on final energy distributions in molecular scattering experiments. Mode specific electronic friction in dissociative chemisorption on metal surfaces: H_2 on Ag(111) John C. Tully December 30, 2023 =================================================================================================The reaction dynamics of elementary processes in heterogeneous catalysis depend sensitively on the energy exchange between adsorbate and substrate degrees of freedom, including substrate phonons and electrons. Specifically in the case of chemical reactions on metal surfaces, nonadiabatic adsorbate-substrate energy exchange via electron-hole pair excitations (EHPs) has been a focus of interest in literature <cit.>. Recent experiments of atomic hydrogen scattering on Au(111) gave evidence of inelastic scattering due to EHP-induced energy loss. <cit.> The ensuing nonadiabatic force acting on the adsorbate and mediating this energy loss is often referred to as electronic friction. Electric currents, induced by electronic friction have been measured for a variety of chemical reactions on surfaces <cit.> and a selective reaction control theory coined "hot-electron chemistry" has emerged from the idea of selectively funneling energy into reaction channels via EHPs <cit.>.Despite the experimental evidence, purely adiabatic theories have reproduced experimental data with reasonable accuracy for molecular hydrogen scattering and dissociation on substrates such as Cu(111) <cit.> and Ru(0001) <cit.>. From this indirect evidence, it has often been argued that EHPs can be safely neglected in studying scattering probabilities and molecular motion leading up to dissociation <cit.>. On the contrary, a recent simulation study has identified EHPs as the dominant energy loss channel after dissociation. Here, the energy loss through EHPs was found to exceed phonon energy loss by a factor of five <cit.>.Much less is known about the role of nonadiabatic effects near the transition state of dissociative chemisorption (DC). Several first-principles simulation studies <cit.> have argued that EHP effects are limited in DC, because the interaction time between the molecule and surface is short. On the other hand, Luntz et al. <cit.> have argued that established models to account for electronic friction based on the local electron density, termed local density friction approximation (LDFA), <cit.> neglect the molecular electronic structure of the adsorbate and misrepresent friction during chemical transformations. The majority of recent studies have employed friction models based on LDFA, assuming that frictional effects on H_2 are isotropic. <cit.> Electronic friction models based on first-order time-dependent perturbation theory (TDPT), <cit.> that account for the adsorbate molecular structure, have revealed significant levels of anisotropy and mode dependence in vibrational energy loss of diatomics on metal surfaces <cit.>. This mode dependence is visible in experiment. <cit.>, with recent molecular scattering experiments adding new evidence for its experimental relevance. <cit.>Unfortunately, the computational cost of TDPT models hitherto did not allow their routineapplication in full-dimensional molecular dynamics simulations (MD), leaving questions relating to EHP-induced effects on dynamical steering or mode-specific energy redistribution unanswered. In this Letter, we combine MD with an efficient implementation of tensorial electronic friction based on TDPT to describe electronic friction effects during chemisorption of molecular hydrogen on a close-packed silver surface. This allows us, for the first time, to overcome restrictions of previous methods by accounting for adsorbate molecular structure, anisotropy, and mode coupling within full-dimensional MD. We select H_2 on Ag(111) as a prototypical example of DC, because EHPs are believed to play a larger role on coinage metal surfaces, where the dissociation barrier lies closer to the surface. <cit.> Indeed, we find electronic friction to act dominantly along the intramolecular stretch motion as the molecule dissociates. As a consequence, electronic friction induces modulated energy loss during scattering events. Whereas reaction outcomes of high-energy molecules are not sensitive to EHP effects, conditions of low incident translational energy or initial vibrational excitation are affected in their reaction outcomes and final energy distributions. The strong frictional mode specificity along the intramolecular stretch is visible in a more effective loss of rovibrational energy than translational energy, serving as a potential experimental signature of mode-specific nonadiabatic effects. <cit.>Electronic friction originates from the coupling of electronic excitations with adsorbate nuclear motion. This effect can be captured within mixed quantum-classical theories. <cit.> The simplest version uses a Langevin expression: <cit.> MR̈_̈ï= - ∂V(R)/∂R_i- ∑_jΛ_ijṘ_̇j̇^F_damp,i + ℛ_i(t).Atomic positions R_i evolve due to forces from an ab-initio potential energy surface (PES), V(𝐑), and frictional forces due to EHPs are captured by an electronic friction tensor Λ and thermal white noise ℛ_i(t). The latter ensures detailed balance <cit.>. In eq. <ref>, Λ is a (3N×3N) matrix for N atoms (in our case 2 hydrogen atoms), wherein each element corresponds to the nonadiabatic relaxation rate due to adsorbate motion along that coordinate or, in the case of off-diagonal elements, the coupling between two coordinates. We calculate nonadiabatic relaxation rates using TDPT within the constant-coupling approximation required by eq. <ref>, where we average over EHP excitations within the relevant energy regime using single particle states from Density Functional Theory (DFT) and a Gaussian envelope of width 0.6 eV <cit.>. This method is implemented in the all-electron code FHI-aims, which employs numerical atomic-orbitals. <cit.> We use the exchange-correlation functional of Perdew, Burke, and Ernzerhof <cit.>. The model consists of an H_2 molecule in a frozen p(2x2) Ag(111) surface unit cell. Further computational details can be found in the Supplemental Material (SM) <cit.>. Fig. <ref> shows the EHP-induced relaxation rates along a minimum energy path of DC, resulting in H adsorption at adjacent fcc and hcp sites. The main features of tensorial electronic friction along these paths are: (1) Electronic friction is strongly mode dependent. As a result, friction along the bond stretch d and the azimuthal and polar angles (θ and ϕ) is considerably larger than along the center of mass translations of the molecule (X, Y, Z). (2) The relaxation rate along the bond stretch d peaks at the transition state and is three times larger than the next largest component. This is in agreement with findings of Luntz et al. for H_2 on Cu(111). <cit.> Both of these findings stand in contrast to the description of electronic friction by LDFA, where the relaxation rate is an isotropic function of the embedding substrate density at the position of the adsorbate atoms. LDFA relaxation rates monotonically increase as the H atoms approach the substrate and they are identical in all directions (see SM <cit.>). Stronger friction at the transition state is well known <cit.> and readily understandable as electronic character changes dramatically during bond breaking. Nevertheless, the computational costs of TDPT-based friction models in full-dimensional MD have prevented further investigations of frictional effects during DC in the past. Our recent TDPT implementation allows us to overcome this computational constraint. <cit.> By explicitly simulating the reaction dynamics of H_2 on Ag(111) using MD, MD with electronic friction (MDEF) based on LDFA, <cit.> and MDEF based on TDPT friction, we study the frictional energy loss of H_2 DC on Ag(111). <cit.> We have selected a total of 14 normal-incidence initial conditions varying in molecular orientation and impingement site with three different energies (see SM, Fig. S1). <cit.> This includes 8 trajectories with a translational energy of 1.8 eV that exceeds the barrier for reaction (1.12 eV), 2 trajectories with a translational energy of 1.0 eV, which barely suffices to overcome the barrier, when accounting for vibrational zero point energy, and 4 vibrationally excited (v=1) trajectories with 0.6 eV translational energy. We integrate eq. <ref> with a precomputed PES <cit.> and evaluate TDPT friction on-the-fly, including off-diagonal elements (see SM <cit.>). We assume a surface temperature of 0 K, effectively neglecting the third term in eq. <ref>. Inclusion of the random force term in eq. 1 to model non-zero temperatures would bestraightforward in our approach. However, in the current case, it would make it more difficult to draw conclusions from a small set of 14 trajectories.Fig. <ref> visualizes the MDEF results as given by two different friction models, the isotropic LDFA <cit.> and the tensorial TDPT model. <cit.> The figures represent two trajectories with 1.8 eV translational energy, one of which scatters from the surface, the other one dissociates.The frictional forces F_damp visualized in panel A correspond to the second term on the r.h.s. of eq. <ref>. Thenonadiabatic relaxation rates due to Λ(t) are given in Panel B of both figures. The corresponding energy loss along a given trajectory is presented in panel C and depends on, both, the velocity profile and the friction tensor at time t. In the case of non-dissociative molecular scattering, electronic friction is only non-zero during a short period of time (ca. 50 fs) in which the molecule-surface distance is small. In the LDFA method, friction is mostly dictated by the atom-surface distance and is isotropic in all directions.In the TDPT method, friction acts most strongly along the bond stretch coordinate with significantly more variation along the trajectory. However, the actual energy loss shows minor differences betwen the two methods and mostly differs in magnitude. We find an average integrated energy loss of 47 meV and 114 meV for the four scattering trajectories with 1.8 eV when calculated with TDPT and LDFA, respectively. This corresponds to an energy loss of only 2% and 6% of the total incidence energy. The reaction outcomes of these high-energy trajectories were not affected by electronic friction.For successful chemisorption (bottom part of Fig. <ref>), friction and energy loss show larger discrepancies between the two models. Upon adsorption, LDFA friction remains constant throughout the dissociation event, whereas TDPT-based friction exhibits more striking variation. The resulting energy-loss profiles are characterized by three spikes in energy loss along the dissociation path, which are weighted differently by the two friction methods. The first one (t=48 fs), corresponds to the initial adsorption event, the second one (t=54 fs) to the dissociation event, and the third (t=72 fs) corresponds to the onset of lateral diffusion. LDFA yields more energy loss for the first and third event, TDPT yields more energy loss for the dissociation event.Throughout the studied trajectories, we found that LDFA yields more energy loss perpendicular to the surface upon adsorption than TDPT. This can be seen from the corresponding force vectors in Fig. <ref>. Overall, we can distinguish between two regions of the energy loss profiles: a highly structured region (t<80 fs) leading up to and including dissociation and a region (t>80 fs) with less variation in energy loss, corresponding to lateral atom diffusion.This suggests that isotropic friction models are appropriate for the description of single atom motion on metal surfaces, <cit.> while they may have problems in representing molecular motion. <cit.>Individual reaction events with high incidence energy reveal significant structure in the energy loss profile of a chemisorption event. Nevertheless, we have not found significant changes in reaction probabilities or final energy distributions upon incorporation of friction. However, in order to unambiguously assess frictional effects on reaction probabilities and final energy distributions, extensive statistical sampling with thousands of independent trajectories would be necessary. This is, however, at the moment computationally challenging.Only when considering initial conditions closer to the dissociation barrier, we find changes in reaction outcomes and energy distributions (see Table <ref>). In many of the studied cases, the inclusion of electronic friction changes the reaction outcome from successful dissociation to scattering or vice versa. The overall energy loss, in all cases is still around 5% of the incidence energy and the total energy always remains sufficient to overcome the barrier in principal. Nevertheless, energy loss and directional steering along specific modes appears to be sufficient to change individual reaction outcomes. Interestingly, we find that absolute energy loss described by the two friction models is closer in magnitude for these trajectories, especially for vibrationally excited cases [see SM, Table S IV]. <cit.> This originates from the dominance of friction along internal modes in TDPT, yielding higher energy loss along rotations and vibrations. In one vibrationally excited case, we find final reaction outcomes that differed between the two friction models [see SM, Figs. S19 and S20]. <cit.> Here, the hydrogen atoms dissociate onto adjacent hollow sites and immediately recombine. The translational energy loss after dissociation is significantly larger when employing LDFA friction than TDPT. As a result, hydrogen atoms cannot recombine and the dissociation remains successful. MD and MDEF(TDPT), on the other hand, both describe recombinative desorption, however with very different final rovibrational and translational energy distributions.This brings us to the discussion of final energy distributions upon scattering. The energy distribution across different modes of a scattered molecule is an important experimental observable and can potentially expose the effects of electronic friction. Table <ref> summarizes relative energy distributions of scattered trajectories before and after interaction with the surface. Coupling between vibration and rotations does not allow independent analysis and we collect them as internal energy (E_int). The first four rows correspond to translationally hot molecules with 1.8 eV, where energy distributions with or without electronic friction are very similar. Energy loss occurs dominantly from translational energy, with LDFA friction yielding higher energy loss than TDPT. Qualitative differences between the two methods become much more obvious in the case of low incidence energy (#10 in Table II) and vibrational excitation (#11, #14) where both friction models yield comparable energy loss. Fig. <ref> visualizes the energy loss of #10 as a function of time. Despite EHP-induced energy loss of 5% and 4% of the total energy, the two friction models differ significantly in the final amount of energy distributed over translation and rovibrational degrees of freedom, with more internal energy loss in the case of TDPT. The result is a difference in relative energy distributions between the two models of almost 10%. In the case of the previously mentioned trajectory # 11, where MD and MDEF(TDPT) both describe recombinative desorption, MD yields a final 2:1 energy distribution of translation and rovibration, MDEF yields desorption of an internally cold molecule. Nonadiabatic surface-adsorbate energy transfer is an essential aspect of gas-surface dynamics on metals and can be decisive in steering chemical reactions. Using an electronic friction model based on TDPT that accounts for the molecular structure of the adsorbate, we studied frictional effects in DC of H_2 on Ag(111). The overall EHP-induced energy loss of scattered trajectories only amounts to about 5% of the total energy in both employed friction models. Nevertheless, for low translational energies and vibrationally excited molecules, significant frictional effects on directional steering and final energy distributions can be found. Measured dissociation probabilities might not necessarily reflect EHP-induced nonadiabatic effects, however, we expect effects on final energy distributions to be measurable in molecular beam scattering. Experimental evidence for this has recently been presented. <cit.> Obvious next steps include a statistically representative determination of the latter, as well as the study of temperature effects on scattering and reaction and the ability to control outcomes through varying electronic temperature. RJM and JCT acknowledge financial support by the US Department of Energy - Basic Energy Science grant DE-FG02-05ER15677. H. G. thanks US National Science Foundation (CHE-1462019). B. J. thanks the National Natural Science Foundation of China (91645202 and 21573203). 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"authors": [
"Reinhard J. Maurer",
"Bin Jiang",
"Hua Guo",
"John C. Tully"
],
"categories": [
"cond-mat.mtrl-sci"
],
"primary_category": "cond-mat.mtrl-sci",
"published": "20170527015739",
"title": "Mode specific electronic friction in dissociative chemisorption on metal surfaces: H$_2$ on Ag(111)"
} |
Department of Chemistry and Biochemistry, University of California San Diego, La Jolla, California 92093, United StatesDepartment of Chemistry and Biochemistry, University of California San Diego, La Jolla, California 92093, United StatesDepartment of Chemistry and Biochemistry, University of California San Diego, La Jolla, California 92093, United StatesDepartment of Chemistry and Biochemistry, University of California San Diego, La Jolla, California 92093, United StatesRecent advancements on the fabrication of organic micro- and nanostructures have permitted the strong collective light-matter coupling regime to be reached with molecular materials. Pioneering works in this direction have shown the effects of this regime in the excited state reactivity of molecular systems and at the same time have opened up the question of whether it is possible to introduce any modifications in the electronic ground energy landscape which could affect chemical thermodynamics and/or kinetics. In this work, we use a model system of many molecules coupled to a surface-plasmon field to gain insight on the key parameters which govern the modifications of the ground-state Potential Energy Surface (PES). Our findings confirm that the energetic changes per molecule are determined by effects which are essentially on the order of single-molecule light-matter couplings, in contrast with those of the electronically excited states, for which energetic corrections are of a collective nature. Still, we reveal some intriguing quantum-coherent effects associated with pathways of concerted reactions, where two or more molecules undergo reactions simultaneously, and which can be of relevance in low-barrier reactions. Finally, we also explore modifications to nonadiabatic dynamics and conclude that, for our particular model, the presence of a large number of dark states yields negligible effects. Our study reveals new possibilities as well as limitations for the emerging field of polariton chemistry. Ultrastrong coupling, ground state, chemical reactivityCan ultrastrong coupling change ground-state chemical reactions? Joel Yuen-Zhou December 30, 2023 ================================================================§ INTRODUCTION The advent of nano- and microstructures which enable strong confinement of electromagnetic fields in volumes as small as 1×10^-7λ^3<cit.>, λ being a characteristic optical wavelength, allows for the possibility of tuning light-matter interactions that can “dress" molecular degrees of freedom and give rise to novel molecular functionalities. Several recent studies have considered the effects of strong coupling (SC) between confined light and molecular states, and its applications in exciton harvesting and transport<cit.>, charge transfer<cit.>, Bose-Einstein condensation <cit.>, Raman <cit.> and photoluminiscence <cit.> spectroscopy, and quantum computing <cit.>, among many others <cit.>. Organic dye molecules are good candidates to explore SC effects due to their unusually large transition dipole moment <cit.>. More recently, it has been experimentally and theoretically shown that the rates of photochemical processes for molecules placed inside nanostructures can be substantially modified <cit.>. The underlying reason for these effects is that the SC energy scale is comparable to that of vibrational and electronic degrees of freedom, as well as the coupling between them <cit.>; this energetic interplay nontrivially alters the resulting energetic spectrum and dynamics of the molecule-cavity system. It is important to emphasize that in these examples, SC is the result of a collective coupling between a single photonic mode and N ≫ 1 molecules; single-molecule SC coupling is an important frontier of current research <cit.>, but our emphasis in this work will be on the N molecule case. Since the energy scale of this collective coupling is larger than the molecular and photonic linewidths, the resulting eigenstates of the system have a mixed photon-matter character. Understanding these so-called polariton states is relevant to develop a physical picture for the emerging energy landscapes which govern the aforementioned chemical reactivities. More specifically, Galego and coworkers <cit.> have recently provided a comprehensive theoretical framework to explain the role of vibronic coupling and the validity of the Born-Oppenheimer (BO) approximation in the SC regime, as well as a possible mechanism for changes in photochemical kinetics afforded by polaritonic systems <cit.>; another theoretical study that focused on control of electron transfer kinetics was given by Herrera and Spano <cit.>. Using a model of one or two molecules coupled to a single mode in a cavity, Galego and coworkers noticed that some effects on molecular systems are collective while others are not; similar findings were reported by Cwik and coworkers using a multimode model and N molecules <cit.>. While prospects of photochemical control seem promising, it is still a relatively unexplored question whether ground-state chemical reactivity can be altered via polaritonic methods, although recently, George and coworkers have shown a proof of concept of such feasibility using vibrational SC <cit.>. Along this line, ultrastrong coupling regime (USC) seems to also provide the conditions to tune the electronic ground-state energy landscape of molecules and in turn, modify not only photochemistry, but ground-state chemical reactivity. Roughly speaking, this regime is reached when Ω/ħω_0≥0.1, Ω being the (collective) SC of the emitter ensemble to the electromagnetic field and ħω_0 the energy gap of the molecular transition<cit.>. Under USC, the “nonrotating" terms of the light-matter Hamiltonian acquire relevance and give rise to striking phenomena such as the dynamical Casimir effect <cit.> and Hawking radiation in condensed matter systems <cit.>. Furthermore, recent experimental advances have rendered the USC regime feasible in circuit QED <cit.>, inorganic semiconductors <cit.>, and molecular systems <cit.>, thus prompting us to explore USC effects on ground-state chemical reactivity.In this article, we address how this reactivity can be influenced in the USC by studying a reactive model system consisting of an ensemble of thiacyanine molecules strongly coupled to the plasmonic field afforded by a metal, where each of the molecules can undergo cis-trans isomerization by torsional motion. The theoretical model for the photochemistry of the single thiacyanine molecule has been previously studied in the context of coherent control <cit.>. As we will show, the prospects of controlling ground-state chemical reactivity or nonadiabatic dynamics involving the ground state are not promising for this particular model, given that the alterations of the corresponding PES are negligible on a per-molecule basis. However, we notice the existence of salient quantum-coherent features associated with concerted reactions that might be worth considering in models featuring lower kinetic barriers. This article is organized as follows: in the Theoretical Model section, we describe the polariton system and its quantum mechanical Hamiltonian. In Methods, we describe the methodology to perform the relevant calculations and understand the effects of polariton states on the ground-state PES of the molecular ensemble. In Results and Discussion we describe our main findings, and finally, in the Conclusions section, we provide a summary and an outlook of the problem.§ THEORETICAL MODEL To begin with, we consider a thiacyanine derivative molecule (Fig. <ref>c) and approximate its electronic degrees of freedom as a quantum mechanical two-level system. To keep the model tractable, this electronic system is coupled to only one vibrational degree of freedom R, namely, the torsion along the bridge of the molecule (Fig. <ref>c) along which cis-trans isomerization occurs. The mathematical description of the PES of the ground and excited states (Fig. <ref>a) as well as the transition dipole moment as a function of the reaction coordinate (Fig. <ref>b) have been obtained from Ref. <cit.>. The adiabatic representation of the electronic states is given by,|g(R)⟩=cos(θ(R)/2)|trans⟩+sin(θ(R)/2)|cis⟩|e(R)⟩= -sin(θ(R)/2)|trans⟩+cos(θ(R)/2)|cis⟩where |e(R)⟩ and |g(R)⟩ are the R-dependent adiabatic excited and ground state respectively. |trans⟩ and |cis⟩ are the (R-independent) crude diabatic electronic states that describe the localized chemical character of each of the isomers. The ground-state PES has a predominant trans (cis) character to the left (right) of the barrier (θ(0)=0, θ(π)=π) in Fig. <ref>a. Our USC model consists of a setup where an orthorhombic ensemble of thyacyanine molecules is placed on top of a thin spacer which, in turn, is on top of a metallic surface that hosts surface plasmons (SPs) <cit.> (see Fig. <ref>). The coupling between molecular electronic transitions and plasmons in the metal give rise to polaritons that are often called plexcitons <cit.>. The ensemble is comprised of N_z single-molecule layers. The location of each molecule can be defined by the Cartesian coordinates 𝐧+(0,0,z_s) where 𝐧=(Δ_xn_x,Δ_yn_y,0) and z_s=z_0+Δ_zs for the s-th layer. Here, the spacing between molecules along the i-th direction is denoted by Δ_i, and z_0 is the width of the spacer (see Fig. <ref>). We chose a SP electromagnetic environment because its evanescent intensity decreases fast enough with momentum 𝐤 (giving rise to vanishing light-matter coupling for large |𝐤|), resulting on a convergent Lamb-shift of the molecular ground-state. As shall be explained below, this circumvents technical complications of introducing renormalization cutoffs, as would be needed for a dielectric microcavity <cit.>. The Hamiltonian of the plexciton setup is given by H=H_el+T_nuc, where T_nuc=∑_i𝐏_i^2/2M_i is the nuclear kinetic energy operator andH_el(𝐑)=∑_𝐤ħω_𝐤a_𝐤^†a_𝐤+∑_𝐧,s(ħω_e(R_𝐧,s)-ħω_g(R_𝐧,s))b_𝐧,s^†(R_𝐧,s)b_𝐧,s(R_𝐧,s) +∑_𝐤∑_𝐧,sg_𝐤^𝐧,s(R_𝐧,s)(a_𝐤^†b_𝐧,s(R_𝐧,s)+a_𝐤b_𝐧,s^†(R_𝐧,s)+a_𝐤b_𝐧,s(R_𝐧,s)+a_𝐤^†b_𝐧,s^†(R_𝐧,s)) +∑_𝐧,sħω_g(R_𝐧,s),corresponds to the Dicke Hamiltonian <cit.>. Here a_𝐤^† (a_𝐤) is the creation (annihilation) operator for the SP mode with in-plane momentum 𝐤 which satisfies [a_𝐤,a_𝐤'^†]=δ_𝐤,𝐤', and 𝐑={R_𝐧s} is an N-dimensional vector that describes the vibrational coordinates of the N=N_xN_yN_z molecules of the ensemble, where N_i is the number of molecules along each ensemble axis. ħω_g(R_𝐧,s) accounts for the ground-state energy of the molecule whose location in the ensemble is defined by 𝐧 and s. We introduce the (adiabatic R-dependent) exciton operator b_𝐧,s^†(R_𝐧,s)(b_𝐧,s(R_𝐧,s)) to label the creation (annihilation) of a Frenkel exciton (electronic excitation) with an energy gap ħω_e(R_𝐧,s)-ħω_g(R_𝐧,s) on the molecule located at 𝐧+z_sẑ. The coefficients ħω_𝐤 and g_𝐤^𝐧,s(R_𝐧,s) stand for the energy of a SP with in-plane momentum 𝐤 and the coupling of the molecule located at 𝐧+z_s𝐳̂ with the latter, respectively. The dipolar SP-matter interaction is described by g_𝐤^𝐧,s(R_𝐧,s)=h_𝐤(R_𝐧,s)f_𝐤(z_s), where h_𝐤(R_𝐧,s)=-μ_𝐧,s(R_𝐧,s)·𝐄_𝐤(𝐧) is the projection of the molecular transition dipole μ_𝐧,s(R_𝐧,s) onto the in-plane component of the SP electric field 𝐄_𝐤(𝐧) and f_𝐤(z_s)=e^-α_𝐤z_s is the evanescent field profile along the z direction, with α_𝐤 being the decay constant in the molecular region (z>0). The quantized plasmonic field 𝐄̂_𝐤f_𝐤(𝐳_s) has been discussed in previous works <cit.> and reads 𝐄̂_𝐤(𝐧)f_𝐤(𝐳_s)=√(ħω_𝐤/2ϵ_0SL_𝐤)a_𝐤χ̂_𝐤e^i𝐤·𝐧e^-α_𝐤z+h.c., where ϵ_0 is the free-space permittivity, S is the coherence area of the plexciton setup, L_𝐤 is the quantization length, and χ̂_𝐤=𝐤̂+i|𝐤|/α_𝐤𝐳̂ is the polarization. Note that the parametric dependence of the exciton operators on R_𝐧,s yield residual non-adiabatic processes induced by nuclear kinetic energy that may be relevant to the isomerization in question. We also highlight the fact that Eq. (<ref>) includes both rotating (“energy conserving”) terms (a_𝐤^†b_𝐧,s and a_𝐤b_𝐧,s^†) where a photon creation (annihilation) involves the concomitant annhilation (creation) of an exciton; and counterrotating (“non-energy conserving”) terms (a_𝐤b_𝐧,s and a_𝐤^†b_𝐧,s^†) where there is a simultaneous annhilation (creation) of photon and exciton. These latter terms are ignored in the widely used Rotating Wave Approximation (RWA)<cit.>, where light-matter coupling is weak compared to the transition energy. Since we are interested in the USC, we shall keep them throughout. § METHODS For simplicity, we assume that all the transition dipoles are equivalent and aligned along x, μ_𝐧,s(R_𝐧,s)=μ(R_𝐧,s)=μ(R_𝐧,s)𝐱̂; a departure of this perfect crystal condition does not affect the conclusions of this article. Furthermore, it is convenient to first restrict ourselves to the cases where all nuclei are fixed at the same configuration (𝐑=𝐑̃, which denotes R_𝐧,s=R for all 𝐧 and s), so that we can take advantage of the underlying translational symmetry to introduce a delocalized exciton basis where the in-plane momentum 𝐤 is a good quantum number. The creation operator of this delocalized state is defined by b_𝐤^†(R)=1/√(𝒩_𝐤(R))∑_𝐧∑_sf_𝐤(z_s)h_𝐤(R)b_𝐧,s^†(R), and the normalization squared is given by 𝒩_𝐤(R)=∑_𝐧∑_s|h_𝐤(R)|^2|f_𝐤(z_s)|^2 which, in the continuum limit, can be seen to be proportional to ρ, the number density of the molecular ensemble. In this collective basis, the previously introduced H_el(𝐑) readsH_el(𝐑̃) =∑_𝐤ħΔ(R)b_𝐤^†(R)b_𝐤(R)+∑_𝐤ħω_𝐤a_𝐤^†a_𝐤 +∑_𝐤√(𝒩_𝐤(R))(a_𝐤^†b_𝐤(R)+a_𝐤b_𝐤^†(R)+a_𝐤b_-𝐤(R)+a_𝐤^†b_-𝐤^†(R))+∑_𝐤H_dark,𝐤(R)+∑_𝐤H_unklapp,𝐤(R)+Nħω_g(R) =∑_𝐤H_𝐤(R)+∑_𝐤H_dark,𝐤(R)+∑_𝐤H_unklapp,𝐤(R)+Nħω_g(R),where Δ(R)=ω_e(R)-ω_g(R) is the exciton transition frequency. H_dark,𝐤(R) =ħΔ(R)𝐏_dark,𝐤(R)accounts for the energy of the (N_z-1)-degenerate exciton states with in-plane momentum 𝐤 that do not couple to SPs, and are usually known as dark states. The latter are orthogonal to the bright exciton b_𝐤^†(R)|G_m(𝐑̃)⟩ that couples to the SP field, where |G_m(𝐑̃)⟩ is the bare molecular ground-state (b_𝐤(R)|G_m(𝐑̃)⟩=0).More specifically, 𝐏_dark,𝐤(R)=𝐈_exc,𝐤(R)-b_𝐤^†(R)b_𝐤(R) is a projector operator onto the 𝐤-th dark-state subspace, with 𝐈_exc(R)=∑_𝐧,sb_𝐧,s^†(R)b_𝐧,s(R)=∑_𝐤,sb_𝐤,s^†(R)b_𝐤,s(R)=∑_𝐤𝐈_exc,𝐤(R) being the identity on the exciton space, and b_𝐤,s^†(R)=1/√(N_xN_y)∑_𝐧e^-i𝐤·𝐧b_𝐧,s^†(R). Finally,H_unklapp,𝐤(R)= ∑_𝐪=(2π q_x/Δ_x,2π q_y/Δ_y)√(𝒩_𝐤(R))(a_𝐤+𝐪^†b_𝐤(R)+a_𝐤+q^†b_-𝐤^†(R)+h.c.)stands for the coupling of excitons with momentum 𝐤 to SP modes with momentum beyond the first excitonic Brillouin zone. H_unklapp,𝐤(R) is usually ignored given the large off-resonance between the SP energy and the exciton states; however, since this work pertains off-resonant effects, we considered it to acquire converged quantities in the calculations explained below. We also note that the normalization constant √(𝒩_𝐤(R)) in Eq. <ref> is precisely the collective SP-exciton coupling. As mentioned in the introduction, the condition √(𝒩_𝐤(R))/ħΔ(R)>0.1 is often used to define the onset of USC <cit.>, and it is fulfilled with the maximal density considered in our model (see Fig. <ref>) taking into account that the largest ħΔ(R) is 3 eV (See Fig. <ref>a). We note, as will be evident later, that our main results do not vary significantly by considering ratios √(𝒩_𝐤(R))/ħΔ(R) below the aforementioned threshold. A Bogoliubov transformation <cit.> permits the diagonalization of the Bloch Hamiltonian H_𝐤 in Eq. <ref> by introducing the polariton quasiparticle operators ξ_𝐤^j(R)=α_𝐤^ja_𝐤+β_𝐤^jb_𝐤(R)+γ_𝐤^ja_-𝐤^†+δ_𝐤^jb_-𝐤^†(R),where j=U,L and U (L) stands for the upper (lower) Bogoliubov polariton state. Notice that this canonical transformation is valid for a sufficiently large number of molecules N, where the collective exciton operators b_𝐤(R), b_𝐤^†(R) are well approximated by bosonic operators <cit.>.The bare molecular ground-state with no photons in the absence of light-matter coupling |G_m(𝐑̃);0⟩, (a_𝐤|G_m(𝐑̃);0⟩=b_𝐤(R)|G_m(𝐑̃);0⟩=0 for all 𝐤) has a total extensive energy with molecular contributions only ⟨ G_m(𝐑̃);0|H_el(𝐑̃)|G_m(𝐑̃);0⟩=Nħω_g(R). Upon inclusion of the counterrotating terms, the ground-state becomes the dressed Bogoliubov vacuum |G(𝐑̃)⟩_d, characterized by ξ_𝐤^j(R)|G(𝐑̃)⟩_d=0 for all 𝐤 and j, with total energy _d⟨ G(𝐑̃)|H_el(𝐑̃)|G(𝐑̃)⟩_d=E_0(𝐑̃), where the zero-point energy is given byE_0(𝐑̃)= Nħω_g(R)+1/2∑_𝐤(∑_j=U,Lħω_j,𝐤(R)-ħω_𝐤-ħΔ(R)),{ħω_j,𝐤(R)} being the eigenvalues of the Bogoliubov polariton branches given byω___L^U,𝐤(R)=√((Δ(R))^2+ω_𝐤^2±√(B(R)^2+16𝒩_𝐤^2(R)Δ(R)ω_𝐤)/2),where we have introduced B(R)=ω_𝐤^2-Δ(R)^2. A hallmark of the SC and USC regimes is the anticrossing splitting of the polariton energies at the 𝐤 value where the bare excitations are in resonance, Δ(R)=ω_𝐤 <cit.> (see Fig. <ref>). The sum in Eq. <ref> accounts for the energy shift from the bare molecular energy Nħω_g(R) due to interaction with the infinite number of SP modes in the setup. Using Eq. (<ref>), it is illustrative to check that this shift vanishes identically when the non-RWA terms are ignored. It is worth describing some of the physical aspects of the Bogoliubov ground-state |G(𝐑̃)⟩_d. With the numerically computed wavefunctions, we can use the inverse transformation of Eq. <ref> to explicitly evaluate its SP and exciton populations <cit.>,n_𝐤^SP=_d⟨ G(𝐑̃)|a_𝐤^†a_𝐤|G(𝐑̃)⟩_d=∑_j|γ_𝐤^j|^2,n_𝐤^exc=_d⟨ G(𝐑̃)|b_𝐤^†b_𝐤|G(𝐑̃)⟩_d=∑_j|δ_𝐤^j|^2,which give rise to humble O(10^-3) values per mode 𝐤, considering a molecular ensemble with ρ=3× 10^8μ m^-3 and W_z=120 nm; this calculation is carried out using N=8×10^7, although results are largely insensitive to this parameter as long as it is sufficiently large to capture the thermodynamic limit. The consequences of the dressing partially accounted for by Eq. (<ref>) (partially since there are also correlations of the form _d⟨ G(𝐑̃)|b_𝐤a_-𝐤|G(𝐑̃)⟩_d) are manifested as energetic effects on |G_m(𝐑̃);0⟩: E_0(𝐑̃)-Nħω_g(R) can be interpreted as the energy stored in |G(𝐑̃)⟩_d as a result of dressing; it is an extensive quantity of the ensemble, but becomes negligible when considering a per-molecule stabilization. For instance, in molecular ensembles with the aforementioned parameters we find E_0(0̃)-Nħω_g(0)=O(10^2) eV, which implies a O(10^-5) eV value per molecule; our calculations show that this intensive quantity is largely insensitive to total number of molecules. This observation raises the following questions: to what extent does photonic dressing would impact ground-state chemical reactivity? What are the relevant energy scales that dictate this impact? With these questions in mind, we aim to study the polaritonic effects on ground-state single-molecule isomerization events. To do so, we map out the PES cross section where we set one “free` molecule to undergo isomerization while fixing the rest at R_𝐧,s=0. A similar strategy has been used before in <cit.>. This cross section, described by E_0(R_𝐧_0,0,0,⋯,0)≡ E_0(R_𝐧_0,0,0̃') (R_𝐧_0,0 being the coordinate of the unconstrained molecule), should give us an approximate understanding of reactivity starting from thermal equilibrium conditions, since the molecular configuration 𝐑̃=0̃ still corresponds to the global minimum of the modified ground-state PES, as will be argued later. By allowing one molecule to move differently than the rest, we weakly break translational symmetry. Rather than numerically implementing another Bogoliubov transformation, we can, to a very good approximation, account for this motion by treating the isomerization of the free molecule as a perturbation on H_el(0̃). More precisely, we write H_el(R_𝐧_0,0,0̃')|G(R_𝐧_0,0,0̃')⟩_d=E_0(R_𝐧_0,0,0̃')|G(R_𝐧_0,0,0̃')⟩_d, where H_el(R_𝐧_0,0,0̃') is the sum of a translationally invariant piece H_el(0̃) plus a perturbation due to the free molecule,H_el(R_𝐧_0,0,0̃')=H_el(0̃)+V(R_𝐧_0,0).The perturbation is explicitly given byV(R_𝐧_0,0) =H_el(R_𝐧_0,0,0̃')-H_el(0̃) =ħΔ(R_𝐧_0,0)b_𝐧_0,0^†(R_𝐧_0,0)b_𝐧_0,0(R_𝐧_0,0)-ħΔ(0)b_𝐧_0,0^†(0)b_𝐧_0,0(0) +∑_𝐤{ g_𝐤^𝐧_0,0(R_𝐧_0,0)[b_𝐧_0,0(R_𝐧_0,0)+b_𝐧_0,0^†(R_𝐧_0,0)]-g_𝐤^𝐧_0,0(0)[b_𝐧_0,0(0)+b_𝐧_0,0^†(0)]}[a_𝐤+a_𝐤^†] +ħω_g(R_𝐧_0,0)-ħω_g(0). Notice that we have chosen the free molecule to be located at an arbitrary in-plane location 𝐧_0 and at the very bottom of the slab at s=0, where light-matter coupling is strongest as a result of the evanescent field profile along the z direction. We write an expansion of the PES cross section as E_0(R_𝐧_0,0,0̃')=∑_q=0^∞E_0^(q)(R_𝐧_0,0,0̃'), where q labels the O(V^q) perturbation correction. The zeroth order term is the Bogoliubov vacuum energy associated to every molecule being at the equilibrium geometry E_0^(0)(R_𝐧_0,0,0̃')=E_0(0̃) as in Eq. (<ref>). The O(V) correction corresponds to ħω_g(R_𝐧_0,0)-ħω_g(0), merely describing the PES of the isomerization of the bare molecule in the absence of coupling to the SP field. The contribution of the SP field on the PES cross-section of interest appears at O(V^2), and it is given byE^(2)(R_𝐧_0,0,0̃')≈∑__i,j=UP,LP^𝐤_1≤𝐤_2|⟨𝐤_1,i;𝐤_2,j|V(R_𝐧_0,0)|G(0̃)⟩_d|^2/E_0(0̃)-E_𝐤_1,𝐤_2,i,j^(0),where |𝐤_1,i;𝐤_2,j⟩≡ξ_𝐤_1^i†(0)ξ_𝐤_2^† j(0)|G(0̃)⟩_d and E_𝐤_1,𝐤_2,i,j^(0)=ħ(ω_i,𝐤_1(0)+ω_j,𝐤_2(0)). As shown in the Appendix, the approximation in Eq. (<ref>) consists of ignoring couplings between |G(0̃)⟩_d and states with three and four Bogoliubov polariton excitations, since their associated matrix elements become negligible in the thermodynamic limit compared to their double excitation counterparts. The remaining matrix elements can be calculated by expressing the operators a_𝐤, a_𝐤^†, b_𝐧_0,0(R_𝐧_0,0), b_𝐧_0,0^†(R_𝐧_0,0) in Eq. (<ref>) in terms of the Bogoliubov operators ξ_𝐤^j(0), ξ_𝐤^† j(0) (see Eq. (<ref>)), leading to ⟨𝐤_1,i;𝐤_2,j|V(R_𝐧_0,0)|G(0̃)⟩_d=F^𝐤_2(R_𝐧_0,0)D_𝐤_1(-δ_-𝐤_1^iα_𝐤_2^j+δ_-𝐤_1^iγ_-𝐤_2^j-β_𝐤_1^iγ_-𝐤_2^j+β_𝐤_1^iα_𝐤_2^j) +F^𝐤_1(R_𝐧_0,0)D_𝐤_2(-δ_-𝐤_2^jα_𝐤_1^i+δ_-𝐤_2^jγ_-𝐤_1^i-β_𝐤_2^jγ_-𝐤_1^i+β_𝐤_2^jα_𝐤_1^i),where F^𝐤(R)=cos(θ(R))g_𝐤^𝐧_0,0(R)-cos(θ(0))g_𝐤^𝐧_0,0(0) depends on the mixing angle that describes the change of character of b_𝐧_0,0^†(R) as a function of R (see Equation (<ref>)); it emerges as a consequence of coupling molecular states at different configurations. D_𝐤=⟨ G_m(𝐑̃);0⟩|b_𝐧_0,0(0)b_𝐤^†(0)|G_m(𝐑̃);0⟩=1/√(N_xN_y)√(1-e^-2α_𝐤Δ_z/1-e^-2α_𝐤Δ_zN_z) accounts for the weight of a localized exciton operator in a delocalized one, such as the participation of b_𝐧_0,0^†(0) in b_𝐤^†(0). Eq. (<ref>) reveals that the maximal contribution of each double-polariton Bogoliubov state to the energetic shift of the considered PES cross section E(R_𝐧_0,0,0̃') is of the order of g_𝐤^𝐧_0,0(0)/√(N_xN_y). Considering macroscopic molecular ensembles with large N≈10^7, we computed Eq. <ref> by means of an integral approximation over the polariton modes 𝐤.§ RESULTS AND DISCUSSION §.§ Energetic effectsWe carry out our calculations with ρ in the range of 10^6 to 10^9 molecules μ m^-3 keeping W_z=120 nm (see Fig. <ref>); to obtain results in the thermodynamic limit, our calculations take N=8×10^7, even though the exact value is unimportant as long as it is sufficiently large to give converged results. The results displayed in Fig. <ref> show that the second order energy corrections to the isomerization PES E^(2)(R_𝐧_0,0,0̃'), and in particular E^(2)(R_𝐧_0,0=R^*,0̃')≈-0.25 meV, are negligible in comparison with the bare activation barrier E_a=ħω_g(R^*)-ħω_g(0)=ħω_g(R^*)≈ 1.8 eV, where R^*≈1.64 rad corresponds to the transition state. From Fig. <ref>b, we notice that there is a substantial difference in SP-exciton coupling between the equilibrium (R_𝐧_0,0=0) and transition state geometries (R_𝐧_0,0=R^*). Since the perturbation in Eq. (<ref>) is defined with respect to the equilibrium geometry, |E^(2)(R_𝐧_0,0,0̃')| maximizes at the barrier geometry. To get some insight on the order of magnitude of the result, we note that the sum shown in Eq. <ref> can be very roughly approximated as E^(2)(R_𝐧_0,0,0̃') =O[-∑_𝐤_1≤𝐤_2[g_𝐤_1^𝐧_0,0(R_𝐧_0,0)]^2D_𝐤_2^2+[g_𝐤_2^𝐧_0,0(R_𝐧_0,0)]^2D_𝐤_1^2/(ħω_𝐤_1+ħω_𝐤_2)/2+ħω_e(R_𝐧_0,0)]=O[-1/N_xN_y∑_𝐤_1≤𝐤_2[g_𝐤_1^𝐧_0,0(R_𝐧_0,0)]^2+[g_𝐤_2^𝐧_0,0(R_𝐧_0,0)]^2/(ħω_𝐤_1+ħω_𝐤_2)/2+ħω_e(R_𝐧_0,0)] =O[-∑_𝐤[g_𝐤^𝐧_0,0(R_𝐧_0,0)]^2/ħω_𝐤+ħω_e(R_𝐧_0,0)] =O(E_LS(R_𝐧_0,0)). In the first line, we used the fact that ⟨𝐤_1,i;𝐤_2,j|V(R_𝐧_0,0)|G(0̃)⟩_d≈ [g_𝐤_1^𝐧_0,0(R_𝐧_0,0)]^2D_𝐤_2^2+[g_𝐤_2^𝐧_0,0(R_𝐧_0,0)]^2D_𝐤_1^2 and averaged the Bogoliubov polariton excitation energies. In the second line, assuming that the 𝐤≫0 values contribute the most, we have D_𝐤≈1/√(N_xN_y). Finally, in the third line, we have used the fact that the sum of terms over 𝐤_1,𝐤_2 is roughly equal to N_xN_y times a single sum over 𝐤 of terms of the same order. The reason why we are interested in the final approximation is because it corresponds to the Lamb shift of a single isolated molecule, which can be calculated to be E_LS(0)=0.16 meV. Typically, Lamb shift calculations require a cutoff to avoid unphysical divergences <cit.>; we stress that in our plexciton model, this is not necessary due to the decaying |g_𝐤^𝐧_0,0(R_𝐧_0,0)| as a function of |𝐤|. The fact that the corrections E^(2)(R_𝐧_0,0,0̃') have a similar order of magnitude to single-molecule Lamb shifts give a pessimistic conclusion of harnessing USC to control ground-state chemical reactions. Note, however, from calculations in Fig. <ref>, that there is variability in E^(2)(R_𝐧_0,0,0̃') as a function of molecular density (since density alters the character of the Bogoliubov polaritons), although the resulting values are always close to E_LS(0). The molecular density cannot increase without bound, since there exists a minimum molecular contact distance determined by a van der Waals radius of the order of 0.3 nm for organic molecules <cit.>, giving a maximum density of ρ≈10^10 molecules/μ m^3.The results discussed so far describe the energy profile of the isomerization of a single molecule keeping the rest at equilibrium geometry. It is intriguing to inquire the effects of the SP field in a concerted isomerization of two or more molecules, while keeping the rest fixed at equilibrium geometry. Generalizing Eqs. (<ref>)–(<ref>) to a two-molecule perturbation V(R_𝐧_0,0,R_𝐧_1,0), we computed the second order energetic corrections to the 2D-PES that describe the isomerization of two neighbouring molecules at 𝐧_0 and at 𝐧_1≡𝐧_0+Δ_x𝐱̂, keeping the other molecules fixed at R_𝐧,s=0. The results are reported in Fig. <ref> for ρ=3×10^8 molecules/μ m^3, although outcomes of the same order of magnitude are obtained for the other densities considered in the one-dimensional case. The two-dimensional PES cross-section E^(2)(R_𝐧_0,0,R_𝐧_1,0,0,⋯,0)≡ E^(2)(R_𝐧_0,0,R_𝐧_1,0,0̃') shows the existence of an energetic enhancement for the concerted isomerization with respect to two independent isomerizations, i.e. E^(2)(R_𝐧_0,0=R^*,R_𝐧_1,0=R^*,0̃')≈4E^(2)(R_𝐧_0,0=R^*,0̃'). This enhacement is due to a constructive interference arising at the amplitude level, ⟨𝐤_1,i;𝐤_2,j|V(R_𝐧_0,0=R^*,R_𝐧_1,0=R^*)|G(0̃)⟩_d≈2⟨𝐤_1,i;𝐤_2,j|V(R_𝐧_0,0=R^*)|G(0̃)⟩_d for values of _1, _2≪1/Δ_x, such that the phase difference between the isomerizing molecules is negligible. Interestingly, choosing the neighbouring molecules along the x direction is important for this argument; if instead we consider neighbours along z (molecular positions 𝐧_0 and 𝐧_0+Δ_zẑ), these interferences vanish and we approximately get the independent molecules result E^(2)(R_𝐧_0,0=R^*,R_𝐧_0,1=R^*,0̃')≈2E^(2)(R_𝐧_0,0=R^*,0̃'). In light of the nontrivial energetic shift of the two-molecule case, it is pedagogical to consider the SP effects on the cross-section of the concerted isomerization of the whole ensemble, even though it is highly unlikely that this kinetic pathway will be of any relevance, especially considering the large barrier for the isomerization of each molecule. Notice that the conservation of translational symmetry in this scenario allows for the exact (nonperturbative) calculation of the energetic shift E_0(𝐑̃)-Nħω_g(R) by means of Eq. <ref>. Our numerical calculations reveal an energetic stabilization profile, which is displayed in Fig. <ref> for a molecular ensemble with ρ=3× 10^8 molecules μ m^-3. As expected, we observe a stabilization of reactant and product regions of the ground-state PES. This is a consequence of the transition dipole moment being the strongest at those regions, as opposed to the transition state, see Fig. <ref>b. However, even though these energetic effects are of the order of hundreds of eV, they are negligible in comparison with the total ground-state PES Nħω_g(R), or more specifically, to the transition barrier NE_a=Nħω_g(R^*) for the concerted reaction. Importantly, the change in activation energy per molecule in the concerted isomerization with respect to the bare case |Δ E_a|=|(E_0(𝐑̃^*)-E_0(0̃)/N-E_a)|≈0.009 meV is more than one order of magnitude smaller than the corresponding quantity |E^(2)(R_𝐧_0,0=R^*,0̃')|≈0.25 meV for the single-molecule isomerization case, see Fig. <ref> and inset of Fig. <ref>. We believe that the reason for this trend is that the isomerization of n molecules, n≪ N, translates into a perturbation which breaks the original translational symmetry of the molecular ensemble. This symmetry breaking permits the interaction of the molecular vacuum with the polaritonic 𝐤-state reservoir without a momentum-conservation restriction. This is reflected in Eq. <ref>, where the sum is carried out over two not necessarily equal momenta. In contrast, in the case of the concerted isomerization of N molecules, thetranslational symmetry of the system is preserved, which in turn restricts the coupling of the vacuum |G(0̃)⟩_d to excited states with 𝐤_exc=-𝐤_phot.Another intriguing observation is that, for this concerted isomerization, the SP energetic effect per molecule E_0(𝐑̃)/N diminishes with the width of the slab W_z. This is the case given that the SP quantization length L_𝐤 decays quickly with |𝐤| so that only the closest layers interact strongly with the field. When we divide the total energetic effects due to the SP modes by N=N_xN_yN_z, we obtain that E_0(𝐑̃)/N=O(1/N_z) for large W_z.The energetic shifts in all the scenarios discussed above are negligible with respect to the corresponding energy barriers and the thermal energy scale at room temperature which, unfortunately, signal the irrelevance of USC to alter ground-state chemical reactivity for this isomerization model. Although there is an overall (extensive) stabilization of the molecular ensemble ground state, this effect is distributed across the ensemble, giving no possibility to alter the chemical reaction kinetics or thermodynamics considerably. However, we highlight the intriguing interferences observed in the concerted isomerization processes. Even though they will likely be irrelevant for this particular reaction, they might be important when dealing with reactions with very low barriers, especially when considering that these concerted pathways are combinatorially more likely to occur than the single-molecule events in the large N limit. This is intriguing in light of the study carried out in <cit.>, which discusses a different but related effect of many reactions triggered by a single photon. §.§ Effects on non-adiabatic dynamicsFinally, we discuss the importance of the nonadiabatic effects afforded by nuclear kinetic energy. Previous works have considered the nonadiabatic effects between polariton states at the level of SC <cit.>. Alternatively, the consideration of nonadiabatic effects in USC for a single molecule in a cavity was provided in <cit.>; here, we address these issues for the many-molecule case and consider both polariton and dark state manifolds. One could expect significantly modified non-adiabatic dynamics about nuclear configurations where the transition dipole moment magnitude |μ_𝐧,s(R_𝐧,s)| is large, given a reduction in the energy gap between the ground and the lower Bogoliubov polariton state. However, as we show below, this energetic effect is not substantial due to the presence of dark states. We consider the magnitude of the non-adiabatic couplings (NACs) for the isomerization of a single molecule with reaction coordinate R_𝐧_0,0. For a region about 𝐑̃=0̃, we estimate the magnitude of the NAC between |G(0̃)⟩_d and a state |𝐤,i⟩=ξ_𝐤^i†(0)|G(0̃)⟩_d as: |A_𝐤,i;g(0)| =|⟨𝐤,i|∂/∂ R_𝐧_0,0|G(0̃)⟩_d|≈|β_𝐤^iD_𝐤⟨ e_𝐧_0,0(0)|∂/∂ R_𝐧_0,0|g_𝐧_0,0(0)⟩|,where |g_𝐧_0,0(0)⟩(|e_𝐧_0,0(0)⟩) is the ground (excited) adiabatic state of the single molecule under consideration (see Eq. (<ref>)) and we have ignored the derivatives of β_𝐤^i and D_𝐤 with respect to R_𝐧_0,0, assuming they are small at 𝐑̃=0̃, where the chemical character of the Bogoliubov polariton states does not change significantly with respect to nuclear coordinate. This is a consequence of the slowly changing transition dipole moment of the model molecule around R_𝐧_0,0=0, see Fig. <ref>b. Notice that we have also assumed ⟨𝐤,i|e_𝐧_0,0(0)⟩≈β_𝐤^iD_𝐤, where we have used the fact that β_𝐤^i≫γ_𝐤^i, thus ignoring counterrotating terms, which as we have seen, give negligible contributions.The time-evolution of a nuclear wavepacket in the ground-state will be influenced by the Bogoliubov polariton states, each of which will contribute with a finite probability of transition out of |G(0̃)⟩_d. From semiclassical arguments <cit.>, we can estimate the transition probability |C_𝐤^i(0)|^2 for a nuclear wavepacket on the ground-state PES at 𝐑̃=0 to the state |𝐤,i⟩, |C_𝐤^i(0)|^2 ≈|ħ v_nucA_𝐤,i;g(0)/ħω_i,𝐤(0)-ħω_g(0)|^2 =|ħ v_nucβ_𝐤^iD_𝐤/ħω_i,𝐤(0)-ħω_g(0)|^2×|⟨ e_𝐧_0,0(0)|∂/∂ R_𝐧_0,0|g_𝐧_0,0(0)⟩|^2,v_nuc being the expectation value of the nuclear velocity. However, the Bogoliubov polariton 𝐤-states are only a small subset of the excited states of the problem. As mentioned right after Eq. <ref>, the plexciton setup contains N_z-1 dark excitonic states for every 𝐤 (eigenstates of H_dark,𝐤(0), see discussion right after Eq. <ref>); we ignore the very off-resonant couplings considered in H_unklapp,𝐤(0). The dark states also couple to |G(0̃)⟩_d non-adiabatically, with the corresponding transition probability out of the ground state being, |C_𝐤^dark(0)|^2 ≈∑_Q|ħ v_nucA_𝐤,Q;g(0)/ħΔ(0)|^2≈ P_bare(0)(1/N_xN_y-|D_𝐤|^2),Here, we have summed over all dark states Q for a given 𝐤 and used P_bare(0)=|v_nuc/Δ(0)|^2|⟨ e_𝐧_0,0(0)|∂/∂ R_𝐧_0,0|g_𝐧_0,0(0)⟩|^2 to denote the probability of transition out of the ground state in the absence of coupling to the SP field. In Eq. (<ref>) we used the fact that the projection |e_𝐧_0,0(0)⟩ onto the dark 𝐤 manifold of exciton states is |𝐏_dark,𝐤(0)|e_𝐧_0,0(0)⟩|^2=⟨ e_𝐧_0,0(0)|𝐈_exc,𝐤(0)|e_𝐧_0,0(0)⟩-|D_𝐤|^2=1/N_xN_y-|D_𝐤|^2, with 𝐏_dark,𝐤(0) being the corresponding projector (see Eq. (<ref>)).We noticed that when |𝐤|→0, the quantization volume L_𝐤 of the plasmonic field spans all the molecular-ensemble volume resulting incompletely delocalized bright and dark exciton states across the different layers of the slab, |𝐏_dark,𝐤|e_𝐧_0,0(0)⟩|^2=N_z-1/N, and the dark states give the major contribution to the nonadiabatic dynamics. On the other hand, when |𝐤|→∞, the plasmonic field interacts with the molecular layer at the bottom of the slab only and |𝐏_dark,𝐤|e_𝐧_0,0(0)⟩|^2→ 0. The dark states do not participate, because the molecule located at 𝐧_0 only overlaps with the bright state which is concentrated across the first layer of the slab (the dark states, being orthogonal to the bright one, are distributed in the upper layers, and do not overlap with |e_𝐧_0,0⟩). With these results, we can compute the probability of transition out of the ground-state P_out as P_out(0) =∑_𝐤[∑_i|C_𝐤^i(0)|^2+|C_𝐤^dark(0)|^2].In view of the large off-resonant nature of most SP modes with respect to ħΔ(0) (see Fig. <ref>) and Eq. (<ref>), we have ∑_i|C_𝐤^i(0)|^2≈ P_bare(0)|D_𝐤|^2, such that P_out(0)≈ P_bare(0).In our model, this is the case, since the plexciton anticrossing occurs at small |𝐤| and the SP energy quickly increases and reaches an asymptotic value after that point (see Fig. <ref>).Using the parameters in <cit.>, we obtain ⟨ e(R_𝐧_0,0)|∂/∂ R_𝐧_0,0|g(R_𝐧_0,0)⟩≈0.01 Å^-1, where we have assumed an effective radius of 1 Å for the isomerization mode of the model molecule. We get an estimate of v_nuc≈ 1Å ω_nuc=1Å√(k_BT/m)=9×10^10 Ås^-1 using k_B=8.62×10^-5 eV K^-1, T=298 K and m=2.5 amu Å^2. Finally, applying Δ(0)=3 eV gives P_bare(0)≈10^-7, which is a negligible quantity. A more pronounced polariton-effect is expected close to the PES avoided crossing. However, the rapid decay of the transition dipole moment in this region (see Fig. <ref>a) precludes the formation of polaritonic states that could have affected the corresponding nonadiabatic dynamics. To summarize this part, even when the USC effects on the nonadiabatic dynamics are negligible for our model, the previous discussion as well as Eq. (<ref>) distill the design principle that controls these processes in other polariton systems: the plexciton anticrossings should happen at large 𝐤 values to preclude the overwhelming effects of the dark states. This principle will be explored in future work in other molecular systems.The negligible polariton effect on the NACs, and the magnitude of the energetic effects on the electronic energy landscape are strong evidence to argue that the chemical yields and rates of the isomerization problem in question remain intact with respect to the bare molecular ensemble.§ CONCLUSIONS We showed in this work that, for the ground state landscape of a particular isomerization model, there is no relevant collective stabilization effect by USC to SPs which can significantly alter the kinetics or thermodynamics of the reaction, in contrast with previous calculations which show such possibilities in the Bogoliubov polariton landscapes <cit.>. The negligible energetic corrections to the ground-state PES per molecule can be approximated and interpreted as Lamb shifts <cit.> experienced by the molecular states due to the interaction with off-resonant plasmonic modes. The key dimensionless parameter which determines the USC effect on the ground-state PES is the ratio of the individual coupling to the transition frequency g_𝐤^𝐧,s/ħΔ. This finding is similar to the conclusions of a recent work <cit.>. In particular, it is shown in <cit.> that the rotational and vibrational degrees of freedom of molecules exhibit a self-adaptation which only depends on light-matter coupling at the single-molecule level. Therefore, more remarkable effects are expected in the regime of USC of a single molecule interacting with an electric field. To date, the largest single molecule interaction energy achievable experimentally is around 90 meV <cit.> in an ultralow nanostructure volume. This coupling strength is almost two orders of magnitude larger that those in our model. Also, previous works have shown <cit.> that this regime is achievable for systems with transition frequencies on the microwave range. Additionally, the experimental realization of vibrational USC has been carried out recently <cit.>. The latter also suggests the theoretical exploration of USC effects on chemical reactivity at the rotational or vibrational energy scales, where the energy spacing between levels is significantly lower than typical electronic energy gaps. We highlighted some intriguing quantum-coherent effects where concerted reactions can feature energetic effects that are not incoherent combinations of the bare molecular processes. These interference effects are unlikely to play an important role in reactions exhibiting high barriers compared to k_BT. However, they might be important for low-barrier processes, where the number of concerted reaction pathways becomes combinatorially more likely than single molecule processes. On the other hand, we also established that, due to the large number of dark states in these many-molecule polariton systems, nonadiabatic effects are not modified in any meaningful way under USC, at least for the model system explored. We provided a rationale behind this conclusion and discussed possibilities of seeing modifications in other systems where the excitonic and the electromagnetic modes anticross at large 𝐤 values.Finally, it is worth noting that even though we considered an ultrastrong coupling regime ( √(𝒩_𝐤(R)) reaches more than 10% of the maximum electronic energy gap in our model <cit.>), the system does not reach a Quantum Phase Transition (QPT) <cit.>. In our model, this regime would require high density (∼ 10^10 molecules μ m^-3) samples, keeping μ≈ 2 eÅ. The implications of this QPT on chemical reactivity have not been explored in this work, but are currently being studied in our group. To conclude, our present work highlights the limitations but also possibilities of USC in the context of control of chemical reactions using polaritonic systems. § ACKNOWLEDGMENTS R.F.R., J.C.A., and J.Y.Z. acknowledge support from the NSF CAREER award CHE-1654732. L.A.M.M is grateful to the support of the UC-Mexus CONACyT scholarship for doctoral studies. All authors acknowledge generous startup funds from UCSD. L.A.M.M. and J.Y.Z are thankful to Prof. Felipe Herrera for useful discussions.§ APPENDIX In this appendix we outline the perturbative methodology that leads to the equations shown in the main text. Under the perturbative approach, it is convenient to express the perturbation in Eq. (<ref>) in terms of the Bogoliubov operators defined by Eq. (<ref>). Notice that Eq. <ref> introduces R-dependent exciton operators, while the zeroth order eigenstates (the polariton quasiparticles) are defined for all molecules at the configuration R_𝐧,s=0. It would be useful to find a relation b_𝐧_0,0^†(R)=f(b_𝐧_0,0^†(0),b_𝐧_0,0(0)) for any R, in order to carry out the aforementioned change of basis. The function f(b_𝐧_0,0^†(0),b_𝐧_0,0(0)) can be found by working on the diabatic basis (see Eq. <ref>). For any operator b_𝐧,s^†(R), using b_𝐧,s^†(R)=|e_𝐧,s(R)⟩⟨ g_𝐧,s(R)|, we haveb_𝐧,s^†(R) =[-sin(θ(R)/2)|trans_𝐧,s⟩+cos(θ(R)/2)|cis_𝐧,s⟩]×[cos(θ(R)/2)⟨trans_𝐧,s|+sin(θ(R)/2)⟨cis_𝐧,s|] =-sin(θ(R)/2)cos(θ(R)/2)b_𝐧,s(0)b_𝐧,s^†(0) -sin(θ(R)/2)^2b_𝐧,s(0)+cos(θ(R)/2)^2b_𝐧,s^†(0) +cos(θ(R)/2)sin(θ(R)/2)b_𝐧,s^†(0)b_𝐧,s(0),where we used b_𝐧,s^†(R_eq)=|cis_𝐧,s⟩⟨trans_𝐧,s|. In light of Eq. <ref> we notice that the perturbation (<ref>) in the second row produces chains with up to four exciton operators. In view of the delocalized nature of the zeroth-order eigenstates and the localized character of the exciton operators b_𝐧,s(0), we have that the matrix elements that appear in E^(2)(R_𝐧_0,0,0̃') are of the form ⟨ G(0̃)|_dξ_𝐤_1^iξ_𝐤_2^j…ξ_𝐤_m+1^lF_m(b_𝐧_0,0^†b_𝐧_0,0)Z(a_𝐤^†,a_𝐤)|G(0̃)⟩_d≈ O(1/(N_xN_y)^m/2), where F_m(b_𝐧_0,0^†b_𝐧_0,0) stands for a chain with 1≤ m≤4 exciton operators and Z(a_𝐤^†,a_𝐤) is a function of a single photonic operator. In the macroscopic limit 1≪ N_xN_y, we can neglect chains F_m(b_𝐧_0,0^†b_𝐧_0,0)Z(a_𝐤^†,a_𝐤) for m≥2. This leads to the simplification,b_𝐧_0,0^†(R)+b_𝐧_0,0(R)≈(cos(θ(R)/2)^2-sin(θ(R)/2)^2)×(b_𝐧_0,0^†(0)+b_𝐧_0,0(0)),and the perturbation acquires the simple form,V(R)≈∑_𝐤,𝐤_1D_𝐤_1^𝐧_0,0(cos(θ(R))g_𝐤^𝐧_0,0(R)-g_𝐤^𝐧_0,0(0))×(b_𝐤_1(0)a_𝐤^†+b_𝐤_1^†(0)a_𝐤^†+a_𝐤b_𝐤_1(0)+b_𝐤_1^†(0)a_𝐤).To write this last expression in terms of the Bogoliubov operators {ξ_𝐤_1^i†(0)ξ_𝐤_2^j†(0)} we start from the transformation ξ⃗=Tb⃗ : [ ξ_𝐤^L(0); ξ_𝐤^U(0); ξ_-𝐤^L†(0); ξ_-𝐤^U†(0) ]=[α_𝐤^Lβ_𝐤^Lγ_𝐤^Lδ_𝐤^L;α_𝐤^Uβ_𝐤^Uγ_𝐤^Uδ_𝐤^U; γ_𝐤^L* δ_𝐤^L* α_𝐤^L* β_𝐤^L*; γ_𝐤^U* δ_𝐤^U* α_𝐤^U* β_𝐤^U* ][ a_𝐤;b_𝐤(0);a_-𝐤^†; b_-𝐤^†(0) ].From the matrix representation of the normalization |α_𝐤^i|^2+|β_𝐤^i|^2-|γ_𝐤^i|^2-|δ_𝐤^i|^2=1 <cit.>, it follows that, TI_-T^†=I_-,where I_-=[1000;0100;00 -10;000 -1 ].We also have that T^-1=I_-T^†I_- and thatT^-1=[α_𝐤^L*α_𝐤^U*-γ_𝐤^L-γ_𝐤^U;β_𝐤^L*β_𝐤^U*-δ_𝐤^L-δ_𝐤^U; -γ_𝐤^L* -γ_𝐤^U* α_𝐤^L α_𝐤^U; -δ_𝐤^L* -δ_𝐤^U* β_𝐤^L β_𝐤^U ].Using Eq. <ref>, we can readily evaluate b⃗=T^-1ξ⃗. From this relationship, the change of the localized operators to the Bogoliubov basis is accomplished. Finally, the matrix elements to compute E^(2)(R_𝐧_0,0,0̃') can be evaluated by means of Wick's theorem,⟨ G(0̃)|_dξ_𝐤_1^lξ_𝐤_2^nξ_𝐤^i†ξ_𝐤'^j†|G(0̃)⟩_d=δ_l,jδ_𝐤_1,𝐤'δ_i,nδ_𝐤_2,𝐤 +δ_n,jδ_𝐤_2,𝐤'δ_m,iδ_𝐤_1,𝐤,leading to Eq. <ref>.achemso | http://arxiv.org/abs/1705.10655v2 | {
"authors": [
"Luis A. Martínez-Martínez",
"Raphael F. Ribeiro",
"Jorge Campos-González-Angulo",
"Joel Yuen-Zhou"
],
"categories": [
"physics.chem-ph",
"cond-mat.mes-hall",
"cond-mat.mtrl-sci"
],
"primary_category": "physics.chem-ph",
"published": "20170525211511",
"title": "Can ultrastrong coupling change ground state chemical reactions?"
} |
On Time-Bandwidth Product of Multi-Soliton PulsesAlexander Span^*, Vahid Aref^†, Henning Bülow^†, and Stephan ten Brink^* ^*Institute of Telecommunications, University of Stuttgart, Stuttgart, Germany ^†Nokia Bell Labs, Stuttgart, GermanyDecember 30, 2023 ======================================================================================================================================================================================================= Multi-soliton pulses are potential candidates for fiber optical transmission where the information is modulated and recovered in the so-called nonlinear Fourierdomain. While this is an elegant technique to account for the channel nonlinearity, the obtained spectral efficiency, so far, is not competitive with the classic Nyquist-based schemes. In this paper, we study the evolution of the time-bandwidth product of multi-solitons as they propagate along the optical fiber.For second and third order soliton pulses, we numerically optimize the pulse shapes toachieve the smallest time-bandwidth product when the phase of the spectral amplitudes is used for modulation. Moreover, we analytically estimate the pulse-duration and bandwidth of multi-solitons in some practically important cases. Those estimations enable us to approximate the time-bandwidth product for higher order solitons.§ INTRODUCTION Advances made over the past decade in coherent optical technology have significantly improved transmission capacities to a point where Kerr nonlinearity once again becomes the limiting factor.The equalization of nonlinear effects is usually very complex and has a limited gain due to the mixing of signal and noise on the channel. The optical channel is usually modeled by the Nonlinear Schrödinger Equation (NLSE)which describes the interplay between Kerr nonlinearity and chromatic dispersion along the fiber.The Nonlinear Fourier Transform (NFT) is a potential way of generating pulsesmatched to a channel governed by the NLSE. It maps a pulse to the nonlinear Fourier spectrum with some beneficial properties. This elegant technique, known also as inverse scattering method <cit.>,has found applications in fiber optics when the on-off keying of first order solitonswas developed in the 1970s <cit.>. Following <cit.>, it has regained attention as coherent technology allows to exploit all degrees of freedom offered by the nonlinear spectrum. Multi-soliton pulses are specific solutions of the NLSE. Using the NFT, an N-th order soliton, denoted here by N-soliton, ismapped to a set of N distinct nonlinear frequencies, called eigenvalues, and the corresponding spectral amplitudes. The key advantage of this representation is that the complex pulse evolution along the fiber can be expressed in terms of spectral amplitudes which evolve linearly in the nonlinear spectrum. Moreover, the transformation is independent of the other spectral amplitudes and eigenvalues. These properties motivate to modulate data using spectral amplitudes. On-off keying of 1-soliton pulses, also called fundamental solitons, has been intensivelystudied two decades ago for different optical applications (see <cit.> and reference therein). To increase spectral efficiency, it has been proposed to modulate multi-solitons <cit.>. One possibility is the independent on-off keying ofN predefined eigenvalues. The concept has been experimentally shown up to using 10 eigenvalues in <cit.>, <cit.>. The other possibility is to modulate thespectral amplitudes of N eigenvalues.The QPSK modulation of spectral amplitudes has been verified experimentally up to 7 eigenvalues in <cit.>. All of these works have a small spectral efficiency.Characterizing the spectral efficiency of multi-soliton pulses is still an open problem.First, the statistics of noisy received pulses in the nonlinear spectrum have not yet been fully understood, even though there are insightful studies for some special cases and under some assumptions <cit.>. Second,the bandwidth and the pulse-duration change as a multi-soliton propagates along a fiber or as spectral amplitudes are modulated. The nonlinear evolution makes it hard to estimate the time-bandwidth product of a multi-soliton.In this paper, we study the evolution of pulse-duration and bandwidth of multi-soliton pulses along an optical fiber link. We numerically optimize the time-bandwidth product of N-soliton pulses for N=2 and 3. The results provide some guidelines for N>3. We focus on scenarios where the phases of N spectral amplitudes are modulated independently. However, our results can also be applied toon-off keying modulation schemes.We assume that the link is long enoughso that the pulse-duration and bandwidth can reach their respective maximum. We also neglect inter-symbol interference. Our results show that the optimization of <cit.>is suboptimal when the evolution along the fiber is taken into account.We further introduce a class of N-solitons which are provably symmetric.A subset of these pulses are already used in <cit.>. We derive an analytic approximation of their pulse-duration. Numerical observations exhibit that the approximation is tight and can serve as a lower-boundfor other N-solitons.To the best of our knowledge, this is the first result on the pulse-duration ofmulti-solitons. We also approximate the time-bandwidth product by lower-bounding the maximal bandwidth.§ PRELIMINARIES ON MULTI-SOLITON PULSESIn this section, we briefly explain the nonlinear Fourier transform (NFT), the characterization of multi-soliton pulses in the corresponding nonlinear spectrum and how they can be generated via the inverse NFT. §.§ Nonlinear Fourier TransformThe pulse propagation along an ideally lossless and noiseless fiber is characterized using the standard Nonlinear Schrödinger Equation (NLSE)∂/∂ zq(t,z)+j∂^2/∂ t^2q(t,z)+2j|q(t,z)|^2q(t,z)=0.The physical pulse Q(τ,ℓ) at location ℓ along the fiber is then described byQ(τ,ℓ)=√(P_0)q(τ/T_0,ℓ|β_2|/2T_0^2)withP_0· T_0^2=|β_2|/γ,where β_2<0 is the chromatic dispersion and γ is the Kerr nonlinearity of the fiber, and T_0 determines the symbol rate. The closed-form solutions of the NLSE (<ref>) can be described in a nonlinear spectrum defined by the following so-called Zakharov-Shabat system <cit.>∂/∂ t(ϑ_1(t;z) ϑ_2(t;z))= ( -jλq(t,z)-q^*(t,z) jλ) (ϑ_1(t;z) ϑ_2(t;z)),with the boundary condition(ϑ_1(t;z) ϑ_2(t;z))→(1 0)exp(-jλ t) fort→ -∞under the assumption that q(t;z)→ 0 decays sufficiently fast as |t|→∞ (faster than any polynomial). The nonlinear Fourier coefficients (Jost coefficients) are defined asa(λ;z) =lim_t→∞ϑ_1(t;z)exp(jλ t) b(λ;z) =lim_t→∞ϑ_2(t;z)exp(-jλ t).The set Ω denotes the set of simple roots of a(λ;z) with positiveimaginary part, which are called eigenvalues as they do not change in terms of z, i.e. λ_k(z)=λ_k. The nonlinear spectrum is usually described by the following two parts: (i) Continuous Part: the spectral amplitude Q_c(λ;z)=b(λ;z)/a(λ;z) for real frequencies λ∈ℝ.(ii) Discrete Part: {λ_k,Q_d(λ_k;z)} whereλ_k∈Ω, i.e. a(λ_k;z)=0, and Q_d(λ_k;z)=b(λ_k;z)/∂ a(λ;z)/∂λ|_λ=λ_k.An N-soliton pulse is described by the discrete part only and the continuous part is equal to zero (for any z).The discrete part contains N pairs of eigenvalue and the corresponding spectral amplitude, i.e. {λ_k,Q_d(λ_k;z)},1≤ k≤ N. An important property of the nonlinear spectrum is its simple linear evolution given by <cit.>Q_d(λ_k;z)=Q_d(λ_k)exp(-4jλ_k^2z), where we define Q_d(λ_k)=Q_d(λ_k;z=0). The transformation is linear anddepends only on its own eigenvalue λ_k. This property motivates for modulation of data over independently evolving spectral amplitudes.Note that there are several methods to compute the nonlinear spectrum by numerically solving the Zakharov-Shabat system. Some of these methods are summarized in <cit.>,<cit.>. §.§ Inverse NFTThe Inverse NFT (INFT) maps the given nonlinear spectrum to the corresponding pulse in time-domain. For the special case of the spectrum without the continuous part, theDarboux Transformation can be applied to generate the corresponding multi-soliton pulse <cit.>.Algorithm <ref> shows the pseudo-code of the inverse transform, as described in <cit.>.It generates an N-soliton q(t) recursively by adding a pair {λ_k,Q_d(λ_k)} in each recursion.The main advantage of this algorithm is that it is exact with a low computational complexity and it can be used to derive some properties of multi-soliton pulses.§.§ Definition of Pulse Duration and BandwidthIn this paper, we consider an N-soliton with the eigenvalues on the imaginary axis, i.e. {λ_k=jσ_k}_k=1^N and σ_k∈ℝ^+.Without loss of generality, we assume that σ_k<σ_k+1. As such an N-soliton propagates along the fiber, the pulse does not disperse and the pulse shape can be repeated periodically. An N-soliton pulse has unbounded support and exponentially decreasing tails in time and (linear) frequency domain. As this pulse is transformed according to the NLSE, e.g. propagation along the ideal optical fiber, its shape can drasticallychange as all Q_d(λ_k;z) are evolved in z. Despite of nontrivial pulse variation and various peak powers, the energy of the pulse remains fixed and equal to E_total=4∑_k=1^NIm{λ_k}.As a result, the pulse-duration and the bandwidth of a multi-soliton pulse are well-defined if they are characterized in terms of energy: the pulse duration T_w (and bandwidth B_w, respectively) is defined as the smallest interval(frequency band) containing E_trunc=(1-ε)E_total of the soliton energy. Note that truncation causes small perturbations of eigenvalues. In practical applications,the perturbations become even larger due to inter-symbol-interference (ISI)when a train of truncated soliton pulses is used for fiber optical communication. Thus,there is a trade-off: ε must be kept small such that the truncation causes only small perturbations, but large enough to have a relatively small time-bandwidth product. Note that truncating a signal in time-domain may slightly change its linear Fourier spectrum in practice. For simplicity, we however computed T_w and B_w with respect to the original pulse as the difference is negligible for ε≪ 1.§ SYMMETRIC MULTI-SOLITON PULSESIn this section, we address the special family of multi-soliton pulseswhich are symmetric in time domain. An application of such solitons for optical fiber transmission is studied in <cit.> where the symmetric 2-solitons are used for data modulation.Let Ω={jσ_1,jσ_2,…,jσ_N} be the set of eigenvalues on the imaginary axis where σ_k∈ℝ^+, for 1≤ k≤ N. The corresponding N-soliton q(t) is a symmetric pulse, i.e. q(t)=q(-t), and keeps this property during the propagation in z, if and only if the spectral amplitudes are chosen as|Q_d,sym(jσ_k)|=2σ_k ∏_m=1;m≠ k^N |σ_k+σ_m/σ_k-σ_m|.Sketch of Proof. The proof is based on Algorithm <ref> with the following steps: (i) g(t)=ρ^*(t)/1+|ρ(t)|^2 is symmetric, ifρ^*(-t)ρ(t)=1.(ii) The update rule (<ref>) preserves the property (<ref>): if ρ(t) and ρ_m^(k-1)(t) satisfy (<ref>), then ρ_m^(k)(t) will satisfy (<ref>) as well. (iii) Because of (<ref>), ρ_k^(0)(t) satisfies (<ref>) for all k. (iv) Using induction, ρ_m^(k) satisfies (<ref>) for all m and k. (v) According to (<ref>) and step (i), q(t) is symmetric. It is already mentioned in <cit.> that (<ref>) leads to a symmetric multi-soliton in amplitude. Theorem <ref> implies that (<ref>)is not only sufficient but also necessary to have q(t)=q(-t). As it is shown in the next section, we numerically observe that these symmetric pulses have the smallest pulse duration[It is correct when ε is small enough.] among all solitons with the same set of eigenvalues Ω (but different |Q_d(λ_k)|). Assuming σ_1=min_k {σ_k}, this minimum pulse-duration can be well approximated byT_sym(ε)≈1/2σ_1(2∑_m=2^Nln(σ_m+σ_1/σ_m-σ_1). .+ln(2/ε)-ln(∑_m=1^Nσ_m/σ_1)),where ε is defined earlier as the energy threshold. The approximation becomes tight as ε→ 0 and is only valid if ε≪σ_1/∑_m=1^Nσ_m. Verification of (<ref>) follows readily by describing an N-soliton by the sum of N terms according to (<ref>), and showing that in the limit |t|→∞, the dominant term behaves as sech(2σ_1 (|t|-t_0)) for some t_0 and all other terms decay exponentially faster. § TIME-BANDWIDTH PRODUCTConsider the transmission of an N-soliton with eigenvalues{jσ_k}_k=1^N over an ideal fiber of length z_L.Each spectral amplitude Q_d(jσ_k;z)=|Q_d(jσ_k;z)|exp(jϕ_k(z))is transformed along the fiber according to (<ref>). Equivalently,|Q_d(jσ_k;z)| =|Q_d(jσ_k;z=0)| ϕ_k(z) =ϕ_k(0)+4σ_k^2zfor z≤ z_L. It means that ϕ_k(z) changes with a distinct speed proportional toσ_k^2. Different phase combinations correspond to different soliton pulse shapes with generally different pulse-duration and bandwidth. It implies that T_w and B_w of a pulse are changing along the transmission. Furthermore, if the ϕ_k(0) are independently modulated for each eigenvalue with a constellation of size M, e.g. M-PSK, this results in M^N initial phase combinations (Nlog_2(M) bits per soliton) associated with different initial pulse shapes. Such transmission scenarios are demonstrated experimentally for M=4, N=2 <cit.> andN=7 <cit.>. To avoid a considerable ISI between neighboring pulses in a train of N-solitons for transmission in time or frequency, we should consider T_w and B_w larger than their respective maximum along the link.For a given set of eigenvalues and fixed |Q_d(jσ_k;z=0)|, the maxima depend on M^N initial phase combinations and the fiber length z_L. To avoid these constraints, we maximize T_w and B_w over all possible phase combinations:T_max=max_ϕ_k, 1≤ k≤ N T_wandB_max=max_ϕ_k, 1≤ k≤ N B_w.These quantities occur in the worst casebut can be reached in their vicinity when N is small, e.g. 2 or 3, or when M is very large, or the transmission length z_L is large enough. In the rest of this section, we address the following fundamental questions: (i) How do T_ max and B_ max change in terms of {jσ_k}_k=1^N and {|Q_d(jσ_k)|}_k=1^N?(ii) What is the smallest time-bandwidth product for a given N, i.e.(T_ maxB_ max)^⋆=min_σ_k,1≤ k≤ Nmin_|Q_d(jσ_k)|,1≤ k≤ NT_ maxB_ maxand which is the optimal choice for{jσ_k^⋆}_k=1^N and{|Q_d^⋆(jσ_k^⋆)|}_k=1^N.The following properties preserving the time-bandwidth product decrease the number of parameters to optimize: (i) If q(t) has eigenvalues {jσ_k}_k=1^N,then 1/σ_1 · q(t/σ_1) will have eigenvalues {jσ_k/σ_1}_k=1^N with the same time-bandwidth product. It implies that T_maxB_max onlydepends on the N-1 eigenvalue ratios σ_k/σ_1.(ii) If {ϕ_k}_k=1^N corresponds to q(t), then {ϕ_k-ϕ_1}_k=1^N corresponds to q(t)exp(jϕ_1). Thus, we assume ϕ_1=0.(iii) Instead of directly optimizing {|Q_d(jσ_k)|}_k=1^N, it is equivalent to optimize η_k>0 defined by|Q_d(jσ_k)|= η_k |Q_d,sym(jσ_k)|.Using {η_k}_k=1^N has two advantages. The first one is the generalization of Theorem <ref>. If {η_k}_k=1^N corresponds to q(t), then {1/η_k}_k=1^N corresponds to q(-t). The proof is similar to the one of Theorem <ref>. Moreover, {e^-2σ_kt_0η_k}_k=1^N corresponds to q(t+t_0). Thus, it suffices to assume η_1=1 andη_2∈(0,1]. §.§ Optimization of Spectral AmplitudesConsider a given set of eigenvalues Ω={jσ_k}_k=1^N. We want to optimize {η_k}_k=2^N to minimize T_ maxB_ max. Recall that {|Q_d(jσ_k)|}_k=1^N, and thus {η_k}_k=1^N do not change along z.We present the optimization method for N=2.In this case, there are two parameters to optimize: ϕ_2 and η_2∈(0,1]. Consider a given energy threshold ε. For each chosen η_2,we find T_ max(ε) andB_ max(ε) by exhaustive search. The phase ϕ_2∈[0,2π) is first quantized uniformly by 64 phases. At each phase,a 2-soliton is generated using Algorithm <ref> and then T_w(ε) and B_w(ε) are computed.To estimate T_ max(ε),another round of search is performed with a finer resolution around the quantized phase with the largest T_w(ε). Similarly, B_ max(ε) is estimated.Fig. <ref> illustrates T_ max(ε) and B_ max(ε)in terms of log(η_2) for different energythresholds ε when Ω={1/2j,1j}. We also depict B_min(ε), the minimum bandwidth of 2-soliton pulses with a given η_2 and various ϕ_2. Fig. <ref> indicates the following features that we observed for any pairs of {jσ_1,jσ_2}. We can see that for any ε, the smallest T_max is attained at η_2=1 (log(η_2)=0) which corresponds to the symmetric 2-soliton defined inSec. <ref>. We also observe that B_max reaches the largest value at η_2=1 while B_min reaches its minimum.As log(η_2) decreases, T_max increases gradually up to some point and then it linearly increases in |log(η_2)|. The behaviour of B_max is the opposite. It decreases very fast in |log(η_2)| up to some η_2 and then converges slowly to the bandwidth defined by the 1-soliton spectrum with λ=jσ_2. In fact, we have two separate 1-solitons without any interaction when η_2=0. As η_2 increases to 1, the distance between these two 1-solitons decreases, resulting in more nonlinear interaction but smaller T_max.The largest B_max-B_min at η_2=1 indicates the largest amount of interaction.The above features seem general for N-solitons. In particular, T_max becomes minimum ifthe N-soliton is symmetric. Moreover, B_max can be lower-bounded by B_sep(ε)=2σ_N/π^2(ln(2/ε)-ln(∑_k=1^Nσ_k/σ_N))with σ_N=max_k {σ_k}. The bound becomes tight when an N-soliton is the linear superposition of N separate 1-solitons.We performed such a numerical optimization for N=2 and N=3 and for different {σ_k/σ_1}_k=2^N. For each ε, we found the optimal {η_k^⋆}_k=2^N with the smallest T_max(ε)B_max(ε).§.§ Optimization of EigenvaluesIn general, an N-soliton has a larger T_maxB_max than a 1-soliton but it has also N times, e.g. Q_d(jσ_k), more dimensions for encoding data.To have a fair comparison,we use a notion of “time-bandwidth product per eigenvalue” defined asT · B_N({σ_k/σ_1}_k=2^N)=1/NT_maxB_max({σ_k/σ_1}_k=2^N,{η_k^⋆}_k=2^N)where T_maxB_max is already optimized in terms of {η_k}, seperately for each eigenvalue combination. This is an important parameter as the spectral efficiency will be𝒪(1/T · B_N). For a 1-soliton with “sech” shape in time and frequency domain, we have T · B_1=T_w(ε)B_w(ε)=π^-2ln^2(2/ε),where ε is the energy threshold defined in Section <ref>.For N=2 and N=3, we numerically optimized T_max(ε)B_max(ε) for different values of {σ_k/σ_1}_k=2^N and {η_k}_k=2^N. Fig. <ref>-(a) shows the numerical optimization of T · B_2 in terms of σ_2/σ_1 for different choices of ε where the best {η_k^⋆}_k=2^N were chosen for each eigenvalue ratio. We normalized T · B_2 by T · B_1 to see how much the “time-bandwidth product per eigenvalue” can be decreased.Fig. <ref>-(b) shows a similar numerical optimization for N=3 and ε=10^-4. We have the following observations:(i) T · B_N is sensitive to the choice of eigenvalues. For instance, equidistant eigenvalues, i.e. σ_k=kσ_1, are a bad choice in terms of spectral efficiency. (ii) The ratio T · B_N/T · B_1 gets smaller as ε vanishes. The intuitive reason is that as ε→ 0, we get T_ max≈1/2σ_1ln(2/ε) (see (<ref>)) which is the pulse-duration of the 1-soliton. (iii) For a practical value of ε∼ 10^-4-10^-3,T · B_N decreases very slowly in N. Moreover, the optimal σ_k^⋆ are close. This can make the detection challenging in presence of noise. For ε=10^-4, T · B_2/T · B_1=0.87for σ_2^⋆/σ_1^⋆=1.11 T · B_3/T · B_1=0.83 for σ_2^⋆/σ_1^⋆=1.28,σ_3^⋆/σ_1^⋆=1.35(iv) Choosing the above optimal {σ_k^⋆/σ_1^⋆}, and the optimal{|Q_d^⋆(σ_k^⋆)}, the resulting solitons for N=2,3 are shown in Fig. <ref> for different phase combinations and two energy thresholds ε. This figure gives some guidelines for a larger N: the optimal N-soliton has eigenvalues close to each other and significantly seperated pulse centers, why the optimum pulse looks similar to a train of 1-solitons with eigenvalues close to each other. The pulse centers should be close to minimize T_ max but not too close to avoid a large interaction which comes along with a growth of B_ max. For ε≪ 1, an estimate on T · B_N at optimal {η_k^*} can be given by (<ref>), where T_ max and B_ max are estimated by T_ sym(ε) and B_ sep(ε), respectively (see Fig. <ref>).T · B_N≈T_ sym(ε) B_sep(ε)/N, For the second order case, these approximations for various ε are plotted in Fig. <ref>-(a) by dashed lines. We see that the approximation becomes better for small ε. This approximation can be used to predict T · B_N for a large N.§ CONCLUSION We studied the evolution of the pulse-durationand the bandwidth of N-soliton pulses along the optical fiber. We focused on solitons with eigenvalues located on the imaginary axis. The class of symmetric soliton pulses was introduced and an analytical approximation of their pulse-duration was derived. The phase of the spectral amplitudes was assumed to be used for modulation while their magnitudes were kept fixed. We numerically optimized the location of eigenvalues and the magnitudes of spectral amplitudes for 2- and 3-solitons in order to minimize the time-bandwidth product. It can be observed thatthe time-bandwidth product per eigenvalue improves in the soliton order N, but very slowly. Another observation is, that the optimal N-soliton pulse looks similar to a train of first-order pulses.There are some remarks about our optimization. As an N-soliton propagates, the phases of the spectral amplitudes change with different speeds. We assumed that all possible combinations of phases occur during transmission. This is the worst case scenario which is likely to happen for N=2 and N=3 but becomes less probable for large N. Moreover, the same magnitudes of spectral amplitudes are used for any phase combination while they can be tuned according to the phases. Without these assumptions, the time-bandwidth productwill decrease. However, it becomes harder to estimate as there are many more parameters to optimize. IEEEtran | http://arxiv.org/abs/1705.09468v1 | {
"authors": [
"Alexander Span",
"Vahid Aref",
"Henning Buelow",
"Stephan ten Brink"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170526080522",
"title": "On Time-Bandwidth Product of Multi-Soliton Pulses"
} |
1Department of Physics and Astronomy, University of California,Riverside, 900 University Avenue, Riverside, CA 92521, USA; [email protected] 2Geneva Observatory, Université de Genève, Chemin des Maillettes 51, 1290 Versoix, Switzerland 3Yale Center for Astronomy and Astrophysics, Yale University, New Haven, CT 06511, USA 4Leiden Observatory, Leiden University, NL-2300 RA Leiden, Netherlands 5UCO/Lick Observatory, University of California, Santa Cruz, 1156 High St, Santa Cruz, CA 95064, USA 6Cahill Center for Astronomy and Astrophysics, California Institute of Technology, MS 249-17, Pasadena, CA 91125, USA 7Institute for Astronomy, ETH Zurich, 8092 Zurich, Switzerland 8Astronomy Centre, Department of Physics and Astronomy, Universityof Sussex, Brighton, BN1 9QH, UK 9Max Planck Institute for Extraterrestrial Physics, Giessenbachstrasse, 85741 Garching bei München, Germany 10Alfred P. Sloan Research FellowDRAFT: December 30, 2023We use a newly assembled large sample of 3,545 star-forming galaxies with secure spectroscopic, grism, and photometric redshifts at z=1.5-2.5 to constrain the relationship between UV slope (β) and dust attenuation (L_ IR/L_ UV≡ IRX).Our sample benefits from the combination of deep Hubble WFC3/UVIS photometry from the Hubble Deep UV (HDUV) Legacy survey and existing photometric data compiled in the 3D-HST survey.Our sample significantly extends the range of UV luminosity and β probed in previous samples of UV-selected galaxies, including those as faint as M_1600=-17.4 (≃ 0.05L^∗_ UV) and having -2.6β 0.0.IRX is measured using stacks of deep Herschel/PACS 100 and 160 μm data, and the results are compared with predictions of the IRX-β relation for different assumptions of the stellar population model and dust obscuration curve.Stellar populations with intrinsically blue UV spectral slopes necessitate a steeper attenuation curve in order reproduce a given IRX-β relation.We find that z=1.5-2.5 galaxies have an IRX-β relation that is consistent with the predictions for an SMC extinction curve if we invoke sub-solar (0.14Z_⊙) metallicity models that are currently favored for high-redshift galaxies, while the commonly assumed starburst attenuation curve over-predicts the IRX at a given β by a factor of 3.The IRX of high-mass M_∗> 10^9.75 M_⊙ galaxies is a factor of >4 larger than that of low-mass galaxies, lending support for the use of stellar mass as a proxy for dust attenuation.Separate IRX-L_ UV relations for galaxies with blue and red β conflate to give an average IRX that is roughly constant with UV luminosity for L_ UV 3× 10^9 L_⊙.Thus, the commonly observed trend of fainter galaxies having bluer β may simply reflect bluer intrinsic UV slopes for such galaxies, rather than lower dust obscurations.Taken together with previous studies, we find that the IRX-β distribution for young and low-mass galaxies at z 2 implies a dust curve that is steeper than that of the SMC, suggesting a lower dust attenuation for these galaxies at a given β relative to older and more massive galaxies.The lower dust attenuations and higher ionizing photon output implied by low metallicity stellar population models point to Lyman continuum production efficiencies, ξ_ ion, that may be elevated by a factor of ≈ 2 relative to the canonical value for L^∗ galaxies, aiding in their ability to keep the universe ionized at z∼ 2. § INTRODUCTIONThe ultraviolet (UV) spectral slope, β, where f_λ∝λ^β, is by far the most commonly used indicator of dust obscuration—usually parameterized as the ratio of the infrared-to-UV luminosity, L_ IR/L_ UV, or “IRX” <cit.>—in moderately reddened high-redshift (z 1.5) star-forming galaxies.The UV slope can be measured easily from the same photometry used to select galaxies based on the Lyman break, and the slope can be used as a proxy for the dust obscuration in galaxies (e.g., ) whose dust emission is otherwise too faint to directly detect in the mid- and far-infrared (e.g., ).Generally, these studies have indicated that UV-selected star-forming galaxies at redshifts 1.5 z 3.0 follow on average the relationship between UV slope and dust obscuration (i.e., the IRX-β relation) found for local UV starburst galaxies (e.g., ; c.f., ), though with some deviations that depend on galaxy age <cit.>, bolometric luminosity (e.g., ), stellar mass <cit.>, and redshift <cit.>.Unfortunately, typical star-forming (L^∗) galaxies at these redshifts are too faint to directly detect in the far-infrared.As such, with the exception of individual lensed galaxy studies <cit.>, most investigations that have explored the relation between UV slope and dust obscuration for moderately reddened galaxies have relied on stacking relatively small numbers of objects and/or used shorter wavelength emission—such as that arising from polycyclic aromatic hydrocarbons (PAHs)—to infer infrared luminosities.New avenues of exploring the dustiness of high-redshift galaxies have been made possible with facilities such as the Atacama Large Millimeter Array (ALMA), allowing for direct measurements of either the dust continuum or far-IR spectral features for more typical star-forming galaxies in the distant universe <cit.>. Additionally, the advent of large-scale rest-optical spectroscopic surveys of intermediate-redshift galaxies at 1.4 z 2.6—such as the 3D-HST <cit.>, the MOSFIRE Deep Evolution Field (MOSDEF; ), and the Keck Baryonic Structure surveys (KBSS; )—have enabled measurements of obscuration in individual high-redshift star-forming galaxies using Balmer recombination lines (e.g., ).While these nebular line measurements will be possible in the near future for z 3 galaxies with the James Webb Space Telescope (JWST), the limited lifetime of this facility and the targeted nature of both ALMA far-IR and JWST near- and mid-IR observations means that the UV slope will remain the only easily accessible proxy for dust obscuration for large numbers of individual typical galaxies at z 3 in the foreseeable future.Despite the widespread use of the UV slope to infer dust attenuation, there are several complications associated with its use. First, the UV slope is sensitive to metallicity and star-formation history (e.g., ).Second, there is evidence that the relationship between UV slope and dust obscuration depends on stellar mass and/or age (e.g., ), perhaps reflecting variations in the shape of the attenuation curve.Third, the measurement of the UV slope may be complicated by the presence of the 2175 Åabsorption feature <cit.>.Fourth, as noted above, independent inferences of the dust attenuation in faint galaxies typically involve stacking mid- and far-IR data, but such stacking masks the scatter in the relationship between UV slope and obscuration.Quantifying this scatter can elucidate the degree to which the attenuation curve may vary from galaxy-to-galaxy, or highlight the sensitivity of the UV slope to factors other than dust obscuration.In general, the effects of age, metallicity, and star-formation history on the UV slope may become important for ultra-faint galaxies at high redshift which have been suggested to undergo bursty star formation (e.g., ).Obtaining direct constraints on the dust obscuration of UV-faint galaxies is an important step in evaluating the viability of the UV slope to trace dustiness, quantifying the bolometric luminosities of ultra-faint galaxies and their contribution to the global SFR and stellar mass densities, assessing possible variations in the dust obscuration curve over a larger dynamic range of galaxy characteristics (e.g., star-formation rate, stellar mass, age, metallicity, etc.), and discerning the degree to which the UV slope may be affected by short timescale variations in star-formation rate.Separately, recent advances in stellar populations models that include realistic treatments of stellar mass loss, rotation, and multiplicity <cit.> can result in additional dust heating from ionizing and/or recombination photons. Moreover, the intrinsic UV spectral slopes of high-redshift galaxies with lower stellar metallicities may be substantially bluer <cit.> than what has been typically assumed in studies of the IRX-β relation.Thus, it seems timely to re-evaluate the IRX-β relation in light of these issues.With this in mind, we use a newly assembled large sample of galaxies with secure spectroscopic or photometric redshifts at 1.5≤ z≤ 2.5 in the GOODS-North and GOODS-South fieldsto investigate the correlation between UV slope and dust obscuration. Our sample takes advantage of newly acquired Hubble UVIS F275W and F336W imaging from the HDUV survey (Oesch et al. 2017, submitted) which aids in determining photometric redshifts when combined with existing 3D-HST photometric data.This large sample enables precise measurements of dust obscuration through the stacking of far-infrared images from the Herschel Space Observatory, and also enables stacking in multiple bins of other galaxy properties (e.g., stellar mass, UV luminosity) to investigate the scatter in the IRX-β relation.We also consider the newest stellar population models—those which may be more appropriate in describing very high-redshift (z 2) galaxies—in interpreting the relationship between UV slope and obscuration.lc0pc Sample CharacteristicsProperty Value Fields GOODS-N, GOODS-S Total area ∼ 329 arcmin^2 Area with HDUV imaging ∼ 100 arcmin^2 UV/Optical photometry 3D-HST Catalogsa and HDUVb F275W and F336W Mid-IR imaging Spitzer GOODS Imaging Programc Far-IR imaging GOODS-Herscheld and PEPe Surveys Optical depth of sample H ≃ 27 UV depth of sample m_ UV≃ 27 Total number of galaxies 4,078 Number of galaxies with far-IR coverage 3,569 Final number (excl. far-IR-detected objects) 3,545 β Range -2.55≤β≤ 1.05 (⟨β⟩ = -1.71) a<cit.>. bOesch et al., submitted. cPI: Dickinson. d<cit.>. ePI: Lutz, <cit.>.The outline of this paper is as follows.In Section <ref>, we discuss the selection and modeling of stellar populations of galaxies used in this study.The methodology used for stacking the mid- and far-IR Spitzer and Herschel data is discussed in Section <ref>.In Section <ref>, we calculate the predicted relationships between IRX and β for different attenuation/extinction curves using energy balance arguments.These predictions are compared to our (as well as literature) stacked measurements of IRX in Section <ref>.In this section, we also consider the variation of IRX with stellar masses, UV luminosities, and the ages of galaxies, as well as the implications of our results for modeling the stellar populations and inferring the ionizing efficiencies of high-redshift galaxies.AB magnitudes are assumed throughout <cit.>, and we use a <cit.> initial mass function (IMF) unless stated otherwise. We adopt a cosmology with H_0=70 km s^-1 Mpc^-1, Ω_Λ=0.7, and Ω_ m=0.3.§ SAMPLE AND IR IMAGING §.§ Parent Sample A few basic properties of our sample are summarized in Table <ref>.Our sample of galaxies was constructed by combined the publicly-available ground- and space-based photometry compiled by the 3D-HST survey <cit.> with newly obtained imaging from the Hubble Deep UV (HDUV) Legacy Survey (GO-13871; Oesch et al. 2017, submitted).The HDUV survey imaged the two GOODS fields in the F275W and F336W bands to depths of ≃ 27.5 and 27.9 mag, respectively (5σ; 04 diameter aperture), with the UVIS channel of the Hubble Space Telescope WFC3 instrument.A significant benefit of the HDUV imaging is that it allows for the Lyman break selection of galaxies to fainter UV luminosities and lower redshifts than possible from ground-based surveys (Oesch et al. 2017, submitted), and builds upon previous efforts to use deep UVIS imaging to select Lyman break galaxies at z∼ 2 <cit.>.The reduced UVIS images, covering ≈ 100 arcmin^2, include previous imaging obtained by the CANDELS <cit.> and UVUDF surveys <cit.>. §.§ Photometry and Stellar Population Parameters Source Extractor <cit.> was used to measure photometry on the UVIS images using the detection maps for the combined F125W+F140W+F160W images, as was done for the 3D-HST photometric catalogs <cit.>.The publicly-available 3D-HST photometric catalogs were then updated with the HDUV photometry—i.e., such that the updated catalogs contain updated photometry for objects lying in the HDUV pointings as well as the original set photometry for objects lying outside the HDUV pointings.This combined dataset was then used to calculate photometric redshifts using EAZY <cit.> and determine stellar population parameters (e.g., stellar mass) using FAST <cit.>.Where available, grism and external spectroscopic redshifts were used in lieu of the photometric redshifts when fitting for the stellar populations.These external spectroscopic redshifts are provided in the 3D-HST catalogs <cit.>.We also included 759 spectroscopic redshifts for galaxies observed during the 2012B-2015A semesters of the MOSDEF survey <cit.>.For the stellar population modeling, we adopted the <cit.> stellar population models for Z=0.019 Z_⊙, a delayed-τ star-formation history with 8.0≤log[τ/ yr] ≤ 10.0, a <cit.> initial mass function (IMF), and the <cit.> dust attenuation curve with 0.0≤ A_ V≤ 4.0.[Below, we consider the effect of stellar population age on the IRX-β relations.In that context, the ages derived for the vast majority of galaxies in our sample are within δlog[ Age/yr] ≃ 0.1 dex to those derived assuming a constant star-formation history.]We imposed a minimum age of 40 Myr based on the typical dynamical timescale for z∼ 2 galaxies <cit.>.The UV slope for each galaxy was calculated both by (a) fitting a power law through the broadband photometry, including only bands lying redward of the Lyman break and blueward of rest-frame 2600 μm; and (b) fitting a power law through the best-fit SED points that lie in wavelength windows spanning rest-frame 1268≤λ≤ 2580 Å, as defined in <cit.>.Method (a) includes a more conservative estimate for the errors in β, but generally the two methods yielded values of the UV slope for a given galaxy that were within δβ≃ 0.1 of each other.We adopted the β calculated using method (a) for the remainder of our analysis, and note that in Section <ref>, we consider the value of β using windows lying strictly blueward of ≈ 1800 Å.§.§ Criteria for Final Sample The photometric catalogs, along with those containing the redshifts and stellar population parameters, were used to select galaxies based on the following criteria.First, the object must have a Source Extractor “class star” parameter < 0.95, or observed-frame U-J and J-K colors that reside in the region occupied by star-forming galaxies as defined by <cit.>—these criteria ensure the removal of stars.Second, the galaxy must have a spectroscopic or grism redshift, or 95% confidence intervals in the photometric redshift, that lie in the range 1.5≤ z≤ 2.5.Note that the high photometric redshift confidence intervals required for inclusion in our sample naturally selects those objects with H 27.Third, the object must not have a match in X-ray AGN catalogs compiled for the GOODS-North and GOODS-South fields (e.g., ).Additionally, we use the <cit.> Spitzer IRAC selection to isolate any infrared-bright AGN.While the X-ray and IRAC selections may not identify all AGN at the redshifts of interest, they are likely to isolate those AGN that may significantly influence our stacked far-IR measurements.Fourth, the object must not have rest-frame U-V and V-J colors that classify it as a quiescent galaxy <cit.>. The object is further required to have a specific star-formation rate sSFR0.1 Gyr^-1.These criteria safeguard against the inclusion of galaxies where β may be red due to the contribution of older stars to the near-UV continuum, or where dust heating by older stars may become significant.Fifth, to ensure that the sample is not biased towards objects with red U-H colors at faint U magnitudes (owing to the limit in H-band magnitude mentioned previously), the galaxy must have an apparent magnitude at [1+z]× 1600 Åof ≤ 27.0 mag. This limit still allows us to include galaxies with absolute magnitudes as faint as M_ 1600≃ -17.4.These criteria result in a sample of 4,078 galaxies. §.§ Spitzer and Herschel Imaging We used the publicly available Spitzer/MIPS 24 μm and Herschel/PACS 100 and 160 μm data in the two GOODS fields for our analysis.The 24 μm data come from the Spitzer GOODS imaging program (PI: Dickinson), and trace the dust-sensitive rest-frame 7.7 μm emission feature for galaxies at 1.5≤ z≤ 2.5 (e.g., ).The observed 24 μm fluxes of z∼ 2 galaxies have been used extensively in the past to derive infrared luminosities (L_ IR) given the superior sensitivity of these data to dust emission when compared with observations taken at longer wavelengths (roughly a factor of three times more sensitive than Herschel/PACS to galaxies of a given L_ IR at z∼ 2; ).However, a number of observations have highlighted the strong variation in L_ 7.7/L_ IR with star-formation rate <cit.>, star-formation-rate surface density <cit.>, and gas-phase metallicity and ionization parameter at high-redshift <cit.>.As such, while we stacked the 24 μm data for galaxies in our sample, we did not consider these measurements when calculating L_ IR.In Appendix <ref>, we consider further the variation in L_ 7.7/L_ IR with other galaxy characteristics.The Herschel data come from the GOODS-Herschel Open Time Key Program <cit.> and the PACS Evolutionary Probe (PEP) Survey (PI: Lutz; ), and probe the rest-frame ≃ 30-65 μm dust continuum emission for galaxies at 1.5≤ z≤ 2.5.We chose not to use the SPIRE data given the much coarser spatial resolution of these data (FWHM18) relative to the 24 μm (FWHM≃ 5 4), 100 μm (FWHM≃ 6 7), and 160 μm (FWHM≃ 11) data.The pixel scales of the 24, 100, and 160 μm images are 1 2, 1 2, and 2 4, respectively.As noted above, only the 100 and 160 μm data are used to calculate L_ IR.Of the 4,078 galaxies in the sample discussed above, 3,569 lie within the portions of the Herschel imaging that are 80% complete to flux levels of 1.7 and 5.5 mJy for the 100 and 160 μm maps in GOODS-N, respectively, and 1.3 and 3.9 mJy for the 100 and 160 μm maps in GOODS-S, respectively.Of these galaxies, 24 (or 0.67%) are directly detected with signal-to-noise S/N>3 in either the 100 or 160 μm images.As we are primarily concerned with constraining the IRX-β relation for moderately reddened galaxies, we removed all directly-detected Herschel objects from our sample—the latter are very dusty star-forming galaxies at the redshifts of interest with L_ IR 10^12 L_⊙.The very low frequency of infrared-luminous objects among UV-faint galaxies in general could have been anticipated from the implied low number density of L_ IR 10^12 L_⊙ objects from the IR luminosity function <cit.> and the high number density of UV-faint galaxies inferred from the UV luminosity function <cit.> at z∼2.The inclusion of such dusty galaxies does not significantly affect our stacking analysis owing to the very small number of such objects.Excluding these dusty galaxies, our final sample consists of 3,545 galaxies with the redshift and absolute magnitude distributions shown in Figure <ref>. §.§ Summary of Sample To summarize, we have combined HDUV UVIS and 3D-HST catalogued photometry to constrain photometric redshifts for galaxies in the GOODS fields and isolate those star-forming galaxies with redshifts z=1.5-2.5 down to a limiting near-IR magnitude of ≃ 27 AB (Table <ref>).All galaxies are significantly detected (with S/N>3) down to an observed optical (rest-frame UV) magnitude of 27 AB.Our sample includes objects with spectroscopic redshifts in the aforementioned range wherever possible.This sample is then used as a basis for stacking deep Herschel data, as discussed in the next section.One of the most beneficial attributes of our sample is that it contains the largest number of UV-faint galaxies—extending up to ≈ 3 magnitudes fainter than the characteristic absolute magnitude at z∼ 2.3 (M^∗_1700=-20.70; ) and z∼ 1.9 (M^∗_1500=-20.16; )—with robust redshifts at 1.5≤ z≤ 2.5 assembled to date (Figure <ref>).The general faintness of galaxies in our sample is underscored by their very low detection rate (S/N>3) at 24 μm—85 of 3,545 galaxies, or ≈ 2.4%—compared to the ≈ 40% detection rate for rest-frame UV-selected galaxies with R≤ 25.5 <cit.>.Consequently, unlike most previous efforts using ground-based UV-selected samples of limited depth, the present sample presents a unique opportunity to evaluate the IRX-β relation for the analogs of the very faint galaxies that dominate the UV and bolometric luminosity densities at z≫3 (e.g., ), but for which direct constraints on their infrared luminosities are difficult to obtain.lccccccccc0pc Stacked Fluxes and Infrared and UV Luminosities⟨ f_24⟩c ⟨ f_100⟩c ⟨ f_160⟩c ⟨ L_7.7⟩d ⟨ L_ IR⟩d ⟨ L_ UV⟩d Sample Na ⟨ z⟩b ⟨β⟩b [μJy] [μJy] [μJy] [10^10 L_⊙] [10^10 L_⊙] [10^10 L_⊙] All 3545 1.94 -1.71 1.54±0.14 29±6 62±17 0.26±0.03 2.1±0.4 0.80 M_1600 bins:M_1600≤ -21 81 2.12 -1.74 4.83±0.96 177±30 377±93 1.00±0.20 17.1±2.4 6.73-21<M_1600≤ -20 575 2.07 -1.68 4.37±0.28 87±13 171±43 0.86±0.06 7.6±1.0 2.92-20<M_1600≤ -19 1390 1.99 -1.67 2.33±0.20 38±8 84±25 0.41±0.03 3.1±0.6 1.26M_1600>-19 1499 1.92 -1.72 1.00±0.16 31±9 54±24 0.16±0.03 2.0±0.5 0.48 β bins:β≤ -1.70 2084 1.96 -2.04 0.52±0.16 5±7 21±18 0.09±0.03 <1.4 0.77-1.70<β≤ -1.40 722 1.92 -1.56 1.89±0.41 43±13 86±37 0.31±0.07 2.9±0.7 0.95-1.40<β≤ -1.10 345 1.94 -1.26 3.92±0.55 52±18 103±56 0.65±0.09 3.7±1.1 0.93-1.10<β≤ -0.80 205 1.93 -0.97 7.07±0.53 80±25 173±73 1.15±0.09 5.7±1.4 0.81β>-0.80 189 1.90 -0.31 5.09±0.62 167±23 340±63 0.80±0.10 11.0±1.2 0.59 M_1600 & β bins: M_1600≤ -19 + β≤ -1.4 1616 2.01 -1.86 1.86±0.21 25±9 51±21 0.33±0.04 1.9±0.5 1.58M_1600≤ -19 + β > -1.4 430 1.97 -1.02 7.20±0.50 117±19 288±46 1.25±0.09 9.5±1.0 1.47M_1600> -19 + β≤ -1.4 1190 1.92 -1.97 0.36±0.21 13±7 26±23 0.06±0.03 <1.0 0.48M_1600> -19 + β > -1.4 309 1.90 -0.79 3.11±0.38 95±19 176±75 0.49±0.06 6.3±1.2 0.48 Stellar Mass & β bins:log[M_∗/ M_⊙]≤ 9.75 2571 1.94 -1.88 0.75±0.14 10±7 17±20 0.13±0.03 <1.2 0.71+β≤ -1.4 2385 1.94 -1.95 0.63±0.19 11±6 28±15 0.10±0.03 <1.0 0.72+β>-1.4 186 1.89 -1.12 2.95±0.76 19±25 72±73 0.47±0.12 <4.0 0.57log[M_∗/ M_⊙]>9.75 974 1.96 -0.92 4.93±0.40 111± 12 229±36 0.84±0.07 8.3±0.7 1.22+β≤ -1.4 421 2.04 -1.61 4.22±0.42 59±14 118±46 0.80±0.07 5.1±1.1 2.26+β>-1.4 553 1.94 -0.72 5.23±0.48 132±14 263±44 0.87±0.09 9.4±0.8 0.90 Age bins:log[ Age/ yr]≤ 8.00 81 1.96 -1.49 0.32±0.92 62±39 135±91 <0.51 <6.3 0.55 log[ Age/ yr]>8.00 3464 1.94 -1.71 1.43±0.22 25±6 52±19 0.23±0.04 1.8±0.4 0.81aNumber of objects in the stack. bMean redshift and UV slope of objects in the stack. cStacked 24, 100, and 160 μm fluxes. dStacked 8 μm and total infrared luminosities, and the mean UV luminosity of objects in the stack.§ STACKING METHODOLOGYTo mitigate any systematics in the stacked fluxes due to bright objects proximate to the galaxies in our sample, we performed the stacking on residual images that were constructed as follows.[As discussed in <cit.>, stacking on the science images themselves yields results similar to those obtained by stacking on the residual images.]We used the 24 μm catalogs and point spread functions (PSFs) included in the GOODS-Herschel data release to subtract all objects with S/N>3 in the 24 μm images, with the exception of the 85 objects in our sample that are directly detected at 24 μm.Objects with S/N>3 in the 24 μm images were used as priors to fit and subtract objects with S/N>3 in the 100 and 160 μm images.The result is a set of residual images at 24, 100, and 160 μm for both GOODS fields.For each galaxy contributing to the stack, we extracted from the 24, 100, and 160 μm residual images regions of 41× 41, 52× 52, and 52× 52 pixels, respectively, centered on the galaxy.The sub-images were then divided by the UV luminosity, L_ UV=ν L_ν at 1600 Å, of the galaxy, and these normalized sub-images for each band were then averaged together using 3 σ clipping for all the objects in the stack.We performed PSF photometry on the stacked images to measure the fluxes.Because the images are normalized by L_ UV, the stacked fluxes are directly proportional to the average IRX.The corresponding weighted average fluxes in each band (⟨ f_24⟩, ⟨ f_100⟩, and ⟨ f_160⟩), where the weights are 1/L_ UV, were computed by multiplying the stacked fluxes by the weighted average UV luminosity of galaxies in the stack.The measurement uncertainties of these fluxes were calculated as the 1 σ dispersion in the fluxes obtained by fitting PSFs at random positions in the stacked images, avoiding the stacked signal itself.While stacking on residual images aids in minimizing the contribution of bright nearby objects to the stacked fluxes, this method will not account for objects that are blended with the galaxies of interest in the Herschel/PACS imaging.This presents a particular challenge in our case, where the galaxies are selected from HST photometry, as a single galaxy (e.g., as observed from the ground) may be resolved with HST into several subcomponents, each of which is of course unresolved in the Herschel imaging but each of which will contribute to the stacked flux.Galaxies that are resolved into multiple subcomponents will contribute more than once to the stack, resulting in an over-estimate of the stacked far-IR flux.This effect is compounded by that of separate galaxies contributing more than once to the stack if they happen to be blended at the Herschel/PACS resolution.This bias was quantified as follows.For a given band, we used the PSF to generate N galaxies, where N is the number of galaxies in the stack, each having a flux equal to the weighted average flux of the stacked signal.These simulated galaxies were added to the residual image at locations that were shifted from those of the real galaxies by offsets δ x and δ y in the x- and y-directions, respectively, where the offsets were chosen randomly.This ensures that the spatial distribution of the simulated galaxies is identical to that of the real galaxies.We then stacked at the locations of the simulated galaxies and compared the simulated and recovered stacked fluxes.This was done 100 times, each time with a different pair of (randomly chosen) δ x and δ y.The average ratio of the simulated and recovered fluxes, or the bias factor, from these 100 simulations varied from ≈ 0.60-0.90, depending on the number of galaxies contributing to the stack and the particular band.These simulations were performed for every band and for every stack in our analysis, and the stacked fluxes of the galaxies in our sample were multiplied by the bias factors calculated from these simulations.To further investigate this bias, we also stacked all galaxies in our sample that had no HST-detected object within 3 35, corresponding to the half-width at half-maximum of the Herschel/PACS 100 μm PSF.While this criterion severely restricts the size of the sample to only 465 objects, it allowed us to verify the bias factors derived from our simulations.Stacking these 465 objects yielded weighted average fluxes at 24, 100, and 160 μm that are within 1 σ of the those values obtained for the entire sample once the bias factors are applied.[While the 160 μm PSF has a half-width at half-maximum that is larger than the exclusion radius of 3 35, the agreement in the average f_100/f_160 ratio, or far-infrared color, between the stack of the full sample and that of the 465 galaxies suggests that the bias factors also recover successfully the average 160 μm stacked flux.]Infrared luminosities were calculated by fitting the <cit.> “main sequence” dust template to the stacked ⟨ f_100⟩ and ⟨ f_160⟩ fluxes.We chose this particular template as it provided the best match to the observed infrared colors f_100/f_160 of the stacks, though we note that the adoption of other templates (e.g., ) results in L_ IR that vary by no more than ≈ 50% from the ones calculated here (seefor a detailed comparison of L_ IR computed using different dust templates).Upper limits in L_ IR are quoted in cases where L_ IR divided by the modeled uncertainty is >3.In a few instances, ∼ 2 σ detections of both the 100 and 160 μm stacked fluxes yield a modeled L_ IR that is significant at the 3σ level.The mean UV slope of objects contributing to the stack was computed as a weighted average of the UV slopes of individual objects where, again, the weights are 1/L_ UV.These same weights were also applied when calculating the weighted average redshift, absolute magnitude, stellar mass, and age of objects contributing to the stack. Table <ref> lists the average galaxy properties and fluxes for each stack performed in our study.§ PREDICTED IRX-Β RELATIONSWe calculated the relationship between IRX and β using the recently developed “Binary Population and Spectral Synthesis” (BPASS) models <cit.> with a stellar metallicity of Z=0.14Z_⊙ on the current abundance scale <cit.> and a two power-law IMF with α = 2.35 for M_∗ > 0.5 M_⊙ and α= 1.30 for 0.1≤ M_∗≤ 0.5 M_⊙.We assumed a constant star formation with an age of 100 Myr and included nebular continuum emission.This particular BPASS model (what we refer to as our “fiducial” model) is found to best reproduce simultaneously the rest-frame far-UV continuum, stellar, and nebular lines, and the rest-frame optical nebular emission line strengths measured for galaxies at z∼ 2 <cit.>.Two salient features of this model are the very blue intrinsic UV continuum slope β_0 ≃ -2.62 relative to that assumed in the <cit.> calibration of the IRX-β relation (β_0 = -2.23), and the larger number of ionizing photons per unit star-formation-rate (i.e., ≈ 20% larger than those of single star models with no stellar rotation; ) that are potentially available for heating dust. For comparison, the BPASS model for the same metallicity with a constant star-formation history and an age of 300 Myr (the median for the sample considered here, and similar to the mean age of z∼ 2 UV-selected galaxies; ) is β_0 = -2.52. Below, we also consider the more traditionally used <cit.> (BC03) models.We calculated the IRX-β relation assuming an energy balance between flux that is absorbed and that which is re-emitted in the infrared <cit.>.The absorption is determined by the extinction or attenuation curve, and we considered several choices including the SMC extinction curve of <cit.>, and the <cit.> and <cit.> attenuation curves.The original forms of these extinction/attenuation curves were empirically calibrated at λ 1200 Å.The <cit.> and <cit.> curves were extended down to λ = 950 Åusing a large sample of Lyman Break galaxy spectra at z∼ 3 and a newly-developed iterative method presented in <cit.>.The SMC curve of <cit.> was extended in the same way, and we used these extended versions of the curves in this analysis.For reference, our new constraints on the shape of dust obscuration curves imply a lower attenuation of λ 1250 Åphotons relative to that predicted from polynomial extrapolations below these wavelengths <cit.>.In practice, because most of the dust heating arises from photons with λ>1200 Å, the implementation of the new shapes of extinction/attenuation curves does little to alter the predicted IRX-β relation.For reference, the following equations give the relationship betweenand β for the fiducial (BPASS) model with nebular continuum emission and the shapes of the attenuation/extinction curves derived above:Calzetti+00: β =-2.616 + 4.684×; SMC: β=-2.616 + 11.259×; Reddy+15: β=-2.616 + 4.594×.The intercepts in the above equations are equal to -2.520 for the 300 Myr BPASS model.For each value of , we applied the aforementioned dust curves to the BPASS model and calculated the flux absorbed at λ > 912 Å.Based on the high covering fraction (92%) of optically-thickinferred for z∼ 3 galaxies <cit.>, we assumed a zero escape fraction of ionizing photons and that photoelectric absorption dominates the depletion of such photons, rather than dust attenuation <cit.>.We then calculated the resultant Lyα flux assuming Case B recombination and the amount of Lyα flux absorbed given the value of the extinction/attenuation curve at λ=1216 Å, and added this to the absorbed flux at λ>912 Å.This total absorbed flux is equated to L_ IR, where we have assumed that all of the dust emission occurs between 8 and 1000 μm.Finally, we divided the infrared luminosity by the UV luminosity of the reddened model at 1600 Åto arrive at the value of IRX. The UV slope was computed directly from the reddened model using the full set of <cit.> wavelength windows.Below, we also consider the value of β computed using the subset of the <cit.> windows at λ < 1740 Å, as well as a single window spanning the range 1300-1800 Å.Formally, we find the following relations between IRX and β given Equation <ref>, where β is measured using the full set of <cit.> wavelength windows:Calzetti+00:IRX = 1.67 × [10^0.4(2.13β + 5.57) - 1]; SMC:IRX = 1.79 × [10^0.4(1.07β + 2.79) - 1]; Reddy+15:IRX = 1.68 × [10^0.4(1.82β + 4.77) - 1].These relations may be shifted redder by δβ = 0.096 to reproduce the IRX-β relations for the 300 Myr BPASS model.For reference, Appendix <ref> summarizes the relations between β andand between IRX and β for different assumptions of the stellar population model, nebular continuum, Lyα heating, and the normalization of the dust curve.Figures <ref> and <ref> conveya sense for how the stellar population and nebular continuum, Lyα heating, UV slope measurements, and the total-to-selective extinction (R_ V) affect the IRX-β relation.Models with a bluer intrinsic UV slope require a larger degree of dust obscuration to reproduce a given observed UV slope, thus causing the IRX-β relation to shift towards bluer β.Relative to the <cit.> relation, the IRX-β relations for the fiducial (BPASS) 100 and 300 Myr models predict a factor of ≈ 2 more dust obscuration at a given β for β -1.7, and an even larger factor for β bluer than this limit (left panel of Figure <ref>).The commonly utilized BC03 model results in a factor of ≈ 30% increase in the IRX at a given β relative to the <cit.> curve, while the 0.28Z_⊙ BC03 model results in an IRX-β relation that is indistinguishable from that of the BPASS model for the same age (right panel of Figure <ref>).These predictions underscore the importance of the adopted stellar population model when using the IRX-β relation to discern between different dust attenuation/extinction curves (e.g., ).Note that the inclusion of nebular continuum emission shifts the IRX-β relation by δβ≃ 0.1 to the right (i.e., making β redder), so that the IRX at a given β is ≈ 0.1 dex lower (leftmost panel of Figure <ref>). The specific treatment of dust heating from Lyα photons has a much less pronounced effect on the IRX-β relation. If none of the Lyα flux is absorbed by dust—also equivalent to assuming that the escape fraction of ionizing photons is 100%—then the resulting IRX is ≈ 10% lower at a given β than that predicted by our fiducial model.Similarly, assuming that all of the Lyα is absorbed by dust results in an IRX-β relation that is indistinguishable from that of the fiducial model.The wavelengths over which β is computed will also effect the IRX-β relation to varying degrees, depending on the specific wavelength ranges and the stellar population model.For the BPASS model, computing β from the reddened model spectrum within a single window spanning the range 1300-1800 Åresults in an IRX-β relation that is shifted by as much as δβ = 0.4 to redder slopes.This effect is due to the fact that the stellar continuum rises less steeply towards shorter wavelengths for λ 1500 Å.Consequently, the log( IRX) is ≃ 0.18 dex lower in this case relative to that computed based on the full set of <cit.> windows.Similar offsets are observed when using the subset of the <cit.> windows lying at λ < 1800 Å, while the offsets are not as large with the BC03 model.Most previous studies of the IRX-β relation adopted a β computed over relatively broad wavelength ranges coinciding with the <cit.> windows.However, the systematic offsets in the IRX-β relation arising from the narrower wavelength range used to compute UV slopes become relevant for very high-redshift (e.g., z 8) galaxies where Hubble photometry is typically used to constrain the UV slope and where such observations only go as red as rest-frame 1800 Å.Finally, the rightmost panel of Figure <ref> shows the effect of lowering the total-to-selective extinction (R_ V), or normalization, of the attenuation/extinction curves by various amounts.Of the physical factors discussed above, the IRX-β relation is most sensitive to the effects of changing the intrinsic UV slope and/or R_ V.To underscore the importance of the assumed stellar population when interpreting the IRX-β relation, we show in Figure <ref> the comparison of our fiducial BPASS model assuming the <cit.> curve and an intrinsic β_0 = -2.23 (accomplished by simply shifting the model to asymptote to this intrinsic value), along with the same model assuming an SMC curve with β_0 = -2.62.As is evident from this figure, the two IRX-β relations that assume different attenuation curves and intrinsic UV slopes have a significant overlap (within a factor two in IRX) over the range -2.1β -1.3.Notably this range includes the typical β≃ -1.5 found for UV-selected galaxies at z∼ 2 (see ).In the next section, we examine these effects further in the context of the stacked constraints on IRX-β provided by the combined HDUV and 3D-HST samples.§ DISCUSSION §.§ IRX-β for the Entire SampleAs a first step in constraining the IRX-β relation at z=1.5-2.5, we stacked galaxies in bins of UV slope.The resulting IRX for each of these bins, as well as for the whole sample, are shown in Figure <ref>.The predicted IRX-β relations for different assumptions of the stellar population (BPASS or BC03) intrinsic UV slope, β_0, and the difference in normalization of the dust curves, δ R_ V, are also shown.To account for the former, we simply shifted the fiducial relation (computed assuming β_0 = -2.62) so that it asymptotes to a redder value of β_0 = -2.23, similar to that assumed in <cit.>.Our stacked results indicate a highly significant (20σ) correlation between IRX and β.However, none of the predicted relations calculated based on assuming an intrinsic UV slope of β_0 = -2.23, as in <cit.>, are able to reproduce our stacked estimates for the full range of β considered here.For example, the upper left panel of Figure <ref> shows that while both the <cit.> and <cit.> attenuation curves predict IRX that are within 3σ of our stacked values for β < -1.2, they over-predict the IRX for galaxies with redder β.Lowering the normalization of the <cit.> attenuation curve by δ R_ V = 1.5 results in a better match to the stacked determinations, but with some disagreement (at the >3σ level) with the stack of the entire sample (lower left panel of Figure <ref>).<cit.> estimated the systematic uncertainty in their determination of R_ V to be δ R_ V≈ 0.4, which suggests that their curve may not have a normalization as low as R_ V = 1.0 given their favored value of R_ V = 2.51. Regardless, without any modifications to the normalizations and/or shapes of the attenuation curves in the literature <cit.>, the corresponding IRX-β relations are unable to reproduce our stacked estimates if we assume an intrinsic UV slope of β_0 = -2.23.At face value, these results suggest that the attenuation curve describing our sample is steeper than the typically utilized <cit.> relation, but grayer than the SMC extinction curve.However, this conclusion depends on the intrinsic UV slope of the stellar population, as we discuss next.Independent evidence favors the low-metallicity BPASS model in describing the underlying stellar populations of z∼ 2 galaxies <cit.>.The very blue intrinsic UV slope characteristic of this model—as well as those of the BC03 models with comparable stellar metallicities (e.g., the 0.28Z_⊙ BC03 model with the same high-mass power-law index of the IMF as the BPASS model has β_0 = -2.65)—is also favored in light of the non-negligible number of galaxies in our sample (≈ 9%) that have β<-2.23 at the 3σ level, the canonical value assumed in <cit.>.Figure <ref> shows that the low-metallicity models with blue β_0 result in IRX-β relations that are significantly shifted relative to those assuming redder β_0.With such models, we find that our stacked measurements are best reproduced by an SMC-like extinction curve (upper right-hand panel of Figure <ref>), in the sense that all of the measurements lie within 3σ of the associated prediction.On the other hand, with such stellar population models, grayer attenuation curves (e.g., ) over-predict the IRX at a given β by a factor of ≈ 2-7.More generally, we find that the slope of the IRX-β relation implied by our stacked measurements is better fit with that obtained when considering the SMC extinction curve, while grayer attenuation curves lead to a more rapid rise in IRX with increasing β.Our stacked measurements and predicted IRX-β curves are compared with several results from the literature in Figure <ref>.In the context of the IRX-β predictions that adopt sub-solar metallicities, we find that most of the stacked measurements for UV-selected galaxies at z∼ 1.5-3.0 suggest a curve that is SMC-like, at least for β -0.5. Several of the samples, including those of <cit.>, <cit.>, and <cit.>, indicate an IRX that is larger than the SMC prediction for β -0.5.Such a behavior is not surprising given that the dust obscuration has been shown to decouple from the UV slope for galaxies with large star-formation rates, as is predominantly the case for most star-forming galaxies with very red β <cit.>.As discussed in a number of studies <cit.>, dusty galaxies in general can exhibit a wide range in β (from very blue to very red) depending on the particular spatial configuration of the dust and UV-emitting stars.Figure <ref> shows that the degree to which such galaxies diverge from a given attenuation curve depends on β_0.Many of the dusty galaxies that would appear to have IRX larger than the <cit.> or <cit.> predictions may in fact be adequately described by such curves if the stellar populations of these galaxies are characterized by very blue intrinsic UV spectral slopes.On the other hand, if these dusty galaxies have relatively enriched stellar populations, and redder intrinsic slopes, then their departure from the <cit.> prediction would be enhanced.Undoubtedly, large variations in IRX can also be expected with different geometries of dust and stars.Regardless, if sub-solar metallicity models are widely representative of the stellar populations of typical star-forming galaxies at z 1.5, then our stacked measurements, along with those in the literature, tend to disfavor gray attenuation curves for these galaxies.The large sample studied here, as well as those of <cit.> and <cit.>, suggest an SMC-like curve.At first glance, this conclusion may appear to be at odds with the wide number of previous investigations that have found that the <cit.> and <cit.> attenuation curves generally apply to moderately-reddened star-forming galaxies at z 1.5 (e.g., ; c.f., ).In the framework of our present analysis, the reconciliation between these results is simple.Namely, our analysis does not call into question previous measurements of IRX-β, but calls for a different interpretation of these measurements.In the previous interpretation, most of the stacked measurements from the literature were found to generally agree with the <cit.> relation if we assume a relatively red intrinsic slope of β=-2.23.In the present interpretation, we argue that sub-solar metallicity models necessitate a steeper attenuation curve in order to reproduce the measurements of IRX-β (e.g., see also ).Our conclusion is aided by the larger dynamic range in β probed by the HDUV and 3D-HST samples, which allows us to better discriminate between different curves given that their corresponding IRX-β relations separate significantly at redder β (Figures <ref> and <ref>).§.§ IRX Versus Stellar MassThe well-studied correlations between star-formation rate and stellar mass (e.g., ), and between star-formation rate and dust attenuation (e.g., ), have motivated the use of stellar mass as a proxy for attenuation <cit.> as the former can be easily inferred from fitting stellar population models to broadband photometry.The connection between reddening and stellar mass can also be deduced from the mass-metallicity relation <cit.>.Motivated by these results, we stacked galaxies in two bins of stellar mass divided at M_∗ = 10^9.75 M_⊙ (and further subdivided into bins of β; Table <ref>) to investigate the dependence of the IRX-β relation on stellar mass.[The stellar masses obtained with <cit.> models (Section <ref>) are on average within 0.1 dex of those obtained assuming the fiducial (BPASS) model with the same <cit.> IMF.]The high-mass subsample (M_∗ > 10^9.75 M_⊙) exhibits a redder UV slope (⟨β⟩ = -0.92) and larger IRX (⟨ IRX⟩=6.8± 0.6) than the low-mass subsample with ⟨β⟩ = -1.88 and ⟨ IRX⟩<1.7 (3σ upper limit).Moreover, the high-mass subsample exhibits an IRX-β relation consistent with that predicted assuming our fiducial stellar population model and the SMC extinction curve (Figure <ref>).Separately, the low-mass subsample as a whole, as well as the subset of the low-mass galaxies with β≤ -1.4, have 3σ upper limits on IRX that require a dust curve that is at least as steep as the SMC. The constraints on the IRX-M_∗ relation from our sample are shown relative to previous determinations in the right panel of Figure <ref>.The “z∼ 2 Consensus Relation” presented in <cit.> was based on the IRX-M_∗ trends published in <cit.>, <cit.>, and <cit.>.Formally, our stacked detection for the high-mass (M_∗>10^9.75 M_⊙) subsample lies ≈ 4σ below the consensus relation, but is in excellent agreement with the mean IRX found for galaxies of similar masses (≃ 2× 10^10 M_⊙) in <cit.>.The upper limit in IRX for the low-mass (M_∗≤ 10^9.75 M_⊙) sample is consistent with the predictions from the consensus relation.Based on these comparisons, we conclude that the IRX-M_∗ relation from the present work is in general agreement with previous determinations, and lends support for previous suggestions that stellar mass may be used as a rough proxy for dust attenuation in high-redshift star-forming galaxies (e.g., ).Moreover, these comparisons underscore the general agreement between our IRX measurements (e.g., as a function of β and M_∗) and those in the literature, in spite of our different interpretation of these results in the context of the dust obscuration curve applicable to high-redshift galaxies (Section <ref>).§.§ IRX Versus UV LuminosityAs alluded to in Section <ref>, quantifying the dust attenuation of UV-faint (sub-L^∗) galaxies has been a longstanding focus of the high-redshift community.While the steep faint-end slopes of UV luminosity functions at z 2 imply that such galaxies dominate the UV luminosity density at these redshifts, knowledge of their dust obscuration is required to assess their contribution the cosmic star-formation-rate density (e.g., ).Several studies have argued that UV-faint galaxies are on average less dusty than their brighter counterparts <cit.>.This inference is based on the fact that the observed UV luminosity is expected to monotonically correlate with star-formation rate for galaxies fainter than L^∗ (e.g., see Figure 13 ofand Figure 10 of ) and that the dustiness is a strong function of star-formation rate <cit.>.While several investigations have shown evidence for a correlation between IRX and UV luminosity <cit.>, others point to a roughly constant IRX as a function of UV luminosity <cit.>.As discussed in these studies, the different findings may be a result of selection biases, in the sense that UV-selected samples will tend to miss the dustiest galaxies, which also have faint observed UV luminosities.Hence, for purely UV-selected samples, IRX would be expected to decrease with L_ UV.Alternatively, the rarity of highly dust-obscured galaxies compared to intrinsically faint galaxies (e.g., as inferred from the shapes of the UV and IR luminosity functions; ) implies that in a number-weighted sense, the mean bolometric luminosity should decrease towards fainter L_ UV.How this translates to the variation of IRX with L_ UV will depend on how quickly the dust can build up in dynamically-relaxed faint galaxies.From a physical standpoint, dust enrichment on timescales much shorter than the dynamical timescale would suggest a relatively constant IRX as a function of L_ UV.The HDUV/3D-HST sample presents a unique opportunity to evaluate the trend between IRX and L_ UV as the selection is based on rest-optical criteria.Consequently, our sample is less sensitive to the bias against dusty galaxies that are expected to be significant in UV-selected samples (e.g., ).Indeed, Figure <ref> shows that our sample includes a large number of UV-faint galaxies that are also quite dusty based on their red β -1.4—these galaxies, while dusty, still have bolometric luminosities that are sufficiently faint to be undetected in the PACS imaging.Figure <ref> shows the relationship between dust attenuation and UV luminosity for our sample (gray points) and those of <cit.> at z∼ 1.5 and <cit.> at z∼ 3. The latter two indicate an IRX that is roughly constant with L_ UV, but one which is a factor of 2-3 offset towards higher IRX than found for our sample.This offset is easily explained by the fact that the IRX-L_ UV relations for the z∼ 1.5 and z∼ 3.0 samples were determined from galaxies that are on average significantly redder than in our sample.In particular, most of the constraints on IRX-L_ UV from these studies come from galaxies with β -1.5.When limiting our stacks to galaxies with β > -1.4 (red points in Figure <ref>), we find an excellent agreement with the IRX-L_ UV relations found by <cit.> and <cit.>.On the other hand, stacking those galaxies in our sample with β≤ -1.4 results in an IRX-L_ UV relation that is not surprisingly offset towards lower IRX than for the sample as a whole.Thus, while the IRX-L_ UV relation appears to be roughly constant for all of the samples considered here, Figure <ref> implies that the β distribution as a function of L_ UV is at least as important as the presence of dusty star-forming galaxies in shaping the observed IRX-L_ UV relation.Furthermore, the trend of a bluer average β with decreasing L_ UV (e.g., Figure <ref>; see also ) suggests that the mean reddening should be correspondingly lower for UV-faint galaxies than for UV-bright ones once the effect of the less numerous dusty galaxies with red β are accounted for.The blue (β≤ -1.4) star-forming galaxies in our sample have IRX1, such that the infrared and UV luminosities contribute equally to the bolometric luminosities of these galaxies.The expectation of rapid dust enrichment from core-collapse supernovae <cit.> implies that it is unlikely that the dust obscuration can be significantly lower than this value when observed for dynamically-relaxed systems.Consequently, the observation that the mean UV slope becomes progressively bluer for fainter galaxies at high-redshift (e.g., ) may simply be a result of systematic changes in metallicity and/or star-formation history where the intrinsic UV slope also becomes bluer but IRX remains relatively constant (e.g., Figure <ref>; ). Thus, the common observation that UV-faint galaxies are bluer than their brighter counterparts may not directly translate into a lower dust obscuration for the former.Moreover, β is relatively insensitive to IRX for β-β_0 0.2 (e.g., Figure <ref>).Our results thus indicate caution when using the IRX-β relation to infer the dust reddening of blue galaxies at high-redshift, as such estimates may be highly dependent on the intrinsic UV slope of the stellar population and even otherwise quite uncertain if the difference in observed and intrinsic UV slopes is small. §.§ Young/Low-Mass GalaxiesALMA has opened up new avenues for investigating the ISM and dust content of very high-redshift galaxies, and a few recent efforts have focused in particular on the [C2]158 μm line in galaxies at z 5 <cit.> and dust continuum at mm wavelengths.<cit.> report on ALMA constraints on the IRX of a small sample of 10 z∼ 5.5 LBGs and find that they generally fall below the SMC extinction curve.The disparity between the SMC curve and their data points is increased if one adopts a bluer intrinsic slope than that assumed in <cit.>, a reasonable expectation for these high-redshift and presumably lower metallicity galaxies.More generally, earlier results suggesting that “young” LBGs (ages 100 Myr) and/or those with lower stellar masses at z 2 are consistent with an SMC curve <cit.> would also require a steeper-than-SMC curve if their intrinsic slopes are substantially bluer than that normally assumed in interpreting the IRX-β relation.Unfortunately, there is only a very small number of galaxies in our sample that have SED-determined ages of <100 Myr (81 galaxies), and stacking them results in an unconstraining upper limit on IRX (Table <ref>).Note that an ambiguity arises because the ages are derived from SED-fitting, which assumes some form of the attenuation curve. Following <cit.>, the number of galaxies considered “young” would be lower under the assumption of SMC extinction rather than <cit.> attenuation, as the former results in lowerfor a given UV slope, translating into older ages.Self-consistently modeling the SEDs based on the location of galaxies in the IRX-β plane results in fewer <100 Myr galaxies, but of course their location in the IRX-β plane is unaffected <cit.>, as is the conclusion that such young galaxies would require a dust curve steeper than that of the SMC if they have blue intrinsic UV slopes. In addition, as noted in Section <ref>, galaxies in our low-mass (M_∗≤ 10^9.75 M_⊙) subsample appear to also require a dust curve steeper than that of the SMC.Figure <ref> summarizes a few recent measurements of IRX-β for young and low-mass galaxies at z∼ 2 <cit.>, low-mass galaxies and LBGs in general at z∼ 4-10 <cit.>, and our own measurements.The compilation from <cit.> and <cit.> includes 24 μm constraints on the IRX of young galaxies.Shifting their IRX by ≈ 0.35 dex to higher values to account for the deficit of PAH emission in galaxies with 400 Myr <cit.> results in upper limits or a stacked measurement of IRX that are broadly consistent with either the SMC curve or one that is steeper.Considering the Herschel measurements here and in <cit.>, and ALMA measurements at z∼ 2-10 <cit.>, we find that young/low-mass galaxies at z 2 follow a dust curve steeper than that of the SMC, particularly in the context of a blue intrinsic slope, β_0 = -2.62.Note that unlike cB58, which has a stellar metallicity of ≃ 0.25 Z_⊙ <cit.>, the Cosmic Eye has a metallicity of ∼ 0.5 Z_⊙, suggesting a relatively red intrinsic UV slope.In this case, the IRX of the Cosmic Eye may be described adequately by the SMC curve.There is additional evidence of suppressed IRX ratios at lower mass from rest-optical emission line studies of z∼ 2 galaxies.In particular, <cit.> found that a significant fraction of z∼ 2 galaxies with M_∗ 10^9.75 M_⊙ have very red β, or , relative to the reddening deduced from the Balmer lines (e.g., see their Figure 17), implying that such galaxies would have lower IRX for a given β than that predicted by common attenuation/extinction curves.Evidence for curves steeper than the SMC average have been observed along certain sightlines within the SMC <cit.>, in the Milky Way and some Local Group galaxies <cit.>, and in quasars (e.g., ).Our results, combined with those in the literature, suggest that such steep curves may be typical of low-mass and young galaxies at high redshift.While the attenuation curve will undoubtedly vary from galaxy-to-galaxy depending on the star-formation history, age, metallicity, dust composition, and geometrical configuration of stars and dust, the fact that young/low-mass galaxies lie systematically below the IRX-β relation predicted with an SMC curve suggests that a steep curve may apply uniformly to such galaxies.An unresolved issue is the physical reason why young and low-mass galaxies may follow a steeper attenuation curve than their older and more massive counterparts.<cit.> suggest a possible scenario in which galaxies transition from steep to gray attenuation curves as they age due to star formation occurring over extended regions and/or the cumulative effects of galactic outflows that are able to clear the gas and dust along certain sightlines. On the other hand, if young and low-mass galaxies have higher ionizing escape fractions as a result of lower gas/dust covering fractions (e.g., ), then one might expect their attenuation curve to exhibit a shallower dependence on wavelength than the SMC extinction curve.In any case, curves steeper than the SMC may be possible with a paucity of large dust grains and/or an over-abundance of silicate grains <cit.>.In particular, large dust grains may be efficiently destroyed by SN shock waves <cit.>, which would have the effect of steepening the dust curve (i.e., such that proportionally more light is absorbed in the UV relative to the optical).If the destruction of large grains is significant in young/low-mass galaxies, then it may explain both their red β and their low IRX.Alternatively, the lower gas-phase [Si/O] measured from the composite rest-frame UV spectrum of z∼ 2 galaxies relative to the solar value indicates significant depletion of Si onto dust grains, while carbon is under-abundant relative to oxygen <cit.>. This result may suggest an enhancement of silicate over carbonaceous grains that may result in a steeper attenuation curve. §.§ Implications for Stellar Populations and Ionizing Production Efficiencies Inferring the intrinsic stellar populations of galaxies based on their observed photometry requires one to adopt some form of the dust attenuation curve.It is therefore natural to ask whether these inferences change in light of our findings of a steeper (SMC-like) attenuation curve for z∼ 2 galaxies with intrinsically blue UV spectral slopes.To address this issue, we re-modeled (using FAST) the SEDs of galaxies in our sample assuming two cases: (a) a 1.4Z_⊙ BC03 model (canonically referred to as the “solar” metallicity model using old solar abundance measurements) with the <cit.> attenuation curve; and (b) a 0.28Z_⊙ BC03 model with an SMC extinction curve.The ages derived in case (b) are on average 30% older than those derived in case (a), primarily because less reddening is required to reproduce a given UV slope with the SMC extinction curve, resulting in older ages (e.g., see discussion in ).Similarly, the stellar masses derived in case (b) are on average 30% lower than those derived in case (a).Perhaps most relevant in the context of our analysis are the changes in inferred reddening and SFR.As shown in Figure <ref>, the reddening deduced from the SMC extinction curve is lower than that obtained with <cit.> attenuation curve, owing to the steep rise in the far-UV of the former relative to the latter.The largest differences inand SFR occur for the reddest objects and those with larger SFRs.We find the following relations between the reddening and SFRs derived for the two cases discussed above:_0.28Z_⊙, SMC = 0.65_1.4Z_⊙, Calzandlog[ SFR_0.28Z_⊙, SMC/M_⊙yr^-1] = 0.79 log[ SFR_1.4Z_⊙, Calz/M_⊙yr^-1] -0.05.The lower SFRs derived in the SMC case result in a factor of ≈ 2 lower SFR densities at z 2, as discussed in some depth in <cit.>.The general applicability of an SMC-like dust curve to high-redshift galaxies is also of particular interest when considering its impact on the ionizing efficiencies of such galaxies, a key input to reionization models <cit.>.Specifically, the ionizing photon production efficiency, ξ_ ion, is simply the ratio of the rate of H-ionizing photons produced to the non-ionizing UV continuum luminosity.This quantity is directly related to another commonly-used ratio, namely the Lyman continuum flux density (e.g., at 800 or 900 Å) to the non-ionizing UV flux density (e.g., at 1600 Å), f_ 900/f_ 1600.The ionizing photon production efficiency is typically computed by combining Balmer emission lines, such as Hα, with UV continuum measurements (e.g., ), but both the lines and the continuum must be corrected for dust attenuation.In the context of our present analysis, the dust corrections for the UV continuum are lower by factors of 1.12, 1.82, and 2.95, for galaxies with Calzetti-inferred SFRs of 1, 10, and 100 M_⊙ yr^-1, respectively.Thus, for given Balmer line luminosities that are corrected for dust using the Galactic extinction curve <cit.>, ξ_ ion would be correspondingly larger by the same factors by which the dust-corrected UV luminosities are lower.A secondary effect that will boost ξ_ ion above the predictions from single star solar metallicity stellar population models is the higher ionizing photon output associated with lower metallicity and rotating massive stars <cit.>.In particular, the fiducial 0.14Z_⊙ BPASS model that includes binary evolution and an IMF extending to 300 M_⊙ predicts a factor of ≈ 3 larger Hα luminosity per solar mass of star formation after 100 Myr of constant star formation relative to that computed using the <cit.> relation, which assumes a solar metallicity Starburst99 model.On the other hand, the UV luminosity is larger by ≈ 30%.Thus, such models predict a ξ_ ion that is elevated by a factor of ≈ 2 relative to those assumed in standard calibrations between Hα/UV luminosity and SFR (e.g., see also ).Consequently, calculations or predictions of ξ_ ion for high-redshift galaxies should take into account the effects of a steeper attenuation curve and lower metallicity stellar populations that may include stellar rotation/binarity.Our results suggest that an elevated value of ξ_ ion is not only a feature of very high-redshift (z 6) galaxies, but may be quite typical for z∼ 2 galaxies as well. § CONCLUSIONSIn this paper, we have presented an analysis of the relationship between dust obscuration (IRX=L_ IR/L_ UV) and other commonly-derived galaxy properties, including UV slope (β), stellar mass (M_∗), and UV luminosity (L_ UV), for a large sample of 3,545 rest-optically selected star-forming galaxies at z=1.5-2.5 drawn from HDUV UVIS and 3D-HST photometric catalogs of the GOODS fields.Our sample is unique in that it significantly extends the dynamic range in β and L_ UV compared to previous UV-selected samples at these redshifts.In particular, close to 60% of the objects in our sample have UV slopes bluer than β = -1.70 and >95% have rest-frame UV absolute magnitudes fainter than the characteristic magnitude at these redshifts, with the faintest galaxies having L_ UV≈ 0.05L^∗_ UV.We use stacks of the deep Herschel/PACS imaging in the GOODS fields to measure the average dust obscuration for galaxies in our sample and compare it to predictions of the IRX-β relation for different stellar population models and attenuation/extinction curves using energy balance arguments.Specifically, we consider the commonly adopted <cit.> stellar population models for different metallicities (0.28 and 1.4Z_⊙), as well as the low-metallicity (0.14Z_⊙) BPASS model.Additionally, we compute predictions of the IRX-β relations for the <cit.> and <cit.> dust attenuation curves, and the SMC extinction curve.The lower metallicity stellar population models result in significant shifts in the IRX-β relation of up to δβ = 0.4 towards bluer β relative to the canonical relation of <cit.>.In the context of the lower metallicity stellar population models applicable for high-redshift galaxies, we find that the strong trend between IRX and β measured from the HDUV and 3D-HST samples follows most closely that predicted by the SMC extinction curve.We find that grayer attenuation curves (e.g., ) over-predict the IRX at a given β by a factor of 3 when assuming intrinsically blue UV spectral slopes. Thus, our results suggest that an SMC curve is the one most applicable to lower stellar metallicity populations at high redshift.Performing a complementary stacking analysis of the Spitzer/MIPS 24 μm images implies an average mid-IR-to-IR luminosity ratio, ⟨ L_7.7/L_ IR⟩, that is a factor of 3-4 lower than for galaxies with reddest (β>-0.5) and the UV-brightest (M_1600 -21) and UV-faintest (M_1600 -19) galaxies relative to the average for all galaxies in our sample (Appendix <ref>).These results indicate large variations in the conversion between rest-frame 7.7 μm and IR luminosity.At any given UV luminosity, galaxies with redder β have larger IRX.IRX-L_ UV relations for blue and red star-forming galaxies average together to result in a roughly constant IRX of ≃ 3-4 over roughly two decades in UV luminosity (2× 10^9L_ UV 2× 10^11 L_⊙).Consequently, the bluer β observed for UV-faint galaxies seen in this work and previous studies may simply reflect intrinsically bluer UV spectral slopes for such galaxies, rather than signifying changes in the dust obscuration.Galaxies with stellar masses M_∗>10^9.75 M_⊙ have an IRX-β relation that is consistent with the SMC extinction curve, while the lower mass galaxies in our sample with M_∗≤ 10^9.75 M_⊙ have an IRX-β relation that is at least as steep as the SMC.The shifting of the IRX-β relations towards bluer β for the lower metallicity stellar populations expected for high-redshift galaxies implies that the low-mass galaxies in our sample, as well as the low-mass and young galaxies from previous studies, require a dust curve steeper than that of the SMC.The low metallicity stellar populations favored for high-redshift galaxies result in steeper attenuation curves and higher ionizing photon production rates which, in turn, facilitate the role that galaxies may have in reionizing the universe at very high redshift or keeping the universe ionized at lower redshifts (z∼ 2).There are several future avenues for building upon this work. Detailed spectral modeling of the rest-UV and/or rest-optical spectra of galaxies (e.g., ) may be used to discern their intrinsic spectral slopes and thus disentangle the effects of the intrinsic β and the attenuation curve relevant for high-redshift galaxies.Second, the higher spatial resolution and depth of X-ray observations (compared to the far-IR) makes them advantageous for investigating the bolometric SFRs and hence dust obscuration for galaxies substantially fainter than those directly detected with either Spitzer or Herschel in reasonable amounts of time, provided that the scaling between X-ray luminosity and SFR can be properly calibrated (e.g., for metallicity effects; ).In addition, nebular recombination line estimates of dust attenuation (e.g., from the Balmer decrement; ) may be used to assess the relationship between IRX and β for individual star-forming galaxies, rather than through the stacks necessitated by the limited sensitivity of far-IR imaging.This work was supported by NASA through grant HST-GO-13871 and from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555.K. 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Β RELATIONSIn Table <ref>, we summarize the relations between β andand between IRX and β for different assumptions of the dust curve, heating from Lyα, inclusion of nebular continuum emission, and the normalization of the dust curve.llcccccc[!h]0pc Summary of β vs.and IRX vs. β RelationsNebular LyαModela Dust Curveb Continuumc Heatingd δ R_ Ve βfIRX BPASS - 0.14Z_⊙ Calzetti+00 Yes Yes 0.0 -2.616 + 4.684 1.67×[10^0.4(2.13β+5.57)-1]Yes Yes 1.5 1.66×[10^0.4(1.81β+4.73)-1]Yes No 0.01.39×[10^0.4(2.13β+5.57)-1]Yes No 1.51.38×[10^0.4(1.81β+4.73)-1]No Yes 0.0 -2.709 + 4.684 1.73×[10^0.4(2.13β+5.76)-1]No Yes 1.5 1.73×[10^0.4(1.81β+4.90)-1]No No 0.01.44×[10^0.4(2.13β+5.76)-1]No No 1.51.42×[10^0.4(1.81β+4.90)-1]Reddy+15Yes Yes 0.0 -2.616 + 4.594 1.68×[10^0.4(1.82β+4.77)-1]Yes Yes 1.5 1.67×[10^0.4(1.50β+3.92)-1]Yes No 0.01.40×[10^0.4(1.82β+4.77)-1]Yes No 1.51.38×[10^0.4(1.50β+3.92)-1]No Yes 0.0 -2.709 + 4.594 1.74×[10^0.4(1.82β+4.94)-1]No Yes 1.5 1.74×[10^0.4(1.50β+4.05)-1]No No 0.01.44×[10^0.4(1.82β+4.94)-1]No No 1.51.43×[10^0.4(1.50β+4.05)-1]SMC (Gordon+03) Yes Yes 0.0 -2.616 + 11.259 1.79×[10^0.4(1.07β+2.79)-1]Yes Yes 1.5 1.80×[10^0.4(0.93β+2.44)-1]Yes No 0.01.47×[10^0.4(1.07β+2.79)-1]Yes No 1.51.47×[10^0.4(0.93β+2.44)-1]No Yes 0.0 -2.709 + 11.259 1.83×[10^0.4(1.07β+2.89)-1]No Yes 1.51.85×[10^0.4(0.93β+2.52)-1]No No 0.0 1.50×[10^0.4(1.07β+2.89)-1]No No 1.51.50×[10^0.4(0.93β+2.52)-1] BC03 - 1.4Z_⊙ Calzetti+00 Yes Yes 0.0 -2.383 + 4.661 1.44×[10^0.4(2.14β+5.10)-1]Yes Yes 1.5 1.41×[10^0.4(1.82β+4.33)-1]Yes No 0.01.35×[10^0.4(2.14β+5.10)-1]Yes No 1.51.32×[10^0.4(1.82β+4.33)-1]No Yes 0.0 -2.439 + 4.661 1.47×[10^0.4(2.14β+5.22)-1]No Yes 1.5 1.44×[10^0.4(1.82β+4.43)-1]No No 0.01.38×[10^0.4(2.14β+5.22)-1]No No 1.51.35×[10^0.4(1.82β+4.43)-1]Reddy+15Yes Yes 0.0 -2.383 + 4.568 1.43×[10^0.4(1.83β+4.37)-1]Yes Yes 1.5 1.39×[10^0.4(1.51β+3.59)-1]Yes No 0.01.40×[10^0.4(1.82β+4.77)-1]Yes No 1.51.30×[10^0.4(1.51β+3.59)-1]No Yes 0.0 -2.439 + 4.568 1.46×[10^0.4(1.83β+4.47)-1]No Yes 1.5 1.42×[10^0.4(1.51β+3.67)-1]No No 0.01.37×[10^0.4(1.83β+4.47)-1]No No 1.51.33×[10^0.4(1.51β+3.67)-1]SMC (Gordon+03) Yes Yes 0.0 -2.383 + 11.192 1.47×[10^0.4(1.07β+2.55)-1]Yes Yes 1.5 1.45×[10^0.4(0.94β+2.23)-1]Yes No 0.01.38×[10^0.4(1.07β+2.55)-1]Yes No 1.51.35×[10^0.4(0.94β+2.23)-1]No Yes 0.0 -2.439 + 11.192 1.50×[10^0.4(1.07β+2.61)-1]No Yes 1.51.49×[10^0.4(0.94β+2.29)-1]No No 0.0 1.40×[10^0.4(1.07β+2.61)-1]No No 1.51.38×[10^0.4(0.94β+2.29)-1] Meurer+99— — — — -2.23 + 5.01 2.07×[10^0.4(1.99β+4.43)-1] aStellar population model, assuming a constant star-formation history and an age of 100 Myr.For reference, we also list the original relations of <cit.>, where their IRX-β relation is shifted to higher IRX by 0.24 dex (see text). bDust attenuation curves from <cit.> and <cit.> and the dust extinction curve for the SMC from <cit.>, where all have been updated for the shape of the curve at λ 1200 Åaccording to the procedure of <cit.>. cIndicates whether the nebular continuum emission is included. dIndicates whether dust heating by Lyα photons is included. eIndicates the value by which the dust attenuation curves are shifted downward in normalization (R_ V). fA blank entry indicates that the relation between β andis the same as that of the previous non-blank entry in this column. § B. INFERENCES OF IRX FROM SPITZER/MIPS 24 ΜM DATAPrior to the launch of Herschel, most non-UV-based inferences of the dust content of L^∗ galaxies at z 1.5 relied on the detection of the redshifted mid-IR emission bands, commonly associated with PAHs, with the Spitzer MIPS instrument.A number of early studies of local and z∼ 1 galaxies suggested that PAH emission correlates with total dust emission (e.g., ), though with some variations with metallicity, ionizing intensity, star-formation-rate surface density, and stellar population age (e.g., ).Recently, <cit.> presented the first statistically significant trends showing lower ratios of the 7.7 μm-to-total IR luminosity, L_ 7.7/L_ IR, with higher ionization intensity, lower gas-phase metallicity, and younger ages for galaxies at z∼ 2 from the MOSFIRE Deep Evolution Field Survey <cit.>.The aforementioned studies have suggested either a delayed enrichment of PAHs in young galaxies or the destruction of PAHs in high ionization and low metallicity environments. In light of these findings, we considered L_ 7.7/L_ IR for the predominantly blue, low luminosity galaxies in our sample.[As virtually all of the galaxies in our sample are undetected in the PACS imaging, we were not able to normalize the 24 μm images by 1/L_ IR before stacking them (e.g., in the same way that we were able to normalize them by 1/L_ UV; Section <ref>).However, as the stacked fluxes are weighted by 1/L_ UV, and L_ UV∝ SFR∝ L_ IR for all but the dustiest galaxies in our sample (Section <ref>), we assumed that the ratio of the average luminosities is similar to the average ratio of the luminosities, i.e., ⟨ L_ 7.7⟩/⟨ L_ IR⟩≈⟨ L_ 7.7/L_ IR⟩ (see discussion in ).Thus, we simply divided L_ 7.7 by L_ IR for each stack presented in Table <ref> in order to deduce the average ratio ⟨ L_ 7.7/L_ IR⟩.]The galaxies in our sample as a whole exhibit ⟨ L_ 7.7/L_ IR⟩ = 0.12±0.03, similar within 1σ to that computed in <cit.>, ⟨ L_ 7.7/L_ IR⟩ = 0.18±0.03, where the latter has been corrected for the difference in this ratio when assuming the <cit.> dust template rather than those of <cit.>.However, when divided into bins of UV slope, we find that ⟨ L_ 7.7/L_ IR⟩ is substantially lower for galaxies with the reddest β, as well as for the brightest (M_1600≤ -21) and faintest (M_1600 > -19) galaxies in our sample (Figure <ref>).The low ratio (⟨ L_ 7.7/L_ IR⟩ = 0.07±0.01) observed for galaxies with the reddest UV slopes may be related to significant 9.9 μm silicate absorption affecting the observed 24 μm flux.An alternative explanation invokes an IR luminosity that is boosted in the presence of AGN, though we consider this possibility unlikely as we have removed obscured AGN from the sample based on their IRAC colors (Section <ref>). Regardless, the lower ⟨ L_ 7.7/L_ IR⟩ found for these red galaxies is partly responsible for a similar low ratio found for the faintest galaxies in our sample, since many of the dustiest galaxies in our sample are also UV-faint (Figure <ref>).Isolating the UV-faint galaxies with slopes bluer than β=-1.4 yields an unconstraining lower limit on ⟨ L_ 7.7/L_ IR⟩.Finally, we note that the brightest galaxies in our sample (M_1600≤ -21) also exhibit a very low ⟨ L_ 7.7/L_ IR⟩ = 0.06±0.01.Such galaxies are on average a factor of 1.6× younger than M_1600>-21 galaxies (≈ 500 vs. ≈ 800 Myr).Thus, their lower mean ratio may be related to a deficit of PAHs due to younger stellar population ages, or may be related to harder ionization fields and/or lower gas-phase metallicities.Unfortunately, we are unable to fully explore how the L_7.7-to-L_ IR ratio varies with age given unconstraining lower limits on this value for galaxies with ages 500 Myr. Irrespective of the physical causes for changes in the PAH-to-infrared luminosity ratio, our results suggest that caution must be used when adopting a single-valued conversion to recover L_ IR from mid-IR measurements.For example, assuming the mean ratio found for our sample would result in 24 μm-inferred L_ IR that are a factor of ≈ 2 lower than the “true” values for the reddest (β >-0.8) and UV-brightest (M_1600≤ -21) galaxies in our sample. | http://arxiv.org/abs/1705.09302v1 | {
"authors": [
"Naveen A. Reddy",
"Pascal A. Oesch",
"Rychard J. Bouwens",
"Mireia Montes",
"Garth D. Illingworth",
"Charles C. Steidel",
"Pieter G. van Dokkum",
"Hakim Atek",
"Marcella C. Carollo",
"Anna Cibinel",
"Brad Holden",
"Ivo Labbe",
"Dan Magee",
"Laura Morselli",
"Erica J. Nelson",
"Steve Wilkins"
],
"categories": [
"astro-ph.GA"
],
"primary_category": "astro-ph.GA",
"published": "20170525180011",
"title": "The HDUV Survey: A Revised Assessment of the Relationship between UV Slope and Dust Attenuation for High-Redshift Galaxies"
} |
Generating Time-Based Label Refinements to Discover More Precise Process Models [===============================================================================We present a new model, Predictive State Recurrent Neural Networks (PSRNNs), for filtering and prediction in dynamical systems. PSRNNs draw on insights from both Recurrent Neural Networks (RNNs) and Predictive State Representations (PSRs), and inherit advantages from both types of models. Like many successful RNN architectures, PSRNNs use (potentially deeply composed) bilinear transfer functions to combine information from multiple sources. We show that such bilinear functions arise naturally from state updates in Bayes filters like PSRs, in which observations can be viewed as gating belief states. We also show that PSRNNs can be learned effectively by combining Backpropogation Through Time (BPTT) with an initializationderived from a statistically consistent learning algorithm for PSRs called two-stage regression (2SR). Finally, we show that PSRNNs can befactorized using tensor decomposition, reducing model size and suggesting interesting connections to existing multiplicative architectures such as LSTMs. We applied PSRNNs to 4 datasets, and showed that we outperform several popular alternative approaches to modeling dynamical systems in all cases.§ INTRODUCTION Learning to predict temporal sequences of observations is a fundamental challenge in a range of disciplines including machine learning, robotics, and natural language processing. While there are a wide variety of different approaches to modelling time series data, many of these approaches can be categorized as either recursive Bayes Filtering or Recurrent Neural Networks. Bayes Filters (BFs) <cit.> focus on modeling and maintaining a belief state: a set of statistics, which, if known at time t, are sufficient to predict all future observations as accurately as if we know the full history.The belief state is generally interpreted as the statistics of a distribution over the latent state of the data generating process conditioned on history. BFs recursively update the belief state by conditioning on new observations using Bayes rule.Examples of common BFs include sequential filtering in Hidden Markov Models (HMMs) <cit.> and Kalman Filters (KFs) <cit.>. Predictive State Representations <cit.> (PSRs) are a variation on Bayes filters that do not define system state explicitly, but proceed directly to a representation of state as the statistics of a distribution of features of future observations, conditioned on history. By defining the belief state in terms of observables rather than latent states, PSRs can be easier to learn than other filtering methods <cit.>. PSRs also support rich functional forms through kernel mean map embeddings <cit.>, and a natural interpretation of model update behavior as a gating mechanism. We note this last is not unique to PSRs, as it is also possible to interpret the model updates of other BFs such as HMMs in terms of gating. Due to their probabilistic grounding, BFs and PSRs possess a strong statistical theory leading to efficient learning algorithms. In particular, method-of-moments algorithms provide consistent parameter estimates for a range of BFs including PSRs <cit.>. Unfortunately, current versions of method of moments initialization restrict BFs to relatively simple functional forms such as linear-Gaussian (KFs) or linear-multinomial (HMMs). Recurrent Neural Networks (RNNs) are an alternative to BFs that model sequential data via a parameterized internal state and update function. In contrast to BFs, RNNs are directly trained tominimize output prediction error, without adhering to any axiomatic probabilistic interpretation. Examples of popular RNN models include Long-Short Term Memory networks <cit.> (LSTMs), Gated Recurrent Units <cit.> (GRUs), and simple recurrent networks such as Elman networks <cit.>. RNNs have several advantages over BFs. Their flexible functional form supports large, rich models. And, RNNs can be paired with simple gradient-based training procedures that achieve state-of-the-art performance on many tasks <cit.>. RNNs have drawbacks however: unlike BFs, RNNs lack an axiomatic probabilistic interpretation, and are therefore difficult to analyze. Furthermore, despite strong performance in some domains, RNNs are notoriously difficult to train; in particular it is difficult to find good initializations.In summary, RNNs and BFs offer complementary advantages and disadvantages: RNNs offer rich functional forms at the cost of statistical insight, while BFs possess a sophisticated statistical theory but are restricted to simpler functional forms in order to maintain tractable training and inference. By drawing insights from both Bayes Filters and RNNs we develop a novel hybrid model, Predictive State Recurrent Neural Networks (PSRNNs).Like many successful RNN architectures, PSRNNs use (potentially deeply composed) bilinear transfer functions to combine information from multiple sources. We show that such bilinear functions arise naturally from state updates in Bayes filters like PSRs, in which observations can be viewed as gating belief states. We show that PSRNNs directly generalize discrete PSRs, and can be learned effectively by combining Backpropogation Through Time (BPTT) with an approximately consistent method-of-moments initialization called two-stage regression (2SR). We also show that PSRNNs can be factorized using tensor decomposition, reducing model size and suggesting interesting connections to existing multiplicative architectures such as LSTMs. Finally, we note that our initialization for PSRNNs is approximately consistent in the case of discrete data.§ RELATED WORKIt is well known that a principled initialization can greatly increase the effectiveness of local search heuristics. For example, <cit.> and <cit.> use subspace ID to initialize EM for linear dyanmical systems, and <cit.> use N4SID <cit.> to initialize GP-Bayes filters. Existing work similar to ours can be organized into two main categories: 1)methods which attempt to use BFs to initialize RNNs <cit.>; and 2) methods which attempt to use BPTT to refine Bayes Filters <cit.>. Researchers have been experimenting with the latter strategy for several decades <cit.>. <cit.> propose an HMM-based pre-training algorithm for RNNs by first training an HMM, then using this HMM to generate a new, simplified dataset, and, finally, initializing the RNN weights by training the RNN on thisdataset. <cit.> propose a two-stage algorithm for learning a KF on text data. Their approach consists of a spectral initialization, followed by fine tuning via BPTT. They show that this approach has clear advantages over either spectral learning or BPTT in isolation. Despite these advantages, KFs make restrictive linear-Gaussian assumptions that preclude their use on many interesting problems. <cit.> propose a two-stage algorithm for learning discrete PSRs, consisting of a spectral initialization followed by BPTT. While this work is similar in spirit, it is still an attempt to optimize a BF using BPTT rather than an attempt to construct a true hybrid model. This results in several key differences: they focus on the discrete setting, and they optimize only a subset of the model parameters.§ BACKGROUND §.§ Predictive State Representations Predictive state representations (PSRs) <cit.> are a class of models for filtering, prediction, and simulation of discrete time dynamical systems. PSRs provide a compact representation of a dynamical system by representing state as a set of predictions of features of future observations.We define a predictive state q_t = q_t| t-1 = E[f_t |h_t], where f_t = f(o_t:t+k-1) is a vector of features of future observations and h_t = h(o_1:t-1) is a vector of features of historical observations. The features are selected such that q_t determines the distribution of future observations P(o_t:t+k-1|o_1:t-1).[For convenience we assume that the system is k-observable: that is, the distribution of all future observations is determined by the distribution of the next k observations. (Note: not by the next k observations themselves.) At the cost of additional notation, this restriction could easily be lifted.] Filtering is the process of mapping a predictive state q_t to q_t+1 conditioned on o_t, while prediction maps a predictive state q_t = q_t| t-1 to q_t+k| t-1 = E[f_t+k|o_1:t-1] without intervening observations.PSRs were originally developed for discrete data as a generalization of existing Bayes Filters such as HMMs <cit.>. However, by leveraging the recent concept of Hilbert Space embeddings of distributions <cit.>, which can be used to embed the PSR in a Hilbert Space, we can generalize the PSR to continuous observations <cit.>. Hilbert Space Embeddings of PSRs (HSE-PSRs) <cit.> represent the state as one or more nonparametric conditional embedding operators in a Reproducing Kernel Hilbert Space (RKHS) <cit.> and use Kernel Bayes Rule (KBR) <cit.> to estimate, predict, and update the state. For a full treatment of HSE-PSRs see <cit.>. Let k_f, k_h, k_o be translation invariant kernels <cit.> defined on f_t, h_t, and o_t respectively. We use Random Fourier Features <cit.> (RFF) to define projections ϕ_t = 𝑅𝐹𝐹(f_t), η_t = 𝑅𝐹𝐹(h_t), and ω_t = 𝑅𝐹𝐹(o_t) such that k_f(f_i, f_j) = ϕ_i^Tϕ_j,k_h(h_i, h_j) = η_i^Tη_j, k_o(o_i, o_j) = ω_i^Tω_j. Using this notation, the HSE-PSR predictive state is q_t = E[ϕ_t |η_t]. Formally an HSE-PSR (hereafter simply referred to as a PSR) consists of an initial state b_1, a 3-mode update tensor W, and a 3-mode normalization tensor Z. The PSR update equation isq_t+1 = (W ×_3 q_t)( Z ×_3 q_t)^-1×_2 o_t. §.§ Two-stage RegressionHefny et al. <cit.> show that PSRs can be learned by solving a sequence of regression problems. This approach, referred to as Two-Stage Regression or 2SR, is fast, statistically consistent, and reduces to simple linear algebra operations. In 2SR the PSR model parameters q_1, W, and Z are learned via the following set of equations:q_1 = 1/T∑_t=1^T ϕ_t W =(∑_t=1^T ϕ_t+1⊗ω_t ⊗η_t )(∑_t=1^T η_t ⊗ϕ_t)^+ Z =(∑_t=1^T ω_t ⊗ω_t ⊗η_t ) (∑_t=1^T η_t ⊗ϕ_t)^+.where ∑_t=1^T ϕ_t+1⊗ω_t ⊗η_t, ∑_t=1^T ω_t ⊗ω_t ⊗η_t and ∑_t=1^T η_t ⊗ϕ_t are all estimated via regression. Here + is the Moore-Penrose pseudo-inverse. We note that multiplying by the pseudo-inverse is also equivalent to solving a least squares regression problem, hence the name 2SR. Finally in practice we use ridge regression in order to improve model stability, and minimize the destabilizing effect of rare events while preserving consistency.Once we learn model parameters, we can apply the filtering equation (<ref>) to obtain predictive states q_1:T. In some settings (such as the case of discrete data) it is possible to predict observations directly from the state; however, in order to generalize to the continuous setting with RFF features we train a regression model to predict ω_t from q_t.With RFF features, linear regression has been sufficient for our purposes.[Note that we can train a regression model to predict any quantity from the state. This is useful for general sequence-to-sequence mapping models. However, in this work we focus on predicting future observations.] §.§ Tensor DecompositionThe tensor Canonical Polyadic decomposition (CP decomposition) <cit.> can be viewed as a generalization of the Singular Value Decomposition (SVD) to tensors. If T ∈ℝ^(d_1×...× d_k) is a tensor, then a CP decomposition of T is:T = ∑_i=1^ma_i^1 ⊗ a_i^2 ⊗ ... ⊗ a_i^kwhere a_i^j ∈^d_j and ⊗ is the Kronecker product. The rank of T is the minimum m such that the above equality holds. In other words, the CP decomposition represents T as a sum of rank-1 tensors. § PREDICTIVE STATE RECURRENT NEURAL NETWORKS In this section we introduce Predictive State Recurrent Neural Networks (PSRNNs), a new RNN architecture inspired by PSRs. PSRNNs allow for a principled initialization and refinement via BPTT. The key contributions which led to the development of PSRNNs are: 1) a new normalization scheme for PSRs which allows for effective refinement via BPTT; 2) the extention of the 2SR algorithm to a multilayered architecture; and 3) the optional use of a tensor decomposition to obtain a more scalable model. §.§ ArchitectureThe basic building block of a PSRNN is a 3-mode tensor, which can be used to compute a bilinear combination of two input vectors. We note that, while bilinear operators are not a new development (e.g., they have been widely used in a variety of systems engineering and control applications for many years <cit.>), the current paper shows how to chain these bilinear components together into a powerful new predictive model. Let q_t and o_t be the state and observation at time t. Let W be a 3-mode tensor, and let q be a vector. The 1-layer state update for a PSRNN is defined as:q_t+1 = W ×_2 o_t ×_3 q_t + b/W ×_2 o_t ×_3 q_t + b_2Here the 3-mode tensor of weights W and the bias vector b are the model parameters.[We define A ×_p B to the be tensor contraction of A and B along the pth mode, e.g., [W ×_2 q]_i,j = ∑_k W_i,k,j q_k.] This architecture is illustrated in Figure <ref>. It is similar, but not identical, to the PSR update (Eq. <ref>); sec <ref> gives more detail on the relationship.This model may appear simple, but crucially the tensor contraction W ×_2 o_t ×_3 q_t integrates information from b_t and o_t multiplicatively, and acts as a gating mechanism, as discussed in more detail in section <ref>.The typical approach used to increase modeling capability for BFs (including PSRs) is to use an initial fixed nonlinearity to map inputs up into a higher-dimensional space <cit.>. However, a multilayered architecture typically offers higher representation power for a given number of parameters <cit.>. We now present a Multilayer PSRNN architecture which benefits from both of the above. To obtain a multilayer PSRNN, we stack the 1-layer blocks of Eq. (<ref>) by providing the output of one layer as the observation for the next layer. (The state input for each layer remains the same.) In this way we can obtain arbitrarily deep RNNs. This architecture is displayed in Figure <ref>.We choose to chain on the observation (as opposed to on the state) as this architecture leads to a natural extension of 2SR to multilayered models (see Sec. <ref>). In addition, this architecture is consistent with the typical approach for constructing multilayered LSTMs/GRUs <cit.>. Finally, this architecture is suggested by the full normalized form of an HSE PSR, where the observation is passed through two layers. §.§ Learning PSRNNsThere are two components to learning PSRNNs: an initialization procedure followed by gradient-based refinement. We first show how a statistically consistent 2SR algorithm derived for PSRs can be used to initialize the PSRNN model; this model can then be refined via BPTT. We omit BPTT equations as they are similar to existing literature, and can be easily obtained via automatic differentiation in a neural network library such as PyTorch or TensorFlow.The Kernel Bayes Rule portion of the PSR update (equation <ref>) can be separated into two terms: (W ×_3 q_t) and ( Z ×_3 q_t)^-1. The first term corresponds to calculating the joint distribution, while the second term corresponds to normalizing the joint to obtain the conditional distribution. In the discrete case, this is equivalent to dividing the joint distribution of f_t+1 and o_t by the marginal of o_t; see <cit.> for details.If we remove the normalization term, and replace it with two-norm normalization, the PSR update becomes q_t+1 = W ×_3 q_t ×_2 o_t/W ×_3 q_t ×_2 o_t, which corresponds to calculating the joint distribution (up to a scale factor), and has the same functional form as our single-layer PSRNN update equation (up to bias).It is not immediately clear that this modification is reasonable. We show in appendix <ref> that our algorithm is consistent in the discrete (realizable) setting; however, to our current knowledge we lose the consistency guarantees of the 2SR algorithm in the full continuous setting. Despite this we determined experimentally that replacing full normalization with two-norm normalization appears to have a minimal effect on model performance prior to refinement, and results in improved performance after refinement. Finally,we note that working with the (normalized) joint distribution in place of the conditional distribution is a commonly made simplification in the systems literature, and has been shown to work well in practice <cit.>. The adaptation of the two-stage regression algorithm of Hefny et al. <cit.> described above allows us to initialize 1-layer PSRNNs; however, it is not immediately clear how to extend this approach to multilayered PSRNNs. But, suppose we have learned a 1-layer PSRNN P using two-stage regression. We can use P to perform filtering on a dataset to generate a sequence of estimated states q̂_1,...,q̂_n. According to the architecture described in Figure <ref>, these states are treated as observations in the second layer. Therefore we can initialize the second layer by an additional iteration of two-stage regression using our estimated states q̂_1,...,q̂_n in place of observations. This process can be repeated as many times as desired to initialize an arbitrarily deep PSRNN. If the first layer were learned perfectly, the second layer would be superfluous, however, in practice, we observe that the second layer is able to learn to improve on the first layer's performance.Once we have obtained a PSRNN using the 2SR approach described above, we can use BPTT to refine the PSRNN. BPPT unrolls the state update over time to generate a feedfoward neural network with tied weights, which can then be used to perform gradient descent updates on the model parameters. We note that one of the we choose to use 2-norm divisive normalization because it is not practical to perform BPTT through the matrix inverse required in PSRs. We observe that 2SR provides us with an initialization which converges to a good local optimum. §.§ Factorized PSRNNs In this section we show how the PSRNN model can be factorized to reduce the number of parameters prior to applying BPTT.Let (W, b_0) be a PSRNN block. Suppose we decompose W using CP decomposition to obtainW = ∑_i=1^n a_i ⊗ b_i ⊗ c_iLet A (similarly B, C) be the matrix whose ith row is a_i (respectively b_i, c_i). Then the PSRNN state update (equation (<ref>)) becomes (up to normalization):q_t+1 = W ×_2 o_t ×_3 q_t + b= (A ⊗ B ⊗ C) ×_2 o_t ×_3 q_t + b= A^T(B o_t ⊙ C q_t ) + bwhere ⊙ is the Hadamard product. We call a PSRNN of this form a factorized PSRNN. This model architecture is illustrated in Figure <ref>. Using a factorized PSRNN provides us with complete control over the size of our model via the rank of the factorization. Importantly, it decouples the number of model parameters from the number of states, allowing us to set these two hyperparameters independently.We determined experimentally that factorized PSRNNs are poorly conditioned when compared with PSRNNs, due to very large and very small numbers often occurring in the CP decomposition. To alleviate this issue, we need to initialize the bias b in a factorized PSRNN to be a small multiple of the mean state. This acts to stabilize the model, regularizing gradients and preventing us from moving away from the good local optimum provided by 2SR. We note that a similar stabilization happens automatically in randomly initialized RNNs: after the first few iterations the gradient updates cause the biases to become non-zero, stabilizing the model and resulting in subsequent gradient descent updates being reasonable. Initialization of the biases is only a concern for us because we do not want the original model to move away from our carefully prepared initialization due to extreme gradients during the first few steps of gradient descent.In summary, we can learn factorized PSRNNs by first using 2SR to initialize a PSRNN, then using CP decomposition to factorize the tensor model parameters to obtain a factorized PSRNN, then applying BPTT to the refine the factorized PSRNN.§ DISCUSSIONThe value of bilinear units in RNNs was the focus of recent work by Wu et al <cit.>. They introduced the concept of Multiplicative Integration (MI) units — components of the form Ax ⊙ By — and showed that replacing additive units by multiplicative ones in a range of architectures leads to significantly improved performance. As Eq. (<ref>) shows, factorizing W leads precisely to an architecture with MI units.Modern RNN architectures such as LSTMs and GRUs are known to outperform traditional RNN architectures on many problems <cit.>. While the success of these methods is not fully understood, much of it is attributed to the fact that these architectures possess a gating mechanism which allows them both to remember information for a long time, and also to forget it quickly. Crucially, we note that PSRNNs also allow for a gating mechanism. To see this consider a single entry in the factorized PSRNN update (omitting normalization).[q_t+1]_i= ∑_j A_ji(∑_k B_jk [o_t]_k ⊙∑_l C_jl [q_t]_l ) + bThe current state q_t will only contribute to the new state if the function ∑_k B_jk [o_t]_k of o_t is non-zero. Otherwise o_t will cause the model to forget this information: the bilinear component of the PSRNN architecture naturally achieves gating.We note that similar bilinear forms occur as components of many successful models. For example, consider the (one layer) GRU update equation:z_t= σ(W_z o_t + U_z q_t + c_z)r_t= σ(W_r o_t + U_r q_t + c_r)q_t+1 = z_t ⊙ q_t + (1-z_t) ⊙σ(W_h o_t + U_h(r_t ⊙ q_t) + c_h)The GRU update is a convex combination of the existing state q_t and and update term W_h o_t + U_h(r_t ⊙ q_t) + c_h. We see that the core part of this update term U_h(r_t ⊙ q_t) + c_h bears a striking similarity to our factorized PSRNN update. The PSRNN update is simpler, though, since it omits the nonlinearity σ(·), and hence is able to combine pairs of linear updates inside and outside σ(·) into a single matrix. Finally, we would like to highlight the fact that, as discussed in section <ref>, the bilinear form shared in some form by these models (including PSRNNs) resembles the first component of the Kernel Bayes Rule update function. This observation suggests that bilinear components are a natural structure to use when constructing RNNs, and may help explain the success of the above methods over alternative approaches. This hypothesis is supported by the fact that there are no activation functions (other than divisive normalization) present in our PSRNN architecture, yet it still manages to achieve strong performance. § EXPERIMENTAL SETUP In this section we describe the datasets, models, model initializations, model hyperparameters, evaluation metrics used in our experiments. All models were implemented using the PyTorch framework in Python. We plan to release code for these experiments soon.We use the following datasets in our experiments: * Penn Tree Bank (PTB) This is a standard benchmark in the NLP community <cit.>. Due to hardware limitations we use a train/test split of 120780/124774 characters.* Swimmer We consider the 3-link simulated swimmer robot from the open-source package OpenAI gym.[<https://gym.openai.com/>] The observation model returns the angular position of the nose as well as the angles of the two joints. We collect 25 trajectories from a robot that is trained to swim forward (via the cross entropy with a linear policy), with a train/test split of 20/5.RLPy <cit.> with the goal of predicting the nose position of the robot. The 2-d action consists of torques applied on the two joints of the links. The observation model returns the angles of the joints and the position of the nose (in body coordinates).The measurements are contaminated with Gaussian noise whose standard deviation is 5% of the true signal standard deviation. We generate test trajectories using a mixed policy: with probability 0.2, we sample a uniformly random action, while with probability 0.8, we sample an action from a pre-specified deterministic policy that seeks a goal point.* Mocap This is a Human Motion Capture dataset consisting of 48 skeletal tracks from three human subjects collected while they were walking. The tracks have 300 timesteps each, and are from a Vicon motion capture system. We use a train/test split of 40/8. Features consist of the 3D positions of the skeletal parts (e.g., upper back, thorax, clavicle).* Handwriting This is a digit database available on the UCI repository <cit.> created using a pressure sensitive tablet and a cordless stylus. Features are x and y tablet coordinates and pressure levels of the pen at a sampling rate of 100 milliseconds. We use 25 trajectories with a train/test split of 20/5. We note that, given the diversity of these datasets, it is a challenge for one method to perform well on all of them. Models compared are LSTMs <cit.>, GRUs <cit.>, basic RNNs <cit.>, KFs <cit.>, PSRNNs, and factorized PSRNNs. All models except KFs consist of a linear encoder, a recurrent module, and a linear decoder. The encoder maps observations to a compressed representation; it can be viewed as a word embedding in the context of text data. The recurrent module maps a state and an observation to a new state and an output. The decoder maps an output to a predicted observation.[This is a standard RNN architecture; e.g., a PyTorch implementation of this architecture for text prediction can be found at <https://github.com/pytorch/examples/tree/master/word_language_model>.]We initialize the LSTMs and RNNs with random weights and zero biases according to the Xavier initialization scheme <cit.>. We initialize the the KF using the 2SR algorithm described in <cit.>. We initialize PSRNN and factorized PSRNN weights as described in section <ref>. We note that if we initialize PSRNNs or Factorized PSRNNs using random weights, BPTT fails and we cannot learn a usable model.In two-stage regression we use a ridge-regression parameter of 10^(-2)n where n is the number of training examples (this is consistent with the values suggested in <cit.>). (Experiments show that our approach works well for a wide variety of hyperparameter values.) We use a horizon of 1 in the PTB experiments, and a horizon of 10 in all continuous experiments. We use 2000 RFFs from a Gaussian kernel, selected according to the method of <cit.>, and with the kernel width selected as the median pairwise distance. We use 20 hidden states, and a fixed learning rate of 1 in all experiments. We use a BPTT horizon of 35 in the PTB experiments, and an infinite BPTT horizon in all other experiments. All models are single layer unless stated otherwise.We optimize models on the PTB using Bits Per Character (BPC) and evaluate them using both BPC and one-step prediction accuracy (OSPA). We optimize and evaluate all continuous experiments using the Mean Squared Error (MSE). § RESULTSIn Figure <ref> we compare performance of LSTMs, GRUs, and Factorized PSRNNs on PTB, where all models have the same number of states and approximately the same number of parameters. To achieve this we use a factorized PSRNN of rank 60. We see that the factorized PSRNN significantly outperforms LSTMs and GRUs on both metrics. In Figure <ref> we compare the performance of 1 and 2 layer PSRNNs on PTB. We see that adding an additional layer significantly improves performance. In Figure <ref> we compare PSRNNs with factorized PSRNNs on the PTB. We see thatPSRNNs outperform factorized PSRNNs regardless of rank, even when the factorized PSRNN has significantly more model parameters. (In this experiment, factorized PSRNNs of rank 7 or greater have more model parameters than a plain PSRNN.) This observation makes sense, as the PSRNN provides a simpler optimization surface: The tensor multiplication in each layer of a PSRNN is linear with respect to the model parameters, while the tensor multiplication in each layer of a Factorized PSRNN is bilinear. One obvious question raised by this result is whether we can represent LSTMs or GRUs in a similar factorized form.In addition, we see that higher-rank factorized models outperform lower-rank ones. However, it is worth noting that even models with low rank still perform well, as demonstrated by our rank 40 model still outperforming GRUs and LSTMs, despite having fewer parameters. In Figure <ref> we compare model performance on the Swimmer, Mocap, and Handwriting datasets. We see that PSRNNs significantly outperform alternative approaches on all datasets. In Figure <ref> we attempt to gain insight into why using 2SR to initialize our models is so beneficial. We visualize the the one step model predictions before and after BPTT. We see that the behavior of the initialization has a large impact on the behavior of the refined model. For example the initial (incorrect) oscillatory behavior of the RNN in the second column is preserved even after gradient descent.§ CONCLUSIONSWe present PSRNNs: a new approach for modelling time-series data that hybridizes Bayes filters with RNNs. PSRNNs have both a principled initialization procedure and a rich functional form. The basic PSRNN block consists of a 3-mode tensor, corresponding to bilinear combination of the state and observation, followed by divisive normalization. These blocks can be arranged in layers to increase the expressive power of the model. We showed that tensor CP decomposition can be used to obtain factorized PSRNNs, which allow flexibly selecting the number of states and model parameters. We showed how factorized PSRNNs can be viewed as both an instance of Kernel Bayes Rule and a gated architecture, and discussed links to existing multiplicative architectures such as LSTMs. We applied PSRNNs to 4 datasets and showed that we outperform alternative approaches in all cases. unsrtnat§ ON THE CONSISTENCY OF INITIALIZATION In this section we provide a theoretical justification for the PSRNN. Specifically, we show that in the case of discrete observations and a single layer the PSRNN provides a good approximation to a consistent model. We first show that in the discrete setting using a matrix inverse is equivalent to a sum normalization. We subsequently show that, under certain conditions, two-norm normalization has the same effect as sum-normalization. .Let q_t be the PSR state at time t, and o_t be the observation at time t (as an indicator vector). In this setting the covariance matrix C_t = E[o_t × o_t | o_1:t-1] will be diagonal. By assumption, the normalization term Z in PSRs is defined as a linear function from q_t to C_t, and when we learn PSRN by 2-stage regression we estimate this linear function consistently. Hence, for all q_t, Z ×_3 q_t is a diagonal matrix, and ( Z ×_3 q_t)^-1 is also a diagonal matrix. Furthermore, since o_t is an indicator vector, ( Z ×_3 q_t)^-1×_2 o_t = o_t/P(o_t) in the limit. We also know that as a probability distribution, q_t should sum to one. This is equivalent to dividing the unnormalized update q̂_t+1 by its sum. i.e.q_t+1 = q̂_t+1/P(o_t) = q̂_t+1/(1^Tq̂_t+1)Now consider the difference between the sum normalization q̂_t+1/(1^Tq̂_t+1) and the two-norm normalization q̂_t+1/q̂_t+1_2. Since q_t is a probability distribution, all elements will be positive, hence the sum norm is equivalent to the 1-norm. In both settings, normalization is equivalent to projection onto a norm ball. Now let S be the set of all valid states. Then if the diameter of S is small compared to the distance from (the convex hull of) S to the origin then the local curvature of the 2-norm ball will be negligible, and both cases will be approximately equivalent to projection onto a plane. We note we can obtain an S with this property by augmenting our state with a set of constant features. | http://arxiv.org/abs/1705.09353v2 | {
"authors": [
"Carlton Downey",
"Ahmed Hefny",
"Boyue Li",
"Byron Boots",
"Geoffrey Gordon"
],
"categories": [
"stat.ML"
],
"primary_category": "stat.ML",
"published": "20170525204013",
"title": "Predictive State Recurrent Neural Networks"
} |
=1Department of Physics, Kent State University, Kent, OH 44242 United StatesInstitute of Nuclear Physics, Polish Academy of Sciences, PL-31342 Kraków, PolandDepartment of Physics, Kent State University, Kent, OH 44242 United States We compare phenomenological results from 3+1d quasiparticle anisotropic hydrodynamics (aHydroQP)with experimental data collected in LHC 2.76 TeV Pb-Pb collisions. In particular, we present comparisons of particle spectra, average transverse momentum, elliptic flow, and HBT radii. The aHydroQP model relies on the introduction of a single temperature-dependent quasiparticle mass which is fit to lattice QCD data. By taking moments of the resulting Boltzmann equation, we obtain the dynamical equations used in the hydrodynamic stage which include the effects of both shear and bulk viscosities. At freeze-out, we use anisotropic Cooper-Frye freeze-out performed on a fixed-energy-density hypersurface to convert to hadrons. To model the production and decays of the hadrons we useTHERMINATOR 2 which is customized to sample from ellipsoidal momentum-space distribution functions. Using smooth Glauber initial conditions, we find very good agreement with many heavy-ion collision observables.12.38.Mh, 24.10.Nz, 25.75.Ld, 47.75.+f, 31.15.xmAnisotropic hydrodynamic modeling of 2.76 TeV Pb-Pb collisions Michael Strickland December 30, 2023 ============================================================== § INTRODUCTION Relativistic hydrodynamics has been quite successful in describing the soft hadron spectra (p_T ≲ 2 GeV) and collective flow observed in ultrarelativistic heavy-ion collision (URHIC) experiments at the Relativistic Heavy Ion Collider (RHIC) and Large Hadron Collider (LHC) <cit.> (see <cit.> for recent reviews).Early on, the success of this program suggested that the quark-gluon plasma (QGP) generated in URHICs was nearly isotropic in the local rest frame (LRF); however, in practice, one finds that there are rather large momentum-space anisotropies, which are driven primarily by the rapid longitudinal expansion of the QGP created in URHICs <cit.>.All studies indicate that at early times after the nuclear impact, the QGP possesses a high degree of momentum-space anisotropy in the fluid LRF, P_T/ P_L ≫ 1, which only slowly relaxes towards unity during the QGP liftetime.Additionally, at all proper times, there are large momentum-space anisotropies near the transverse/longitudinal “edges” of the fireball where the system is nearly free streaming.The situation only gets worse as one goes from AA to pA and pp collisions, since gradients are larger and system lifetimes are considerably shorter.As a consequence, larger non-equilibrium deviations are expected for these systems, which push traditional viscous hydrodynamics to its limits <cit.>.This has motivated the investigation of alternative formulations of dissipative relativistic hydrodynamics which can be applied to systems which might possess a high degree of momentum-space anisotropy at all points in spacetime <cit.>.One way to proceed is to reorganize the expansion of the one-particle distribution function around a leading-order form which possesses intrinsic momentum-space anisotropies but still guarantees positivity <cit.>.This method has become known as anisotropic hydrodynamics (aHydro) and there are now many groups pursuing this idea <cit.>.One of the selling points for the aHydro approach has been that it better reproduces exact solutions to the Boltzmann equation compared to traditional near-equilibrium viscous hydrodynamic approaches, even in the limit of very large shear viscosity to entropy density ratio and/or initial momentum-space anisotropy <cit.>.Given this success, the focus has recently turned to making aHydro a practical phenomenological tool with a realistic equation of state (EoS) and self-consistent anisotropic hadronic freeze-out.In a previous short paper <cit.>, we presented the first comparisons of experimental data with phenomenological results obtained using generalized 3+1d aHydro including:(1) three momentum-space anisotropy parameters in the underlying distribution function, (2) the quasiparticle aHydro (aHydroQP) method for implementing a realistic EoS <cit.>, and (3) anisotropic Cooper-Frye freeze-out <cit.> using the same form for the distribution function as was assumed for the dynamical equations.To the best of our knowledge, all previous phenomenological applications of aHydro have either relied on the approximate conformal factorization of the energy-momentum tensor, see e.g. <cit.>, and/or have used isotropic freeze-out <cit.>.To address the issue with freeze-out, we use a customized version of THERMINATOR 2 <cit.> which has been modified to accept ellipsoidally-anisotropic distribution functions.In this paper, we extend our previous work <cit.> by (1) giving the details of the formalism of 3+1d aHydroQP and (2) presenting comparisons with a wider set of heavy-ion observables. Here we present comparisons of charged-hadron multiplicity, identified-particle spectra, identified-particle average transverse momentum, charged-particle elliptic flow, identified-particle elliptic flow, the integrated elliptic flow vs pseudorapidity, and the HBT radii. For some observables, such asthe spectra, compared to Ref. <cit.> we present comparisons in more centrality classes and with higher statistics. The structure of the paper is as follows. In Sec. <ref>, we present our general setup. In Sec. <ref>, weintroduce the formalism for quasiparticle anisotropic hydrodynamics and then derivethe 3+1d dynamical equations. In Sec. <ref>, we discuss anisotropic Cooper-Frye freeze-out. In Sec. <ref>, we compare our model results obtained using the 3+1d aHydroQP model for Pb-Pb collisions at LHC energies with data from the ALICE collaboration. Sec. <ref> contains our conclusions and an outlook for the future. In App. <ref>, we list the derivatives used in the body of the paper. App. <ref> contains details concerning the thermodynamic integrals introduced in the body of the text and our optimized scheme for their evaluation. § SETUP§.§ Conventions and notationHerewe define some conventions and notation that will be used in the body of this paper. The metric is taken to be “mostly minus” with x^μ = (t, x, y, z)where the line element is ds^2=g_μν dx^μ dx^ν=dt^2-dx^2-dy^2-dz^2 with g^μν being metric tensor in Minkowski space. The longitudinal proper time is τ = √(t^2-z^2) and the longitudinal spacetime rapidity is ς = tanh^-1 (z/t).The basis vectors for a general 3+1d system in the laboratory frame can be obtained by the following parametrization, which is based on a set of Lorentz transformations applied to the LRF basis vectors <cit.>. The set of successive transformations correspond to a longitudinal boost ϑ along the beam line, a rotation φ≡tan^-1(u_y/u_x) around the beam line, and a transverse boost θ_⊥, which together yield u^μ ≡(u_0 coshϑ,u_x,u_y,u_0 sinhϑ) , X^μ ≡ (u_⊥coshϑ,u_0 u_x/u_⊥,u_0 u_y/u_⊥,u_⊥sinhϑ) ,Y^μ ≡ (0,-u_y/u_⊥,u_x/u_⊥,0), Z^μ ≡(sinhϑ,0,0,coshϑ ) ,where u^μ is the fluid four-velocity and u_⊥≡√(u_x^2+u_y^2)=√(u_0^2-1) = sinhθ_⊥.§.§ Distribution Function The leading order distribution function in aHydro is assumed to be of generalized Romatschke-Strickland form <cit.>f(x,p) = f_ eq(1/λ√(p_μΞ^μν p_ν)) ,where λ is an energy scale which becomes the temperature in the isotropic equilibrium limit. The anisotropy tensor has the form Ξ^μν≡ u^μ u^ν + ξ^μν - Δ^μνΦ where ξ^μν is a symmetric traceless tensor obeying u_μξ ^μν = 0 and ξ^μ_μ = 0, Φ is the bulk degree of freedom, and Δ^μν = g^μν - u^μ u^ν is the transverse projector <cit.>. Using the ellipsoidal form (<ref>) and the tracelessness of ξ^μν, we are left with three independent parameters out of the four original parameters Φ and ξ = (ξ_x,ξ_y,ξ_z). In thermal equilibrium, the distribution function f_ eq(x) can be identified as Fermi-Dirac, Bose-Einstein, or Maxwellian distribution. Herein, ignoring the quantum statistics, we take the Boltzmann form with zero chemical potential. We note that, in order to perform the integrals, it is useful to define three parameters α_i as the system's independent anisotropy parametersα_i ≡ (1 + ξ_i + Φ)^-1/2,such that the distribution function can be written asf(x,p)= f_ eq(1/λ√(∑_i p_i^2/α_i^2+m^2)) . §.§ The equation of state We use an analytic parameterization of lattice QCD (LQCD) data for the trace anomaly taken from the Wuppertal-Budapest collaboration <cit.>.This analytic parameterization can be used to calculate the energy density, pressure, and entropy density using standard thermodynamic identities.In order to implement the equation of state in the quasiparticle model, we fit a single temperature-dependent mass, m(T), to the LQCD entropy density.For details, we refer the reader toRef. <cit.>.§.§ Shear and bulk viscosity in the quasiparticle (QP) model The relaxation time τ_ eq can be related to the shear viscosity, η, for the system of quasiparticles.The shear viscosity for a quasiparticle gas can be found in Refs. <cit.> and <cit.>, Eqs. (4.3) and (46), respectively, with both giving the same resultη/τ_ eq = 1/T I_3,2(m̂_ eq),with m̂_ eq≡ m/T and I_3,2(x)= N_ dof T^5x^5/30 π^2[ 1/16(K_5(x)-7K_3(x)+22 K_1(x) )-K_i,1(x) ],K_i,1(x) = π/2[1-x K_0(x)S_-1(x)-xK_1(x)S_0(x)] ,where N_ dof is the number of degrees of freedom, K_n are modified Bessel functions of the second kind, and S_n are modified Struve functions.Using Eq. (<ref>), the relaxation time can be written asτ_ eq(T)= η̅E+P/I_3,2(m̂_ eq),where η̅≡η/s with s being the entropy density, E is the energy density, and P is the pressure. Using the quasiparticle model, one can extract the bulk viscosity in a similar manner.Expressions for the bulk viscosity for a quasiparticle gas can be found in Refs. <cit.> and <cit.>, Eqs. (4.4) and (45), respectively, with, both again giving the same resultζ/τ_ eq =5/3 T I_3,2 (m̂_ eq) - c_s^2 ( E + P)+ c_s^2 m dm/dT I_1,1(m̂_ eq) , whereI_1,1(x) = N_ dof T^3 x^3/6 π^2[ 1/4( K_3(x)-5K_1(x))+K_i,1 (x)] . In Fig. <ref>-a, we plot the result for the bulk viscosity to entropy density ratio ζ/s in the quasiparticle model as a black solid line. For comparison, we plot, where c_s is the speed of sound.This is an often-used small-mass expansion result <cit.>. For both curves, we assume that η/s = 2/(4π). In Fig. <ref>-b we plot m/T obtained by fitting to the Wuppertal-Budapest LQCD results for the entropy density <cit.>.We point out that, at small temperatures, the value of m/T necessary to fit the LQCD data <cit.> is not small, invalidating the often used small m/T expansion used to compute effective transport coefficients. The quasiparticle model has a finite bulk viscosity to entropy density ratio as shown in Fig. <ref>-a. It peaks in the vicinity of the phase transition temperature from QGP to a hadronic gas. By comparing to other ansätze for ζ/s used in other studies we see that, in our quasiparticle model, the peak value is much smaller ∼ 0.05, compared to prior works.For example, in Ref. <cit.> the respective peak value of ζ/s is approximately 0.3. § DYNAMICAL EQUATIONS For a system of quasiparticles with a temperature-dependent mass, the Boltzmann equation is <cit.>p^μ∂_μ f+1/2∂_i m^2∂^i_(p)f =- C[f] ,with i ∈{x,y,z}.Herein, we take the collisional kernel C[f] in the relaxation time approximation, C[f] = p_μ u^μ ( f - f_ eq)/τ_ eq.By taking moments of this equation we can generate the necessary dynamical equations.The kinetic part of the energy-momentum tensor can be obtained from the second moment of distribution function, however, when the quasiparticle mass is temperature dependent, this quantity is not conserved by itself.In order to enforce thermodynamic consistency and energy-momentum conservation, one must introduce the background field contribution <cit.>, B(T), which is fixed through comparison with LQCD data <cit.>T^μν=T^μν_ kinetic+B(T) g^μν,where T^μν_ kinetic = ∫ dP p^μ p^ν f(x,p) with dP ≡ E^-1 d^3p/(2π)^3 being the Lorentz-invariant integration measure.For the case considered here, namely a diagonal anisotropy tensor, the full energy-momentum tensor can be expanded as T^μν= Eu^μ u^ν+ P_x X^μ X^ν+ P_y Y^μ Y^ν+ P_z Z^μ Z^ν.The resulting energy density and pressures are E =H_3(α,m̂)λ^4+B ,P_i=H_3i(α,m̂)λ^4-B ,with i ∈{x,y,z} and the H-functions appearing above defined in App. <ref>. Taking the first moment of Boltzmann equation we obtain four equations D_u E+ Eθ_u+P_x u_μ D_xX^μ+P_y u_μ D_yY^μ + P_z u_μ D_zZ^μ = 0, D_xP_x+ P_xθ_x - EX_μ D_uu^μ - P_y X_μ D_yY^μ -P_z X_μ D_zZ^μ =0 , D_yP_y+ P_y θ_y- EY_μ D_uu^μ - P_x Y_μ D_xX^μ -P_z Y_μ D_zZ^μ =0 , D_zP_z+ P_z θ_z- EZ_μ D_uu^μ-P_x Z_μ D_xX^μ -P_y Z_μ D_yY^μ =0 .The second moment of Boltzmann equation involves a rank-3 tensor I which is the third moment of the distribution function I^μνλ≡p^μ p^ν p^λf(x,p) .Expanding I over the basis vectors one has I = I_u [ u⊗ u ⊗ u]+ I_x [ u⊗ X ⊗ X +X⊗ u ⊗ X + X⊗ X ⊗ u] +(X→ Y) +(X→ Z), with <cit.>I_i= α α_i^2 I_ eq(λ,m) , I_ eq(λ,m)= 4 πÑλ^5 m̂^3 K_3(m̂) ,where α = ∏_i α_i.Since we haveeight independent variables, α_x, α_y, α_z, u_x, u_x, ϑ, T, and λ, we need eight equations to solve the full 3+1d aHydroQP system. With ten equations obtained from the second moment of Boltzmann equation, the system is overdetermined. Therefore, we have to come up with a selection rule to choose a subset of these equations in order to close the set of dynamical equations. The final equations are taken from the three diagonal projections of the equation of motion of the third moment, X_μ X_ν∂_α I^αμν, Y_μ Y_ν∂_α I^αμν, and Z_μ Z_ν∂_α I^αμν giving <cit.>D_uI_x +I_x (θ_u + 2 u_μ D_x X^μ)= 1/τ_ eq[I_ eq(T,m) -I_x ] ,D_uI_y +I_y (θ_u + 2 u_μ D_y Y^μ)= 1/τ_ eq[I_ eq(T,m) -I_y ] , D_uI_z +I_z (θ_u + 2 u_μ D_z Z^μ)= 1/τ_ eq[I_ eq(T,m) -I_z ] .Finally, in order to compute the local effective temperature T, we match the non-equilibrium kinetic energy density with the equilibrium kinetic energy density H_3(α,m̂) λ^4 =H_3, eq(1,m̂_ eq) T^4. To summarize, to perform the numerical simulations reported herein we use the eight equations resulting from the first(<ref>)and second (<ref>)moments of the Boltzmann equation, together with the matching condition (<ref>).§ ANISOTROPIC FREEZE-OUT The QGP undergoes freeze-out at late times/low temperatures and the degrees of freedom need to be changed from hydrodynamical variables to hadronic positions and momenta. In this work, we perform “anisotropic Cooper-Frye freeze-out” using Eq. (<ref>) as the form for the one-particle distribution function. The anisotropic distribution function used in the freeze-out is guaranteed to be positive-definite, by construction, in all regions in phase space, avoiding the usual problems encountered within standard viscous hydrodynamic freeze-out. In practice, we construct a constant energy-density hypersurface, defined through T_ FO= E^-1( E_ FO). Then, by computing the number of particles that cross this hypersurface, one can determine the number of hadrons produced in heavy-ion collisions at freeze-out using (p^0dN/d^3p)_i= N_i/(2π)^3∫ f_i(x,p) p^μ d^3Σ_μ,where i labels the hadronic species, N_i≡ 2s_i+1 is the degeneracy factor with s_i being the spin of particle species i, and f_i is the distribution function for particle species i taking into account the appropriate quantum statistics.[In THERMINATOR 2, different isospin states are treated separately negating the need for an explicit isospin degeneracy factor.] For more details we refer the reader to <cit.>. § NUMERICAL RESULTS In this section, we presentcomparisons of our aHydroQP modelresults with √(s_NN) = 2.76 TeV Pb-Pb collision data available from the ALICE collaboration.To set the initial conditions, we assume the system to be initially isotropic in momentum space (α_i(τ_0)=1), with zero transverse flow (u_⊥(τ_0) =0), and Bjorken flow in the longitudinal direction (ϑ(τ_0) = η). The initial energy density distribution in the transverse plane is computed from a “tilted” profile <cit.>.The distribution used is a linear combination of smooth Glauber wounded-nucleon and binary-collision density profiles, with a binary-collision mixing factor of χ = 0.15.In the longitudinal direction, we used a profile with a central plateau and Gaussian “tails”, resulting in a longitudinal profile function of the form ρ(ς) ≡exp[ - (ς - Δς)^2/(2 σ_ς^2)Θ (|ς| - Δς) ] .The parameters entering (<ref>) were fitted to the pseudorapidity distribution of charged hadrons with the results being Δς = 2.3 and σ_ς = 1.6.The first quantity sets the width of the central plateau and the second sets the width of the Gaussian “tails”.The resulting initial energy density at a given transverse position x_⊥ and spatial rapidity ς was computed usingE( x_⊥,ς) ∝ (1-χ) ρ(ς) [ W_A( x_⊥) g(ς) + W_B( x_⊥) g(-ς)] + χρ(ς) C( x_⊥) , where W_A,B( x_⊥) is the wounded nucleon density for nucleus A or B, C( x_⊥) is the binary collision density, and g(ς) is the “tilt function”.The tilt functionis defined throughg(ς) = {[0ς < -y_N ,; (ς+y_N)/(2y_N)-y_N ≤ς≤ y_N ,;1ς > y_N, ]. where y_N = log(2√(s_NN)/(m_p + m_n)) is the nucleon momentum rapidity <cit.>.We solved the aHydroQP dynamical equations on a 64^3 lattice with lattice spacings Δ x = Δ y = 0.5 fm and . To compute spatial derivatives we used fourth-order centered-differences and, for temporal updates, we used fourth-order Runge-Kutta with step size of Δτ = 0.02 fm/c. To regulate potential numerical instabilities associated with the centered-differences scheme, we used a weighted-LAX smoother <cit.>. In most cases, we set the weighted-LAX fraction to be 0.005, however, for large impact parameters we used 0.02.[This does not affect the evolution considerably since, for high impact parameters, the system reaches T_ FO at times ≲ 4 fm/c.] The aHydroQP evolution was started at τ_0 = 0.25 fm/c and stopped when the highest effective temperature in the entire volume was sufficiently below T_ FO. Using aHydroQP, we first ran the full 3+1d evolution of the system, then we extracted a freeze-out hypersurface based on the effective temperature. We assumed that all hadronic species were in chemical equilibrium and had the same fluid anisotropy tensor (Ξ_μν) and scale parameter (λ). The distribution function parameters on the freeze-out hypersurface were fedinto a customized version of THERMINATOR 2 which allows for an ellipsoidal distribution function of the form given in Eq. (<ref>). THERMINATOR 2 performs sampled event-by-event hadronic production from the exported freeze-out hypersurfaceusing Monte-Carlo sampling. It then performs hadronic feed down (resonance decays) for each sampled event. Depending on the observables under consideration and the centrality class considered, one may need to generate more hadronic events for the purposes of improved statistics.For all plots shown herein we used between 7,400 and 36,200 hadronic events per centrality class. We indicate the statistical uncertainty of our model results associated with the hadronic Monte-Carlo sampling by a shaded band surrounding the hadronic event-averaged value (the central line).In our model we have three remaining free parameters: (1) the initial central temperature T_0 obtained in a perfectly central collision at x_⊥=0 and ς=0, (2) the freeze-out temperature T_ FO, and (3) η/s which is assumed to be a (temperature-independent) constant. In order to fix these parameters we scanned over them and compared the theoretical predictions resulting from this scan with experimental data from the ALICE collaboration for the differential spectra of pions, kaons, and protons in both the 0-5% and 30-40% centrality classes. The fitting error was minimized across species, with equal weighting for the three particle types.The parameters obtained from this procedure are T_0 = 600 MeV, η/s = 0.159, and .We first present our comparisons of thetransverse momentum spectra of π^±, K^±, and p+p̅ in six centrality classes0-5%, 5-10%, 10-20%, 20-30%, 30-40%, and 40-50% in Fig. <ref>. These comparisons show that our model provides a very good simultaneous description of the ALICE data for the pion, kaon, and proton spectra <cit.>, with largest differences at p_T ≳ 1.5 GeV and relatively high centrality classes 30-40%, and 40-50%. We note that our model slightly underpredicts the pion spectrum at low transverse momentum which is similar to what is observed in other hydrodynamic models (see e.g. Ref. <cit.>). One possible explanation for this discrepancy that has been suggested is pion condensation <cit.>.In Fig. <ref>,we show the charged-hadron multiplicity in different centrality classes as a function of pseudorapidity, η. In panel (a), we show the 0-5%, 5-10%, 10-20%, 20-30%, and 30-40% centrality classes, and in panel (b) we show the 40-50%, 50-60%, 70-80%, 80-90%, and90-100% centrality classes.As can be seen from both panels, our model is able to describe the charged hadron multiplicity as a function of pseudorapidity <cit.> quite well in all centrality classes. Another observable to consider isthe average transverse momentum of pions, kaons, and protons as a function of centrality. This is shown in Fig. <ref>, where our model is again able to reproduce the data reasonably well. Next, in Fig. <ref>, we showthe integrated elliptic flow coefficient v_2 for charged hadrons as a function of centrality.Our model predictions were computed using the geometrical definition of the elliptic flow coefficient, v_2 ∼⟨cos(2ϕ) ⟩, for all charged hadrons.The experimental data were obtained using second- and fourth-order cumulants v_2{2} and v_2{4}<cit.>. From this figure we see that our model agrees well with v_2{4} measurements at low centrality, but agrees better with v_2{2} at higher centrality.One would expect better agreement with v_2{4} than v_2{2}, since the former has non-flow effects subtracted.The fact that we agree better with v_2{2} at high centrality could be due to the fact that our smooth initial condition is too simple or that we have not included the off-diagonal components of the anisotropy tensor in the evolution and freeze-out. In Fig. <ref>, we present comparisons of the identified-particle v_2 as a function of p_T obtained using our model with experimental data reported by ALICE collaboration <cit.>.Our model provides a quite reasonable description of the identified-particle elliptic flow as can be seen in panels (b) and (c), 20-30% and 30-40% centrality classes, respectively.In panel (b), the 20-30% centrality class, we see that our model reproduces the data very well for the pion, kaon, and proton data out to p_T ∼ 1.5, 1.5, and 2.5 GeV, respectively.A very similar agreement is seen in panel (c), the 30-40% centrality class, where the model is in goodagreement with the pion, kaon, and proton data out to p_T ∼ 1, 1, and 2 GeV, respectively. However, in panels (a) and (d), 10-20% and 40-50% centrality classes, respectively, we see less agreement than panels (b) and (c). For example, we underpredict the pion elliptic flow in the 10-20% centrality class as can be seen from panel (a).Again, as in the case of Fig. <ref>, this is related to our use of smooth Glauber initial conditions.In order to furtherexamine how well our model describes various observables, we look at thepseudorapidity dependence of v_2 for different centrality classes in Fig. <ref>. As can be seen from Fig. <ref> our model results do not fall fast enough at large pseudorapidity compared to the experimental data <cit.>.One possible remedy for this may be including temperature-dependent η/s, since this has been shown to improve agreement with this observable in the context of viscous hydrodynamics <cit.>.In Fig. <ref>wecompare the HBT radii predicted by aHydroQPwith experimental data from the ALICE collaboration <cit.>. To compute the HBT radii, we used exactly the same parameters used to describe other observables including the same number of hadronic events in each centrality class.In this set of figures, in the left, middle, and right panels we show R_ out, R_ side, and R_ long, respectively, as a function of the mean transverse momentum of the pair π^+ π^+ in four different centrality classes, 0-5%, 5-10%, 10-20%, and 20-30%.From the left column of this set of figures, we see that our model reproduces the data quite well for R_ out out to k_T ∼ 0.6 GeV. In the middle column, we presentcomparisons of R_ side where our model shows a good agreement out to k_T ∼ 0.9 GeV. Lastly, in the right column wecompare results for R_ long which show poorer agreementwith the data when compared to the first two columns. However, in most cases, our model predictions are within the error bars of the experimental data, with the biggest differences at low k_T ∼ 0.2 GeV. This isopposite to what we see in R_ out and R_ side where we observegood agreement with the experimental data at low k_T. For more comparisons, we compare also the ratios of the HBT radii in Fig. <ref>. In this set of figures, in the left, middle, and right panels we show R_ out/R_ side, R_ out/R_ long, and R_ side/R_ long, respectively, as a function of the mean transverse momentum of the pair π^+ π^+ in four different centrality classes, 0-5%, 5-10%, 10-20%, and 20-30%. We see that our model was able to reproduce the data quite well in all three panels for all centrality classes shown here.§ CONCLUSIONS AND OUTLOOK In this paper we presented phenomenological comparisons of aHydroQP with LHC experimental data collected in 2.76 TeV Pb-Pb collisions. This work is an extension of a previous letter <cit.>.Herein, we gave more details about the formalism used and presented a more thorough comparison between our model and LHC data for a variety of observables. InaHydroQP, we included three momentum anisotropy parameters in the underlying distribution function, both in the dissipative hydrodynamic stage and at freeze-out. We also used a quasiparticle implementation of the LQCD EoS in order to take into account the non-conformality of the system.At freeze-out, we used a customized version of THERMINATOR 2 which was modified to accept anisotropic distribution functions of generalized Romatschke-Strickland form.As a first test, in this work, we used smooth Glauber initial conditions which were obtained froma linear combination of wounded-nucleon and binary-collision profiles. We additionally assumed the system to be initially isotropic in momentum space with no initial transverse flow. To fix the remaining phenomenological parameters, we performed a parameter scan and compared our results with experimentally observed identified-particle spectra in the 5-10% and 30-40% centrality classes. The resulting set of best fit parameters was T_0 = 600 MeV, η/s = 0.159, and .After this fitting was complete, we computed an array of different heavy-ion observables, finding quite good agreement between our model and experimental data despite our simple smooth initial condition. We looked at particle multiplicity and spectra, average transverse momentum, v_2, and HBT radii.Compared to Ref. <cit.>, we have added additional centrality classes in some cases and increased the statistics associated with the hadronic Monte-Carlo sampling where necessary.Combined with what was reported in <cit.>, the phenomenological results presented herein represent the first aHydro results to include three separate anisotropy parameters together with the quasiparticle method for imposing the EoS and self-consistent anisotropic freeze-out.Compared to prior results which used a single anisotropy parameter and/or an approximate conformal-factorization implementation of the equation of state <cit.> we see much better agreement with the pion, kaon, and proton spectra and, relatedly, the total multiplicity as a function of pseudorapidity.Prior studies which used the approximate conformal-factorization implementation of the equation of state dramatically underestimated the low p_T spectra <cit.>, making this the first phenomenological study within the context of aHydro which is able to reproduce both the experimentally observed spectra and elliptic flow.Looking to future, there is certainly room for improvements in our model. For example, we are working on including a temperature-dependent shear viscosity to entropy density ratio since in this study it was assumed to be constant. Based on prior studies in the context of viscous hydrodynamics <cit.>, there is some hope that this will improve the agreement between our model and the experimental data, in particular, with regards to the pseudorapidity dependence of v_2. We also plan to include realistic fluctuating initial conditions, realistic initial momentum anisotropy profiles, and more realistic collisional kernels. Additionally, we are planning to look at different collision energies, e.g.RHIC 200 GeV collisions and LHC 5.023 TeV collisions, and different colliding systems, e.g. pA and pp, in the near future.The application of aHydro to pA and pp is of particular interest, since in these systems viscous hydrodynamics is being pushed to its limits, especially at freeze-out <cit.>.Finally, it would be interesting to apply this formalism also to even lower energy collisions where it is critical to take into account the finite net baryon density, heat flow, etc.In this context, it would also be interesting to extend the formalism to multicomponent fluids, e.g. two- and three-fluid models, similar to what has been done by the Los Alamos <cit.>, Kurchatov Institute <cit.>, Frankfurt <cit.>, and GSI <cit.> groups.Along these lines, there have already been prior anisotropic hydrodynamics studies which have considered multi-component fluids with explicit quark and gluon components <cit.>; however, to the best of our knowledge there have thus far not been any attempts to do this in the context of multicomponent hadronic fluids.We thank Piotr Bożek for useful comments. M. Alqahtani was supported by a PhD fellowship from the Imam Abdulrahman Bin Faisal University, Saudi Arabia.M. Nopoush and M. Strickland were supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Award No. DE-SC0013470.R. Ryblewski was supported by the Polish National Science Center grant No. DEC-2012/07/D/ST2/02125.§ EXPLICIT FORMULAS FOR DERIVATIVES In this section, we introduce the notation used in our formulation of the general moment-based hydrodynamics equations. Using the definitions D ≡ cosh(ϑ-ς)∂_τ+1/τsinh(ϑ-ς)∂_ς ,D̃ ≡ sinh(ϑ-ς)∂_τ+1/τcosh(ϑ-ς)∂_ς ,∇_⊥· u_⊥ ≡ ∂_x u_x+∂_y u_y, u_⊥·∇_⊥ ≡u_x∂_x+u_y∂_y,u_⊥×∇_⊥ ≡u_x ∂_y-u_y∂_x ,and four-vectors defined in Eq. (<ref>) one obtains D_u ≡ u^μ∂_μ=u_0 D+ u_⊥·∇_⊥ ,D_x ≡ X^μ∂_μ=u_⊥ D+u_0/u_⊥( u_⊥·∇_⊥) ,D_y ≡ Y^μ∂_μ=1/u_⊥( u_⊥×∇_⊥) ,D_z ≡ Z^μ∂_μ=D̃ .The divergences can be defined asθ_u ≡ ∂_μ u^μ= Du_0+u_0D̃ϑ+∇_⊥· u_⊥ , θ_x ≡ ∂_μ X^μ= Du_⊥+u_⊥D̃ϑ+u_0/u_⊥(∇_⊥· u_⊥)-1/u_0 u_⊥^2( u_⊥·∇_⊥) u_⊥ , θ_y ≡ ∂_μ Y^μ=-1/u_⊥( u_⊥·∇_⊥) φ ,θ_z ≡ ∂_μ Z^μ= Dϑ ,where φ=tan^-1(u_y/u_x). Finally, we list the non-vanishing contractions appearing in the second moment equations u_μ D_α X^μ = 1/u_0D_α u_⊥ , u_μ D_α Y^μ =u_⊥ D_αφ , u_μ D_α Z^μ =u_0 D_αϑ ,X_μ D_α Y^μ =u_0 D_αφ ,X_μ D_α Z^μ =u_⊥ D_αϑ , Y_μ D_α Z^μ =0 ,where α∈{u,x,y,z}. Note that, using the orthogonality of the basis vectors, i.e. D_α(X^μ u_μ) = 0, contractions such as X^μ D_α u_μ can be related to the contractions listed above, e.g. X^μ D_α u_μ = -u_μ D_α X^μ.§ SPECIAL FUNCTIONS AND DERIVATIVES The H-functions appearing in the body of the paper can be written as H_3(α,m̂)≡ Ñα∫ d^3p̂Rf_ eq(√(p̂^2 + m̂^2)) , H_3i(α,m̂)≡ Ñα α_i^2 ∫ d^3p̂R_if_ eq(√(p̂^2 + m̂^2)) , H_3B(α,m̂)≡ Ñα∫ d^3p̂R^-1f_ eq(√(p̂^2 + m̂^2)) , where α = (α_x,α_y,α_z), m̂=m/λ, i ∈{x,y,z}, α≡∏_i α_i, and Ñ≡ N_ dof/(2π)^3 with N_ dof being the number of degrees of freedom.The R and R_i functions appearing above are R ≡ √(α_x^2p̂_x^2+α_y^2p̂_y^2+α_z^2p̂_z^2+m̂^2), R_i≡ p̂_i^2 R^-1.More details concerning the H-functions and the manner in which they appear in the dynamical equations can be found in Refs. <cit.> and <cit.>.To the best of our knowledge, it is not possible to analytically evaluate the H-functions listed above. In practice, only one integral can be done analytically and we are left with integrals over ϕ and p. Evaluation of these 2d integrals is numerically intensive, making it infeasible to evaluate them in real-time during 3+1d simulations.One might consider interpolating them, however, they are functions of 4 variables (α,m̂) and, in practice, one must separately interpolate these five functions and all derivatives necessary.As a consequence, one quickly runs into memory limitations, even on modern computers.A more efficient technique, which does not require a great deal of memory either, is needed.In the next subsection, we present a method for doing this. §.§ Series expansions Since, in practice, the α's do not evolve too far from α_x = α_y = α_z =1, it makes sense to expand these integrals around such an isotropic point.After expanding around an isotropic point, the angular part of the integrals become trivial and one is left only with the p integral which can easily be interpolated since it is only function of the mass.Before proceeding, we note that many of the H-functions are related to each other by symmetries. As a result, we will present the method for evaluating H_3 and H_3x and use symmetries to find the other H-functions and their derivatives necessary.To proceed, we expand around an arbitrary isotropic point defined by α_i^2 ∼δ_0 using α_i^2 = δ_0 + δ_iϵ where δ_0 is the point around which the expansion is performed, and ϵ is used to keep track of the order of the expansion.Based on this, we expand R and R_x as R= ∑ _n=0^∞ 1/2n(m̂^2+δ_0p̂^2)^1/2-np̂^2n[sin ^2θ(δ_xcos ^2ϕ+δ_ysin ^2ϕ)+δ_zcos ^2θ]^n , R_x= ∑ _n=0^∞ -1/2n(m̂^2+δ_0p̂^2)^-1/2-np̂^2+2ncos^2 ϕsin^2 θ[sin ^2θ(δ_xcos ^2ϕ+δ_ysin ^2ϕ)+δ_zcos ^2θ]^n ,where δ_i=α_i^2-δ_0.As a result, H_3(α,m̂) can be written as H_3(α,m̂) = Ñα∑_n=0^∞1/2n Ω( δ,n)G(1/2-n,m̂,δ_0) ,where Ω( δ,n) is the angular part of the integral which is trivial to evaluateΩ( δ,n) = ∫ dΩ[sin ^2θ(δ_xcos ^2ϕ+δ_ysin ^2ϕ)+δ_zcos ^2θ]^n ,and G(a,m̂,δ_0) = ∫dp̂ p̂^3-2 a (m̂^2+δ_0 p̂^2)^a f_ eq(√(p̂^2 + m̂^2)). Similarly, H_3x(α,m̂) can be written as H_3x(α,m̂) = Ñα α_x^2 ∑_n=0^∞-1/2n Ω_x( δ,n)G(-1/2-n,m̂,δ_0) ,where Ω_x( δ,n) is the angular part of the integral which is trivial Ω_x( δ,n) = ∫ dΩ cos^2 ϕsin^2 θ [sin ^2θ(δ_xcos ^2ϕ+δ_ysin ^2ϕ)+δ_zcos ^2θ]^n .The derivatives of both H_3 and H_3x with respect to α's are straightforward since the G(a,m̂,δ_0) integral is independent of α. There are symmetries of each H-function that can be used for efficiently computing all derivatives necessary, for example, H_3 is symmetric under the exchange of α's, i.e., H_3(α_x,α_y,α_z,m̂) = H_3(α_y,α_x,α_z,m̂) = H_3(α_z,α_y,α_x,m̂) .In a similar way,H_3x(α_x,α_y,α_z,m̂) = H_3x(α_x,α_z,α_y,m̂) . Using these identities, once one calculates one of these derivatives, the other ones can be determined using symmetry arguments. For example, once ∂ H_3/∂α_x is known, the other derivatives with respect to α_y and α_z are related by the exchange symmetry∂ H_3(α_x,α_y,α_z,m̂) /∂α_y = ∂ H_3(α_y,α_x,α_z,m̂) /∂α_x,∂ H_3(α_x,α_y,α_z,m̂) /∂α_z = ∂ H_3(α_z,α_y,α_x,m̂)/∂α_x.Unlike H_3, for H_3x we have only one identity to use, so we need two derivatives,∂ H_3/∂α_x and∂ H_3/∂α_y and thenEq. (<ref>) can be used to find ∂ H_3/∂α_z ∂ H_3x(α_x,α_y,α_z,m̂) /∂α_z = ∂ H_3x(α_x,α_z,α_y,m̂)/∂α_y.We now turn to the derivative with respect to the fourth argument, m̂. The only part in H_3 and H_3x that involves m̂ is the integral G(a,m̂,δ_0). Taking its derivative gives another integral which can be easily interpolated and used G_m(a,m̂,δ_0) = - ∫dp̂ p̂^3-2 a(m̂^2+δ_0p̂^2)^a-1/2 √(m̂^2+p̂^2)(m̂^2-2 a √(m̂^2+p̂^2)+δ_0p̂^2) f_ eq(√(p̂^2 + m̂^2)) .So,∂ H_3/∂m̂ =2 m̂αÑ∑_n=0^∞1/2n Ω( δ,n)G_m(1/2-n,m̂,δ_0) ,∂ H_3x/∂m̂ =2 m̂α_x^2 αÑ∑_n=0^∞-1/2n Ω_x( δ,n)G_m(-1/2-n,m̂,δ_0) .Using the symmetries obeyed by the H functions, one can evaluate H_3y, H_3z, and H_3B and their derivatives similarly H_3y(α_x,α_y,α_z,m̂) =H_3x(α_y,α_x,α_z,m̂) , H_3z(α_x,α_y,α_z,m̂) =H_3x(α_z,α_y,α_x,m̂) , H_3B(α,m̂)= 1/m̂^2( H_3(α,m̂)- H_3x(α,m̂)- H_3y(α,m̂)- H_3z(α,m̂)) .§.§ Expansion points Finally, we must specify which value(s) of δ_0 to use and the order of the expansion.Since the α_i's are typically in a region 0 < α_i ≲ 3 during the dynamical evolution, we expand around two points corresponding to δ_0 =1,4 and interpolate between these two expansions in the intermediate region.For the interpolation, we define r_ min=1.75 and r_ max=1.85 where r= √(α_x^2+α_y^2+α_z^2) and, for H_3, use H_3(α,m̂,δ_0)= {[ H_3(α,m̂,1) r<r_ min,; H_3(α,m̂,4) r>r_ max,; r_ max-r/r_ max-r_ min H_3(α,m̂,1)+r-r_ min/r_ max-r_ min H_3(α,m̂,4) . ]. In a similar way, H_3x and all derivatives necessary can be calculated. In all cases, we expand up to 12^ th order (n ≤ 12) which was found to reproduce the direct numerical evaluation of all H-functions very well. | http://arxiv.org/abs/1705.10191v2 | {
"authors": [
"Mubarak Alqahtani",
"Mohammad Nopoush",
"Radoslaw Ryblewski",
"Michael Strickland"
],
"categories": [
"nucl-th",
"hep-ph"
],
"primary_category": "nucl-th",
"published": "20170525232746",
"title": "Anisotropic hydrodynamic modeling of 2.76 TeV Pb-Pb collisions"
} |
Intermodal Four-Wave-Mixing and Parametric Amplification in km-long Fibers Massimiliano Guasoni, Francesca Parmigiani, Peter Horak, Julien Fatome, and David J. Richardson Manuscript received XXX. M. Guasoni is supported through an Individual Marie Sklodowska- Curie Fellowship (H2020 – MSCA – IF – 2015 , project AMUSIC - Grant Agreement 702702). This research is sponsored by EPSRC grant EP/P026575/1. M. Guasoni, F. Parmigiani, P. Horak, and D. J. Richardson are with the Optoelectronics Reserch Centre, University of Southampton, Southampton SO17 1BJ, United Kingdom. J. Fatome is with the Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB), UMR 6303 CNRS - Université Bourgogne Franche-Comté, 9 Avenue Alain Savary, BP 47870, 21078 Dijon, France M. Guasoni's email is [email protected] December 30, 2023 ========================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================We theoretically and numerically investigate intermodal four-wave-mixing in km-long fibers, where random birefringence fluctuations are present along the fiber length. We identify several distinct regimes that depend on the relative magnitude between the length scale of the random fluctuations and the beat-lengths of the interacting quasi-degenerate modes. In addition, we analyze the impact of polarization mode-dispersion and we demonstrate that random variations of the core radius, which are typically encountered during the drawing stage of the fiber, can represent the major source of bandwidth impairment. These results set a boundary on the limits of validity of the classical Manakov model and may be useful for the design of multimode parametric amplifiers and wavelength converters, as well as for the analysis of nonlinear impairments in long-haul spatial division multiplexed transmission. Four-wave mixing (FWM), nonlinear optics, optical amplifiers, optical wavelength conversion.§ INTRODUCTION The last decade has been characterized by intense research in space-division multiplexing (SDM) schemes <cit.> and novel all-optical devices for signal processing <cit.>. Both these hot topics aim to develop new generation high-capacity internet networks capable of responding to the exponential growth of data demand. Within this framework, intermodal four-wave mixing (IM-FWM) in km-long multi-mode fibers (MMFs) is a key nonlinear process to be investigated for two main reasons. First, FWM is one of the main impairments affecting SDM transmissions <cit.>. Second, the use of long fibers leads to large degrees of nonlinearity even at low input powers. This increases the overall efficiency of FWM-based devices and paves the way for the development of all-optical devices that may overcome the main limits associated with single-mode fiber-based devices. Specifically, the phase-matching condition in IM-FWM processes can be achieved far away from both the zero dispersion wavelength and the bandwidth of spontaneous Raman scattering, thus reducing the impact of the nonlinear cross-talk and of the Raman noise contribution <cit.>.When analyzing light propagation in km-long fibers, it is important to take into account random birefringence fluctuations that occur on a length scale ranging from a few meters to several tens of meters <cit.>. These fluctuations are caused by manufacturing imperfections, environmental variations or local stress mechanisms and impair the FWM dynamics by inducing linear coupling among quasi-degenerate modes. Recently, the experimental demonstration of IM-FWM in km-long fibers has been reported <cit.>. However, while several theoretical works have addressed this issue in single-mode fibers <cit.>, there are currently very few theoretical studies for MMFs <cit.>.In this paper we aim to provide a complete overview of the impact of random perturbations on the IM-FWM dynamics. We address several issues whose understanding will give useful guidelines for boththe mitigation of FWM for SDM transmission and for the design of all-optical devices for signal processing. The paper is organized as follows. In Section <ref> we introduce the characteristic lengths that define the main fiber features in the presence of random perturbations. In Section <ref> we model the random birefringence fluctuations and the induced linear coupling among quasi-degenerate modes of a MMF. In Section <ref> we highlight the existence of different FWM regimes related to the aforementioned characteristic lengths. Depending on their relative magnitude, the fiber may exhibit an "isotropic"-like behavior or a fully random coupling dynamics between quasi-degenerate modes which is described by the Generalized Manakov Model for MMFs <cit.>. These resultsset a boundary on the limits of validity of the classical Manakov model and therefore provide new perspectives for multimode long-haul transmission, similarly to what has recently been observed in single-mode fibers <cit.>. In Section <ref> the impact of polarization mode dispersion on IM-FWM is discussed. Finally, in Section <ref>, we provide evidence that random fiber perturbations lead not only to a coupling between quasi-degenerate modes but also to fluctuations of the dispersion parameters of the different propagating spatial modes which can strongly impair the IM-FWM dynamics. § MODELING OF RANDOM BIREFRINGENCE FLUCTUATIONSCircular core isotropic fibers are characterized by groups of modes that are two-fold (groups LP_0m) or four-fold (groups LP_nm, with n>1 integer) degenerate. Degenerate modes of the same group possess identical dispersion parameters. However, in real optical fibers various random imperfections break the circular symmetry and isotropy of the fiber. Each degenerate mode is affected in a different way by these perturbations: as a result degenerate modes of the same group separate into a set of distinct quasi-degenerate modes, each one characterized by its own dispersive properties.The exact modeling of each source of perturbation is cumbersome and is still an active topic of research <cit.>. On the other hand, their global effect is that of a local and asymmetric weak variation of the fiber cross-section shape and size, as well as of the refractive index, giving rise to a weak local birefringence whose axes move randomly along the fiber <cit.>. We take as a reference the fast axis of the fiber and indicate with α(z) its angular orientation at the position z along the fiber. As the perturbations are typically weak, we can safely assume that the shape of the modes is preserved. What changes instead is the orientation of their electric field, which is aligned to the local axes of birefringence and is thus either parallel or orthogonal to α(z) (see Fig. <ref>). Furthermore, each mode has its own propagation constant β(z), inverse group velocity β_1(z)=∂β/∂ω and chromatic dispersion β_2(z)=∂^2β/∂ω^2 that are generally z-dependent.In the following, for the sake of simplicity, we neglect higher-order dispersion and we assume the two groups of modes involved in the IM-FWM process are the LP_01 and the LP_11. Note however that the main outcomes in this paper can be easily generalized to include the interaction between different groups of modes and the presence of higher-order dispersion terms. We denote by 0p and 0o the two quasi-degenerate modes of group LP_01 that are respectively parallel (p) or orthogonal (o) to α. Similarly, we denote by 1ap, 1bp, 1ao and 1bo the four quasi-degenerate modes of the group LP_11, as illustrated in Fig. <ref>.The angle α(z) changes randomly along the fiber length and is characterized by a correlation length, L_C, which defines the length-scale over which random perturbations become uncorrelated. As previously outlined, in typical standard fibers, L_C varies from a few meters to some tens of meters.For each pair of quasi-degenerate modes m and n we can define a corresponding beat-length L_B(m-n)=2π/(β_m-β_n). In the problem under analysis there are 4 independent beat-lengths, L_B(0p-0o), L_B(1ap-1ao), L_B(1ap-1bp) and L_B(1ap-1bo), from which the 3 remaining beat-lengths can be computed (e.g., L_B(1bp-1ao)^-1= L_B(1ap-1ao)^-1-L_B(1ap-1bp)^-1). While beat-lengths are generally z-dependent, the FWM dynamics is mainly sensitive to their spatial average (see Section <ref>). Therefore, in what follows we refer to their spatial average.The beat-length is indicative of the length scale over which the two quasi-degenerate modes acquire a significant phase-difference and thus of the minimum fiber length L which is necessary to distinguish each other. Typically, the stronger the local perturbations, the larger the difference between the two propagation constants is, thus the shorter the corresponding beat-length. Therefore, the correlation length L_C is a measure of "how fast" random perturbations occur, whereas the beat-lengths among quasi-degenerate modes measure "how strong" these perturbations are. If the fiber length L is much shorter than all the beat-lengths, that is L ≪min{|L_B|}, then modes within the same group propagate together in phase. In other words: in this instance random perturbations are weak enough so that the fiber can be considered perfectly circular and isotropic along its whole length. It is worth noting that beat-lengths can vary across a range of values from a few meters to tens of meters. For this reason, typical isotropic fibers are a few tens of meters long, so that relevant degrees of nonlinearity can only be achieved at the expenses of a large amount of input power. In the following, however, we are interested in km-long fibers, for which even small power levels may give rise to significant nonlinear effects. Therefore, in the following we can safely assume L ≫max{|L_B|}, where max{|L_B|} is the largest beat-length.The relative magnitude between the characteristic lengths discussed here gives rise to different FWM regimes that will be analyzed in the next sections. § RANDOM COUPLING INDUCED BY PERTURBATIONSTo understand the coupling mechanism induced by random perturbations, it is useful to represent the fiber as a concatenation ofshort segments of length Δ z (see Fig. <ref>). Each segment is short enough to preserve, along its whole length, both the direction α of the birefringence axes and the dispersion parameters of all modes. Let us consider light propagation in two consecutive segments s_n and s_n+1. Segment s_n (s_n+1) is characterized by its own direction α_n (α_n+1), with respect to which the electric field of the modes is parallel or orthogonal. We indicate with A(z)=[A_0p(z),A_0o(z),A_1ap(z),A_1ao(z),A_1bp(z),A_1bo(z)] the vector of the corresponding 6 modal amplitudes. Modes of segment s_n, immediately before entering s_n+1, are projected onto the modes of s_n+1. The projection is described by the following linear relation: A_n+1^(in) =P A_n^(out), where A_n^(out)≡ A(z^-) is the vector of amplitudes at point z^-, at the end of s_n and just before entering s_n+1, and A_n+1^(in)≡ A(z^+ ) is the vector at point z^+, just after entering s_n+1 (Fig. <ref>). Theprojection matrix reads: P=[ [C -S0000;SC0000;00C -S -S0;00SC0 -S;00S0C -S;000SSC ]] where C=cos(Δα) and S=sin(Δα), with Δα = α_n+1-α_n. According to this model, coupling among different quasi-degenerate modes is thus induced by the random variation Δα of the birefringence axes. If no variation occurs, i.e. Δα=0, then there is no energy exchange within quasi-degenerate modes (A_n+1^(in)= A_n^(out)), which is consistent with the assumption that the modal shape is largely preserved along the fiber length. Note also that according to matrix Pthere is no coupling between a mode of group LP_01 and a mode of group LP_11, which is typically the case in real fibers due to the large difference in their propagation constants.In order to describe the propagation in the fiber, we study the evolution of light from point (z-Δ z)^+, at the entry of segment s_n, to the point z^+, at the entry of s_n+1. First, light propagates through the segment s_n from (z-Δ z)^+ to z^-, undergoing both dispersion (operator D̂) and nonlinearity (operator N̂): A_n^(out) -A_n^(in) = Δ z[D̂{ A_n^(in)} + N̂{ A_n^(in)}], where A_n^(out) -A_n^(in) indicates the mode amplitude variation, with A_n^(in)≡ A((z-Δ z)^+). Then, modes of segment s_n are projected onto modes of s_n+1 according to the relation A_n+1^(in) =P A_n^(out). From the two aforementioned relations, we finally evaluate the derivative ∂ A/∂ z = lim_Δ z→ 0 ( A_n+1^(in) -A_n^(in))/Δ z, which after some algebra takes the form of the following Nonlinear Schrödinger Equation (NLSE): ∂_z A =Q̅ A + (β̅_̅1̅-v_r^-1)∂_t A + i(1/2)β̅_̅2̅∂_tt A+ + N̂{ A}+β̃ A + β̃_̃1̃∂_t A + i(1/2)β̃_̃2̃∂_tt A In Eq. (<ref>) each dispersion coefficient x=x̅+x̃ is separated into the sum of its spatial average x̅ and its z-varying part x̃. This separation is introduced because, as will be discussed later, the averages and varying parts play a different role in the IM-FWM dynamics.The 6x6 matrix Q̅=2πL̅_B^-1 + ∂_zα U, where L̅_B = diag[0, L̅_B(0p-0o),0,L̅_B(1ap-1ao), L̅_B(1ap-1bp),L̅_B(1ap-1bo) ] and U is identical to P, Eq. (<ref>), for Δα=π/2. Matrix β̅_̅1̅=diag[β̅_1,0p,β̅_1,0o,β̅_1,1ap,β̅_1,1ao,β̅_1,1bp,β̅_1,1bo] includes the average inverse group velocities, whereas v_r is a free parameter that represents the velocity of a reference frame and can be conveniently chosen as v_r=1/β̅_1,0p. Matrix β̅_̅2̅=diag[β̅_2,0p,β̅_2,0o,β̅_2,1ap,β̅_2,1ao,β̅_2,1bp,β̅_2,1bo] includes the averagechromatic dispersion coefficients. Thematrices β̃_̃1̃ and β̃_̃2̃ are formed analogously to β̅_̅1̅ and β̅_̅2̅ by replacing the average parameter x̅ with the z-varying part x̃. Finally, matrix β̃=diag[β̃_0p,β̃_0o,β̃_1ap,β̃_1ao,β̃_1bp,β̃_1bo]. The operator N̂ accounts for all nonlinear Kerr and Raman multimode interactions, as discussed in <cit.>. In the following we assume that the chromatic dispersion of modes within the same group is the same, indicating with β_ 2,0 the coefficients β_ 2,0p=β_ 2,0o and with β_ 2,1 the coefficients β_ 2,1ap=β_ 2,1a0=β_ 2,1bp=β_ 2,1bo. It is worth noting that Eq. (<ref>) represents a generalization of the approach introduced in <cit.> to describe the effects of birefringence fluctuations in single-mode fibers. Note also that vector A in Eq. (<ref>) describes the modal amplitudes in the local reference frame, which is defined by the orientation α(z).Similarly to previous work <cit.>, in Eq. (<ref>) the overall effect of linear coupling is described by a 6x6 matrix Q̅. However, a major advantage of our approach is that this matrix is explicitly written in terms of the main real fiber parameters, that are the average beat-lengths among quasi-degenerate modes and the function α(z) which accounts for the random evolution of the birefringence axes. We can therefore study light propagation versus different profiles of α(z) and of beat-lengths, and identify different regimes that are discussed in the next Section.§ FROM UNCOUPLED TO THE MANAKOV REGIMEIn this Section we study the impact of random perturbations on two important IM-FWM processes, namely Bragg scattering (BS) and phase conjugation (PC) <cit.>. The configuration of the corresponding processes is represented in Fig. <ref>, where two input pumps P_0 and P_1 are coupled to modes LP_01 and LP_11, respectively, and an input seed signal S_0 is coupled to mode LP_01. Due to IM-FWM new idlers in the corresponding LP11 mode group are generated for both the BS (I_1,BS) and the PC processes (I_1,PC). All the waves are monochromatic and we indicate with f_P0, f_S0, f_P1 and f_I1 their corresponding frequency.We analyze the idler growth as a function of the system parameters, computing the idler power as the sum of the powers in the 4 quasi-degenerate LP_11 modes. We initially assume that the dispersion coefficients are constant along the fiber length (that is, β̃=β̃_̃1̃=β̃_̃2̃=0 in Eq. (<ref>)); the effects of their z-dependence will be discussed later. We also assume the pumps and signal are linearly copolarized, which maximizes the idler growth. For the fiber parameters used here we refer to the graded-index fiber employed in <cit.>. We fix v_0p-v_1ap=100 ps/km, whereas the chromatic dispersion coefficients are fixed respectively to 19.8 ps/(nm km) for both modes of group LP_01 and to 21.1 ps/(nm km) for all modes of group LP_11. The nonlinear overlap coefficients <cit.> are indicated here as C_abcd. Indices {a,b,c,d} are employed to refer to the modes: the index 0 refers to one of the modes 0p and 0o; 1 refers to one of the modes 1ap and 1ao ; 2 refers to one of the modes 1bp and 1bo. Due to the symmetries of the modes under consideration, the coefficients with subscripts of the kind aabb or aaaa are the only non-zero ones and are invariant with respect to permutations of the indices (e.g. C_aabb=C_abab)<cit.>. The nonlinear coefficients (related to the fiber discussed in <cit.>) are: C_0000=0.63 km^-1W^-1; C_0011=C_0022=0.39 km^-1W^-1; C_1111=C_2222=0.60 km^-1W^-1; C_1122=0.18 km^-1W^-1. Input powers are 22.5 dBm for each pump and 3.5 dBm for the signal. In order to get some realistic value for the strength of the random perturbations, we assume the average residual birefringence (that is the difference between the refractive indices of the birefringence axes) to be Δ n=1.5 · 10^-7, which is a typical value for standard optical fibers used in telecommunications. This provides an estimate for the beat-length L_B(0p-0o) = λ/Δ n = 10 m and the inverse group velocity mismatch β_1,0p-β_1,0o=Δ n/c =0.5 ps/km, where λ=1550 nm is the wavelength of the pump in mode 0p. We use values of the same order for the beat-lengths and inverse group velocity mismatches of the group LP_11: L_B(1ap-1ao)=25 m; L_B(1ap-1bp)=50 m; L_B(1ap-1bo)=8 m; β_1,ap-β_1,1ao= 0.2 ps/km; β_1,ap-β_1,1bp=0.4 ps/km; β_1,1ap-β_1,1bo=0.6 ps/km.The phase matching condition of IM-FWM processes in an isotropic fiber is fulfilled when the sum of the inverse group velocities of the pump and signal in group LP_01 equates to the sum of the inverse group velocities of the pump and signal in group LP_11 <cit.>. Therefore, phase matching is essentially related to the dispersion properties of the different mode groups. In randomly perturbed fibers, the small differences of group velocity among quasi-degenerate modes do not significantly affect the IM-FWM phase matching. Therefore the aforementioned phase-matching condition can be safely rewritten as β_1,0p(f_P0)+β_1,0p(f_S0) = β_1,ap(f_P1)+β_1,ap(f_I1).We first study the BS process (Fig. <ref>a). We simulate Eq. (<ref>) using the system parameters illustrated above and with a pump-to-pump detuning f_P0-f_P1=0.575 THz, which corresponds to the phase-matching condition for the BS process. The signal-to-pump detuning f_S0-f_P0 spans from -0.5 THz to -0.1 THz. Moreover, we generate random smooth profiles for α(z) with vanishing spatial average and different values of correlation length L_C. In our simulations L_C is defined on the basis of the correlation function C_α(z)=|∫α(z')α(z'-z)dz'|/∫α(z')^2dz'; it indicates the length beyond which the correlation function remains below 0.1, that is C_α(z>L_C)<0.1. Simulation results are displayed in Fig. <ref> and show the existence of 3 distinct regimes depending on the relative magnitude between L_C and the beat-lengths. For values of L_C > 5·max{|L_B|}, as in the case of L_C=260 m in Fig. <ref>, the idler dynamics does not depend on the particular value of L_C and resembles the dynamics found for L_C=∞, i.e. when the angle α(z) does not vary along the fiber length. In this instance, here named the Uncoupled Regime, random perturbations evolve slowly enough to prevent any significant linear coupling among quasi-degenerate modes. Therefore, the fiber can be considered as a birefringent fiber with fixed axes of birefringence. As such, the idler growth strictly depends on the polarization direction of thecopolarized input waves and is maximized when they are aligned to one of the birefringence axes.In the other extreme, when random fluctuations are fast and take place on a length scale shorter than the beat-lengths (L_C < min{|L_B|}, as in the case of L_C=7 m in Fig. <ref>),the idler growth computed by Eq. (<ref>) turns out to be in excellent agreement with the growth obtained by the solution of the multimode Manakov equations in the weak-coupling limit<cit.>. Therefore we refer to this instance as the Manakov Regime where, differently from the Uncoupled Regime, the system dynamics is independent of the polarization direction of the copolarized input beams. Following considerations similar to those discussed in <cit.> we analytically estimate an idler amplification impairment of about -3.5 dB between the Uncoupled Regime and the Manakov Regime, which isconfirmed by our numerical results displayed in Fig. <ref>. Note that this impairment is almost independent of the signal frequency and the fiber length L. However, it is important to notice that this estimate applies only when, in the Uncoupled Regime, the input beams are aligned with one of the axes of birefringence, so that idler growth is maximized.For intermediate values min{|L_B|} < L_C < 5max{|L_B|}(L_C= 50 m in Fig. <ref>) we find an Intermediate Regime where the idler amplification depends on the specific value of L_C.Differently from the BS process, where phase matching is essentially governed by the pump-to-pump detuning, in the PC process (Fig. <ref>b) it is mainly related to the signal wavelength <cit.>. When both pumps are centered at the same frequency (degenerate FWM, f_P0=f_P1) we find that phase matching is optimized for f_S0-f_P0=0.605 THz.Simulation results are displayed in Fig. <ref> when largepump powers (32.5 dBm) are employed to get efficient idler amplification; the input signal power is -9 dBm. These results demonstrate once again the existence of the 3 distinct regimes observed in the BS process; on the other hand they also clearly highlight some particular differences with respect to the BS process, which are mainly related to the instability of the PC process. First, the idler power can significantly exceed the input signal power; and second, the idler amplification impairment induced by quick randomperturbations is not a constant value but is instead proportional to the fiber length as well as to the input pump powers. More precisely, we notice that for any value of L_C the idler power (in dBm) versus fiber length is well approximated by a line with a slope that depends on L_C. Analytical considerations similar to those discussed in <cit.> allow us to estimate an impairment of about (2/3)C_0011(P_0P_1)^1/2 between the slope in the Uncoupled Regime (when input waves are aligned to one of the birefringence axes) and the slope in the Manakov Regime.We conclude this section by highlighting that in these simulations the beat-lengths have been chosen in such a way to achieve a strong linear coupling between the modes LP_11a and LP_11b. However, as noticed in Ref. <cit.>, the system dynamics is almost independent of their linear coupling. More specifically, in our study we find the same regimes and outcomes when the linear coupling between the modes LP_11a and LP_11b is null, which is one of the assumptions underlying the Manakov model <cit.> . Consequently, our results indicate that the Manakov model can correctly describe the full FWM dynamics only when L_C is of the same order as the shortest beat-length. On the other hand, even for realistic values of a few tens of meters (see e.g. L_C=50 m in Figs. <ref>, <ref>) the Manakov model fails in describing the idler dynamics, which sets important boundaries on its limits of applicability.§ IMPACT OF POLARIZATION MODE DISPERSIONSo far we have neglected polarization mode dispersion (PMD), i.e., that the random dynamics discussed in the previous sections is in reality frequency dependent, similar to the case of single-mode fibers.The main issue related to PMD is that waves at two different frequencies undergo a different randomization: as a result, the relative state of polarization (SOP) oftheir spatial modes cannot be indefinitely preserved along the fiber. In the following analysis, the notion of spatial modes is related to the spatial shape of the transverse mode, therefore the group LP_11 is split into two distinct spatial modes LP_11a and LP_11b.The diffusion length L_D=3/(4π^2D_p^2Δ f^2) indicates the length scale beyond which the relative SOP is not maintained, where Δ f is the frequency detuning between the two waves and D_p the PMD coefficient that is related to the beat-length through the relation D_p=(2L_e)^1/2/(L_Bf), where f is the carrier frequency of one of the two waves and L_e the polarization correlation length <cit.>. The latter is related to L_C and L_B according to a relation that varies depending on the regime under analysis <cit.>. Note that in the multimode system considered here a PMD coefficient and the related diffusion length can be defined for each one of the beat-lengths. Here we are mainly interested in the smallest, min{L_D}, and largest, max{L_D}, diffusion lengths. These two lengths allow us to distinguish two regimes: a low-PMD regime, when the fiber length L ≪min{L_D}, where the relative SOP of spatial modes is preserved along the fiber;and a high-PMD regime, when L ≫max{L_D}, where the relative SOP varies randomly along the fiber.An exhaustive analysis of the PMD impact is complex and out of the scope of the current paper, therefore in the following we limit our study to the Manakov Regime, which is important for km-long fibers, and we focus on the degenerate PC process introduced in Section <ref>. According to the system parameters, when L_C=7 m (Manakov Regime), the minimum diffusion length is several tens of km, therefore results displayed in Figs. <ref>, <ref> concern the low-PMD regime. In order to move towards the high-PMD regime of the PC process we keep the system parameters unchanged except for the chromatic dispersion coefficients that are reduced to 0.78 ps/(nm km) for modes of group LP_01 and to 1.18 ps/(nm km) for modes of group LP_11. Consequently, the phase matching detuning Δ f=f_S0-f_PO increases to 12.73 THz and the corresponding largest diffusion length max{L_D} decreases to 450 m. In Fig. <ref>(a) the idler power versus fiber length is displayed for different realizations of the orientation angle α(z) that are all characterized by the same correlation length L_C=7 m. Differently from the low-PMD regime, where the idler growth is almost independent of the particular realization, the high-PMD regime is characterized by severe variations of the idler amplification from one realization to another. Note that a similar dynamics has been previously observedin single-mode fibers <cit.>.Furthermore, from Fig. <ref>(b) we notice that the idler growth, averaged over a consistent number of different realizations, is generally reduced by several dB due to PMD. The impairment is proportional to the fiber length and can be as large as several tens of dB at the output of a km-long fiber. These results clearly indicate that PMD puts a strong limit on the maximum bandwidth of multimode parametric devices based on km-long fibers.§ IMPACT OF THE VARIATION OF DISPERSIVE PARAMETERSThe study of random perturbations portrayed in previous sections was based on the assumption that the dispersion parameters were constant along the fiber length (β̃=β̃_̃1̃=β̃_̃2̃=0 in Eq. (<ref>)). More realistically however, random local perturbations affect these dispersion parameters and we will study the impact of this z-dependence in the following.Towards this, we distinguish the intragroup dispersive parameters, which are the beat-lengths (L_B(0p-0o), L_B(1ap-1ao), L_B(1ap-1bp), L_B(1ap-1bo)) and the relative inverse group velocities (β_1,0p-β_1,0o, β_1,ap-β_1,ao, β_1,ap-β_1,bp, β_1,ap-β_1,bo) among quasi-degenerate modes of the same group, andthe intergroup dispersive parameters β_1,0p, β_1,ap, β_2,0 and β_2,1, which describe the different dispersive properties of different modal groups.As pointed out in Section <ref>, the IM-FWM dynamics atphase-matching depends on the intragroup parameters, more precisely the beat-lengths, so that three distinct regimes can be distinguished which we called the Uncoupled, Manakov, and Intermediate Regimes. On the other hand, the phase matching condition for the BS and PC nonlinear processes between LP01 and LP11 modes depends on the intergroup parameters but is practically unaffected by intragroup parameters. Therefore, we proceed by studying separately the effect of the z-dependence of the intergroup and intragroup parameters.Initially,Eq. (<ref>) is solved by keeping the intergroup parameters fixed while varyingthe intragroup parameters β̃ and β̃_̃1̃ with z. In order to implement the z-dependence, each intragroup parameter is defined as a random function p(z) with spatial average p̅, standard deviation σ(p) and correlation length L_C.Our numerical simulations show that the IM-FWM dynamics is not sensitive to local variations of the intragroup parameters, but only to their average value. We verified numerically that this is true even for standard deviations which are as large as the average value. It is worth noting that this outcome is in line with previous studies in single-mode fibers <cit.>. Therefore, we still recognize the three regimes found in Section <ref>, provided that the values of the beat-lengths are replaced by their spatial averages, so that the thresholds for the Uncoupled and Manakov regimes become L_C > 5max{L̅_B} and L_C < min{L̅_B}, respectively.Contrary to the case of intragroup parameters, even small variations of the intergroup parameters can strongly impact the phase-matching condition and then severely affect the IM-FWM dynamics. This issue has already been addressed in single-mode fibers, where small fluctuations of thechromatic dispersion along the fiber can lead to a remarkable reduction of the idler amplification bandwidth <cit.>. In the multimode dynamics this issue is even more critical, asthe idler amplification band depends on all four intergroup parameters, that is, not only on the chromatic dispersion coefficients but also on the group velocities of groups LP_01 and LP_11 <cit.>.As a practical example, we consider here a 1-km long MM silica step-index fiber whose core radius R(z) varies randomly in z, thereby inducing fluctuations of the intergroup parameters. Note that radius fluctuation is a typical perturbation occurring on a length-scale L_C of a few meters during the drawing stage of the fiber, where the standard deviation of the radius can be as large as 1% <cit.>. Here we assume the core radius R(z) to have average R̅=40 μm and correlation length L_C=5 m. We simulate two distinct instances where its standard deviation is either 0.5%(σ(R)=2 μm) or 1%(σ(R)=4 μm), respectively.The core-cladding index difference is fixed at 0.0025, independently of the wavelength. Several groups of modes can propagate in this fiber, but we assume only modes 0p and 1ap are excited and propagate. Note that in order to isolate the impact of the variation of intergroup parameters here we do not introduce random linear coupling among quasi-degenerate modes (i.e. we set ∂_zα=0 in Eq. (<ref>)). We calculate the intergroup parameters and nonlinear coefficients at each position z of the fiber, and solve Eq. (<ref>) accordingly. In order to compute the intergroup parameters at position z, we first derive the propagation constants β_0p(z) and β_1ap(z) of modes 0p and 1ap by solving the modal characteristic equation for a circular-core fiber of radius R(z); we then directly infer the intergroup parameters β_1,0p(z)=∂β_0p/∂ω, β_1,1ap(z)=∂β_1ap/∂ω, β_2,0=∂^2 β_0p/∂ω^2 and β_2,1=∂^2 β_1ap/∂ω^2 by also taking into account the material dispersion of silica. Similarly, in order to compute the nonlinear coefficients we first calculate the transverse mode profiles as a function of R(z) and then the nonlinear overlap integrals. In this way, our numerical simulations also account for the z-dependence of the nonlinear coefficients. We do this for completeness, however we anticipate that these fluctuations of the nonlinearity have very little effect on the IM-FWM dynamics, so that in practice their average value could safely be used in simulations.When solving Eq. (<ref>) for the BS process, we fix the pump-to-pump detuning to f_P0-f_P1=1.65 THz, which corresponds to the phase matching condition for a fiber with constant radius R=R̅=40 μm. In Fig. <ref>(a) the output idler power is plotted versus the signal-to-pump detuning in both the cases of fixed and varying radius. We note that the idler amplification bandwidth is only slightly impaired by fluctuations of the intermodal parameters. When repeating the same analysis with a bimodal fiber of average radius R̅=10 μm (phase-matching at f_P0-f_P1=5.095 THz) and standard deviation0.5% or 1% (that is σ(R)=0.05 μm or σ(R)=0.1 μm, see Fig. <ref>(b)), we find instead that the amplification bandwidth is severely reduced. This is explained by the fact that the smaller the radius, the more the modes spread out in the outer core, such that their dispersion parameters become strongly sensitive to variations of the core size. Consequently, the phase matching condition cannot be preserved along the fiber length, which causes the drastic reduction of bandwidth. Note that almost the same results are found when introducing linear coupling among quasi-degenerate modes, except for an amplification impairment of about -3.5 dB, as pointed out in Section <ref>.It is worth noting that in Ref. <cit.> the authors have studied the BS process in a 1 km-long bimodal fiber and found that the experimental bandwidth at -3 dB was about 4 times narrower than the bandwidthestimated in numerical simulations when considering the propagation in a totally uniform fiber. They then conjectured that fluctuations of the dispersive parameters may be the principal source of the observed discrepancy. The plots in Fig.<ref>(b), related to the bimodal fiber of average R̅=10 μm, give support to this interpretation.More in general, the results displayed in Fig.<ref> demonstrate that the analysis of the device robustness against fluctuations of the relative intergroup dispersive parameters is an essential step when designing multimode parametric devices in km-long fibers. It is worth noting from Fig. <ref>(b) that these fluctuations may completely suppress the idler growth even when the frequency detuning among waves is low (that is, in a low-PMD regime). Therefore they may constitute the dominant factor of bandwidth impairment in parametric amplifiers. On the other hand, the same effect may be an interesting tool to exploit in order to reduce FWM impairments in SDM transmissions. § CONCLUSIONIn this paper we studied the IM-FWM dynamics taking place between different groups of modes in a km-long MMF. We identified three distinct regimes which depend on the relative magnitude between the correlation length L_C of random longitudinal fiber fluctuations and the beat-lengths of the interacting quasi-degenerate modes. We then demonstrated that the Manakov model can reproduce the FWM dynamics only when L_C is of the same order of or shorter than the smallest beat-length. On the contrary, when L_C is much longer than all beat-lengths, the fiber acts as a birefringent fiber with fixed axes of birefringence where the IM-FWM dynamics strictly depends on the relative polarization of the input waves with respect to the axes of birefringence (Uncoupled Regime). The maximum amplification impairment between the Uncoupled and the Manakov regime varies depending on the kind of FWM process being considered: for BS processes it is about -3.5 dB almost independently of the fiber length and input pump powers, whereas for PC processes it is directly proportional to both the fiber length and the pump powers. PMD is a further source of impairment: in the high-PMD regime not only is the amplification substantially reduced, but it also depends on the particular longitudinal profile of the fiber perturbations. Therefore, two different profiles α(z) with the same correlation length may lead to strongly different FWM amplification. Finally, we highlighted that random fiber fluctuations induce a random evolution of the dispersion parameters of the different groups of modes. This can result in a severe reduction of the FWM bandwidth and thus constitutes one of the major issues when addressing the design of efficient multimode devices for parametric amplification/conversion.Overall, these results shed light on the FWM dynamics in km-long MMFs, and as such could find useful application in the study of SDM transmission as well as in the development of multimode devices for all-optical signal processing, which are building-blocks of future all-optical networks. Finally, the finding of different FWM regimes that are not captured by the Manakov model raises important questions on its limits of validity and paves the way towards novel and robust transmission formats in multimode systems, similar to what was recently demonstrated insingle-mode fibers <cit.>.§ ACKNOWLEDGMENTSThe data for this work is accessible through the University of Southampton Institutional Research Repository (DOI:xxxxxx) 99Richardson13 D. J. Richardson, J. M. Fini and L. E. Nelson, Space-division multiplexing in optical fibres, Nature Photon. 7, 354-362 (2013). Wabnitz15 S. Wabnitz and B. J. Eggleton, All-optical signal processing, Springer (2015). Ellis13 A. D. Ellis, N. Mac Suibhne, F. C. Garcia Gunning, and S. Sygletos, Expressions for the nonlinear transmission performance of multi-mode optical fiber, Opt. Exp. 21, 22834-22846 (2013).Friis16 S. M. M. Friis, I. Begleris, Y. Jung, K. Rottwitt, P. Petropoulos, D. J. Richardson, P. Horak and F. Parmigiani, Inter-modal four-wave mixing study in a two-mode fiber, Opt. Exp. 24, 30338-30349 (2016). Wuilpart01 M. Wuilpart, P. Mégret, M. Blondel, A. J. Rogers and Y. Defosse, Measurement of the spatial distribution of birefringence in optical fibers, IEEE Photon. Technol. Lett. 13, 836-838 (2001).Galtarossa04 A. Galtarossa and L. Palmieri, Spatially resolved PMD measurements, J. Lightw. Technol. 22, 1103-1115 (2004).Essiambre13 R.-J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. Jiang, and R. Lingle, Experimental investigation of inter-modal four-wave mixing in few-mode fibers, IEEE Photon. Technol. Lett. 25, 539-542 (2013).McKinstrie04 C. J. McKinstrie, H. Kogelnik, R. M. Jopson, S. Radic and A. V. Kanaev, Four-wave mixing in fibers with random birefringence, Opt. Exp. 12, 2033-2055 (2004).Karlsson98 M. Karlsson, Four-wave mixing in fibers with randomly varying zero-dispersion wavelength, J. Opt. Soc. Am. 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Wai and C. R. Menyuk, Polarization mode dispersion, decorrelation, and diffusion in optical fibers with randomly varying birefringence, J. Lightw. Technol. 14, 148-157 (1997).Poletti08 F. Poletti and P. Horak, Description of ultrashort pulse propagation in multimode optical fibers, J. Opt. Soc. Am. B 25, 1645-1654 (2008).McKinstrie02 C. J. McKinstrie, S. Radic, and A. R. Chraplyvy, Parametric amplifiers driven by two pump waves,IEEE J. Sel. Top. Quantum Electron. 8, 538-547 (2002).AgrawalBook G. P. Agrawal, Nonlinear Fibre Optics, 4th Edition, Elsevier (2007).Lin04 Q. Lin and G. P. Agrawal, Effects of polarization-mode dispersion on fiber-based parametric amplification and wavelength conversion, Opt. Lett. 29, 1114-1116 (2004).Yaman04 F. Yaman, Q. Lin, S. Radic and G. P. Agrawal, Impact of dispersion fluctuations on dual-pump fiber-optic parametric amplifiers, IEEE Photon. Techol. Lett. 16, 1292-1294 (2004). | http://arxiv.org/abs/1705.09106v1 | {
"authors": [
"Massimiliano Guasoni",
"Francesca Parmigiani",
"Peter Horak",
"Julien Fatome",
"David J. Richardson"
],
"categories": [
"physics.optics"
],
"primary_category": "physics.optics",
"published": "20170525093232",
"title": "Intermodal Four-Wave-Mixing and Parametric Amplification in km-long Fibers"
} |
Holomorphic foliations tangent to Levi-flat subsets Jane Bretas & Arturo Fernández-Pérez & Rogério Mol Received: date / Accepted: date ========================================================We study Segre varieties associated to Levi-flat subsets in complex manifolds and apply them to establish local and global results on the integration of tangent holomorphic foliations. [1] 2010 Mathematics Subject Classification. Primary 32S65 ; Secondary 32V40.[2]Keywords. Holomorphic foliation, CR-manifolds, Levi-flat varieties. [3]First author partially financed by a CNPq Ph.D. fellowship. Second an third authors partially financed by CNPq-Universal § INTRODUCTIONLetH ⊂ M be a real analytic hypersurface, where M is a complex manifold of _ M = N.Let H_reg denote its regular part, that is, the collection of all points near which His a manifold of maximal dimension. For each p ∈ H_reg, there is a unique complex hyperplane L_p contained in the tangent space T_pH_reg. Thisdefines a real analytic distribution p ↦L_p of complex hyperplanes in T H_reg.When this distribution is integrable in the sense of Frobenius, we say that H is a Levi-flat hypersurface. The resulting foliation in H, denoted by L, is known asLevi foliation. A normal form forsuch an object was given by E. Cartan <cit.>: for each p ∈ H_reg, there are holomorphic coordinates (z_1,…,z_N) in a neighborhood U of p such thatH_reg∩ U = {(z_N) = 0}. As a consequence, the leaves of Lhave local equations z_N = c, for c ∈.Cartan's local trivialization allows the extension the Levi foliationto a non-singular holomorphic foliationin a neighborhood ofH_reg in M, which is unique as a germ around H_reg. In general, it is not possible to extendL to a singular holomorphic foliation in a neighborhood of H_reg. There are examples of Levi-flat hypersurfaces whose Levi foliations extend to k-webs in the ambient space <cit.>.However there is an extensionin some “holomorphic lifting” of M <cit.>. If afoliation F in the ambient space M coincides with the Levi foliation on H_reg, we say either that H is invariant by F or that F is tangent to H.Locally, germs of codimension one foliations at (^N,0) tangent to real analytic Levi-flat hypersurfacesare given by the levels ofmeromorphic functions — possibly holomorphic — according to atheorem by D. Cerveau and A. Lins Neto<cit.>.Questions involving the global integrability of codimension one foliations in ℙ^N tangent to Levi-flat hypersurfaces where addressed by J. Lebl in <cit.>. For instance,if H is a real algebraic Levi-flat hypersurface tangent to a codimension one foliationinℙ^N, thenadmits a rational first integral R and there is a real algebraic curve S ⊂ such that H ⊂R^-1(S).Our goal in this paper is to establish local and global integrability results for foliations tangent to real analyticLevi flat subsets. A real analytic subset H ⊂ M, where M is an N-dimensional complex manifold, is called Levi-flat if it has real dimension 2n + 1 and its regular part H_reg is foliated by complex varieties of dimension n (Section <ref>, Definition <ref>). This is called Levi foliation and nisthe Levi dimension of H.When N = n+1, we recover the definition of Levi-flat hypersurface. This object appears inM. Brunella'sstudy on Levi-flat hypersurfaces <cit.>, as the result of the lifting of a Levi-flat hypersurfaceto thethe projectivized cotangent bundle of the ambient space by means of the Levi foliation(see Section <ref>).Key ingredients in the study of integrability properties of Levi-flat hypersurfaces are Segre varieties. Their structureisusedin Brunella's geometricproof for the local integrability of foliations tangent to Levi-flat hypersurfaces <cit.> as well asin Lebl's global integrability results<cit.>. Segre varieties for Levi-flat subsets are the cornerstone of our work. Their definition, along with main properties,are presented in Section <ref>. Recently, a research paper on Levi flat subsets, also founded on the study of Segre varieties,has been released<cit.>. It has anapproach to Segre varieties slightly different from ours, although leading to equivalent constructions.Given a Levi-flat subset H of Levi dimension n, there is a unique complex variety H^ of dimension n+1, called intrinsic complexification or -complexification,defined in a neighborhood of H_regcontaining H_reg <cit.>. If H is tangent to a foliationof dimension n in the ambient space, then H^ is invariant by . Our integration results are stated in terms of the -complexification H^ andthe foliation ^, the restriction ofto H^. For real analyticLevi-flat subsets in projective spaces, we can state the following theorem, to be proved along Sections <ref> and <ref>:Let H⊂ℙ^N, N>3, be a real analytic Levi-flat subsetof Levi dimension n invariant by a n-dimensional holomorphic foliation ℱ on ℙ^N. Suppose that n>(N-1)/2. If the Levi foliation ℒ has infinitely many algebraic leaves, then: * the -complexification H^ of H extends to an algebraic variety in ℙ^N;* the foliation ℱ^ι=ℱ|_H^ has a rational first integral R in H^;* there exists a real algebraic curve S⊂ℂ such that H⊂R^-1(S). In particular H is semialgebraic.For a real algebraic Levi-flat subset H⊂ℙ^N, the -complexification H^ is algebraic. If further H is invariant by a global n dimensional holomorphic foliation, then the same elementsof the proof of Theorem <ref>give that assertions (2) and (3) are also true in this case.In the local point of view,we have the following integrability result: Let ℱ be a germ of holomorphic foliation of dimension nat (ℂ^N,0) tangent to a germ of real analytic Levi-flat subsetH of Levi dimension n.Suppose that (H^), the singular set of the -complexification of H,has codimension at least two. Then ℱ^ admits a meromorphic first integral.The proof of this theorem, in Section <ref>, relies on theintegration techniques of Brunella'sgeometric proof for Cerveau-Lins Neto'slocal integrability theorem<cit.>. Lastly,we illustrate our main results with some examples in Section <ref>.This article is apartial compilation of the results of the Ph.D. thesis of the first author <cit.>, written under the supervision of the second and third authors. They all express their gratitude toR. Rosas and B. Scárdua forsuggestions in the development of this work.§ MIRRORING AND COMPLEXIFICATIONConsider coordinates z = (z_1,…,z_N) in ℂ^N, where z_j = x_j + i y_j, and the complex conjugation z̅ = (z̅_1,…,z̅_N), where z̅_j = x_j - i y_j. We will employ the standard multi-indexnotation. For instance, if μ = (μ_1,…μ_N) ∈_≥ 0^N then z^μ = (z_1^μ_1,…,z_N^μ_N).We also fix the following notation for rings of germs at (^N,0): * 𝒪_N ={z_1,…,z_N} is the ring of germ of complex analytic functions;* _N = {z_1,…,z_N,z̅_1,…,z̅_N} = {x_1,y_1,…,x_N,y_N} is the ring of germs of real analytic functions with complex values;* _N ⊂_N is the ring of germs of real analytic functions with real values.Agerm of functionϕ(z)=∑_μ,ν a_μν z^μz̅^ν in _N is in _N if and only if ϕ(z) = ϕ(z) for all z, which is equivalent to a_μν = a̅_νμ for all μ, ν. Let ℂ^N* be thespace with the opposite complex structure ofℂ^N, having complex coordinates w = (w_1,…,w_N) = z̅. The conjugation map Γ: z=x+iy ↦ x-iy = w defines a biholomorphism between ℂ^N and ℂ^N*. This correspondence is referred to as mirroring. In general, given a subset A ⊂^N, itsmirror is the setA^* = Γ(A) = {z̅ ;z∈ A}⊂ℂ^N*.Given a complex function ϕ in A ⊂ℂ^N, its mirror ϕ^* is thefunction in A^*⊂ℂ^N* given by ϕ^*(w) =ϕ(w). For instance, if ϕ(z)=∑_μa_μz^μ is complex analytic, then its mirrorϕ^*(w) = ∑_μa_μw̅^μ = ∑_μa̅_μw^μis complex analytic. In the same way, ifϕ∈_Nhas adevelopment in power series ϕ(z)=∑_μ,νa_μνz^μz̅^ν, where z ∈ℂ^N, then its mirror function ϕ^*∈_N has a power seriesexpansionϕ^*(w)= ∑_μ,νa_μνw̅^μw^ν =∑_μ,νa̅_μνw^μw̅^ν = ∑_μ,ν a_νμw^μw̅^ν, where w ∈ℂ^N*. It follows from this discussion that, if A ⊂ℂ^N isa (real or complex) analytic subset, so is its mirror A^*⊂ℂ^N*. Thismirroring procedure canbe applied to other geometric objects. For example, to ananalytic p-form η=∑_Iα_I(z)dz_I, where I=(i_1,…,i_p) and dz_I=dz_i_1∧⋯∧ dz_i_p, weassociate thep-form η^*=∑_Iα_I^*(w)dw_I. A germ of holomorphic foliation ℱ of codimensionpat (ℂ^N,0), defined by a p-form η — that is integrable and locally decomposable outside the singular set — engenders its mirror ℱ^*, which is the foliationof codimension p definedbyη^* whose leaves arethe mirroring ofthose of ℱ (see the Appendix for the definition of holomorphic foliation).We consider ℂ^N×ℂ^N*≃ℂ^2N with coordinates (z,w), the embeddingi:ℂ^N → ℂ^N×ℂ^N*z ↦ (z,z̅)and the diagonal subsetΔ:=i(ℂ^N)={(z,w)∈ℂ^N×ℂ^N*; w=z̅}. Given a germ of analytic function ϕ∈_N we say that a connected neighborhood U of 0 ∈ℂ^N isreflexive for ϕ or ϕ-reflexive ifϕ(z,w) converges in U× U^*⊂ℂ^N×ℂ^N*.For a germ of map ϕ=(ϕ_1,...,ϕ_k) ∈(_N )^k, a ϕ-reflexiveneighborhood is one that is ϕ_j-reflexive for every j=1,...,k.Let ϕ∈_N be a real function withdevelopment in power series ϕ(z)=∑_μ,νa_μνz^μz̅^ν. The complexification of ϕ is the germof complex function ϕ^∈O_2N defined at the origin 0∈ℂ^N×ℂ^N* by the seriesψ^(z,w)=∑_μ,νa_μνz^μ w^ν.If U ⊂ℂ^N is a ϕ-reflexive neighborhood, then this series converges in U × U^*. The complexification of a germ of map ϕ=(ϕ_1,...,ϕ_k) ∈(_N )^k is the germ of complex map ϕ^∈ (O_2N)^k defined by ϕ^=(ϕ_1^,...,ϕ_k^).Let H be a germ of real analytic variety at (ℂ^N,0). As before, we denote by H_reg its regular part. The singular part of H, denoted by H_sing,consists of the points in H ∖H_reg. Letℐ(H) denote the ideal of H in _N. Since_N is Noetherian, we can take a system of generators ϕ_1,...,ϕ_k of ℐ(H) and associate a map ϕ=(ϕ_1,...,ϕ_k) ∈(_N )^k that is calledgenerating map of H. We have the definition:The extrinsic complexification or simply complexification H^ℂ of H is the germ of complex analytic variety at (ℂ^N×ℂ^N*,0) defined by the equation ϕ^(z,w)=0.If U is ϕ-reflexive neighborhood, then H^ℂ is realized asH^ℂ = {(z,w)∈ U × U^*; ϕ(z,w)=0}.The set H^ℂ is the smallest germ of complex analytic subset at (ℂ^N×ℂ^N*,0) containingH_Δ := i(H) = H^ℂ∩Δ. It is evident from the definition that the complexification respects inclusions: if H_1⊂ H_2 are germs of real analytic varieties then H_1^⊂H_2^. This notion of complexification, introduced by H. Cartan in <cit.>, has the following properties: * H^ℂ⊃ H_Δ;* every germ of holomorphic function vanishing over H_Δ also vanishes over H^ℂ;* the irreducible components of the real analytic variety H are in correspondence, by complexification, to the irreducible components of thecomplex analytic variety H^ℂ. In particular, H is irreducible if and only if H^ℂ is irreducible.Let us examine the effect of the complexification procedure oncomplex varieties. Take X ⊂ (ℂ^N,0) a germ of complex analytic variety whose ideal in O_N is generated by f_1, …, f_k. Seen as a real analytic variety, the corresponding generators ofthe ideal of X in _N are ϕ_j = (f_j) = (f_j + f̅_j)/2 and ψ_j = (f_j) = (f_j - f̅_j)/2 i, for j=1,…,k. Thus, the complexification X^ in (ℂ^N×ℂ^N*,0) is the complex analytic variety defined by the system of equationsϕ_j(z,w) =f_j(z) + f̅_j(w̅)/2 = f_j(z) + f_j^*(w)/2 = 0andψ_j(z,w) =f_j(z) - f̅_j(w̅)/2i = f_j(z) - f_j^*(w)/2i = 0, for j=1,…,k, which is equivalent tof_j(z) = 0and f_j^*(w) =0 forj=1,…,k.We therefore conclude that X^ = X × X^*. In particular, we have that _ X^= 2 _ X. § LEVI-FLAT SUBSETS, LOCAL ASPECTS Essentially, real analytic Levi-flat subsets are real analytic subsetsof odd real dimension 2n+1 foliated by complex varieties of complexdimension n. When the real codimension is one, we are in the case ofLevi-flat hypersurfaces. We give the precise definition:Let H⊂ M be areal analytic subset of real dimension 2n+1, whereM is an N-dimensional complex manifold, N ≥ 2 and 1≤ n≤ N-1. We say that H is a Levi-flat subset if the distribution of tangent spacesℒ: H_reg⊂ℂ^N →Tℂ^N≃ℂ^Np ↦ T_pH_reg∩ J(T_pH_reg)has dimension n and is integrable in the sense of Frobenius.Theregular part of H is a CR-variety, of CR-dimension n+1, carryingan n-dimensional foliation with complex leaves. We use the qualifier “Levi” for the foliation, its leaves and its dimension. The foliation itself is also denoted byL, its dimension is called L-dimension and denotedby L or H.The leaf through by p ∈ H_reg is denotedby L_p. Also, we say that N a the ambient dimension ofH. Most of the time we are concerned with local properties of Levi-flat subsets. In this case, an open set U ⊂^N plays the role of M in the definition. The notion ofLevi-flat subset germifies and,in general, we do not distinguish a germat (^N,0) from its realization in some neighborhood U of 0 ∈^N.A trivial model for a Levi-flat subset of L-dimension n in ^N is provided byH={z = (z',z”) ∈^n+1×^N-n-1; (z_n+1)=0,z”=0},where z'=(z_1, ... ,z_n+1) and z”=(z_n+2,...,z_N). The Levi foliation is given by{z = (z',z”) ∈^n+1×^N-n-1;z_n+1=c , z”=0, with c ∈ℝ}.This trivial model is in fact a local form for Levi-flat subsets.This was mentioned in <cit.> without an explicit proof, which we give for the sake of completeness: Let H be a Levi-flat subset of L-dimension n and ambient dimension N. Then, at each p∈ H_reg, there are local holomorphic coordinates (z',z”) ∈^n+1×^N-n-1such that H has the local form (<ref>).Since H_reg is a CR-subvariety, for some k with 2 ≤ k ≤ N, there arelocal holomorphic coordinatest=(t',t”)∈ℂ^k×ℂ^N-k at p such that H_reg⊂{t” =0}≅ℂ^k is a generic subvariety, that is, H_reg is defined by d real functions in ℂ^k whosecomplex differentials are ℂ-linearly independent <cit.>. This givesdim_ℝH_reg+d=2k and dim_ℂT^(1,0)H_reg+d=k.Combining these equations, we obtain k=dim_ℝH_reg-dim_ℂT^(1,0)H_reg= (2n +1) - n =n+1.We found that H_reg is as a real analytic Levi-flat hypersurface in the complex variety {t”=0}. It then sufficesto apply E. Cartan's normal form (<ref>)to the coordinates t' in order to get thecoordinates z' and take z” = t”. Inthe local form (<ref>), {z” = 0} corresponds to the unique local(n+1)-dimensional complex subvariety of the ambient space containing the germ of H_reg at p.These local subvarieties glue together forming a complex variety defined in a whole neighborhood of H_reg.It is analytically extendableto a neighborhood of H_reg by the following theorem:[Brunella <cit.>]Let M be an N-dimensional complex manifold and H ⊂ M be areal analytic Levi-flat subset of -dimension n.Then, there existsa neighborhood V ⊂ M of H_reg anda unique complexvariety X ⊂ V of dimension n+1 containing H.The variety X is the realization in the neighborhood V of a germ of complex analytic variety around H. Wedenote it — or its germ — by H^ and call itintrinsic complexification or-complexification of H. It plays a central role in the theory of Levi-flat subsets we develop. The notion of intrinsic complexification also appears in <cit.> with the name of Segre envelope.In this article we are mostly interestedin real analytic Levi-flat subsets which are invariant byholomorphic foliations in the ambient space. A real analytic Levi-flat subset H ⊂ M isinvariant by an n-dimensionalsingular holomorphic foliationℱ on M ifthe Levi leaves areleaves of ℱ. We also say that ℱ is tangent to H. If H is invariant by a foliation, the same holds for its -complexification:Let H ⊂ M be areal analytic Levi-flat subset of -dimension n, whereM is a complex manifold of dimension N. If H is invariant by an n-dimensionalholomorphic foliation ℱ on M, then its -complexification H^ is also invariant by ℱ. We have ℱ|_H_reg=ℒ, where ℒ is the Levi foliation. The problem is local, so we can work ina local trivialization(<ref>), in which the -complexification is defined by z” = 0 and the Levi leaves are given by{ z_n+1=c, z”=0}, where c ∈ℝ.Let v⃗=(v_1,...,v_n+2,...,v_N) be alocal vector fieldtangent to ℱ. For each i=n+2,..., N, every ζ=(z_1,...,z_n)∈ℂ^n and z_n+1∈ℂ sufficiently small, it holdsv_i(ζ, Re(z_n+1),0,...,0)≡0,and thusv_i(ζ,z_n+1,0,...,0) ≡ 0.This says that H^ is invariant by v⃗.When H is invariant by the foliation , we denote by ℱ^=ℱ|_H^ the restriction of ℱ to H^. Note that ℱ^ has codimension one in H^. Let H be a germ of real analytic Levi-flat subset.Then H^ℂ is a subset of H^× H^* of complex codimension one. Since H ⊂ H^, it is a consequence of the comments in Section <ref> thatH^⊂ (H^)^ = H^× H^*.Now, this inclusion must be proper since, otherwise, given a defining mapϕ for H, the complexification ϕ^ would vanish over H^× H^*, which would imply that ϕ itselfwould vanish overH^i. Finally, if L is the closure of a Levi leaf of H, which is an analytic set of dimension H (see Proposition <ref> below), then L × L^* = L^⊂ H^C. That is, H^ contains infinitely many complex varieties of codimension two in H^× H^*. This implies that the codimension of H^ℂ in H^× H^*is strictly lower than two, which gives the result. Denote by π_1:H^ℂ→ H^ and π_2:H^ℂ→ H^ * the restrictions of the two canonicalprojections to H^ℂ⊂ℂ^N×ℂ^N*≃ℂ^N×ℂ^N→ℂ^N. The following fact appeared in the proof of Theorem <ref>. Its usefulness motivates anexplicit statement:Let H be a germ of real analytic Levi-flat subset. Then given p∈H_reg, we have π_1(H^ℂ_(p,p̅))=H_p^andπ_2(H^ℂ_(p,p̅))=H_p̅^ *,where the sets involved are germs of H^ℂ, H^ and H^* at (p,p̅), p and p̅, respectively.§ SEGRE VARIETIES OF LEVI-FLAT SUBSETSLet H be a germ of real analytic Levi-flat subset at (^N,0), ϕ=(ϕ_1,...,ϕ_k) ∈(_N )^kbe a generating map and U be a ϕ-reflexive neighborhood. For each p ∈ H^∩ U, the setΣ_p(U, ϕ):={z∈ U;ϕ(z,p̅)=0}∩ H^⊂ U ∩ H^is called Segre variety at p associated to thegenerating map ϕ and to the ϕ-reflexive neighborhood U. The Segre variety Σ_p(U, ϕ) ⊂ U is a closed analytic setthat contains p if and only if p ∈ H. It does not depend on the generating map and on the neighborhood U of 0 ∈ℂ^N in the following sense: if ψ is another generating map of H and V is a ψ-reflexive neighborhood of 0 ∈ℂ^N, then there exists a neighborhood of the originW⊂ V ∩ U such that whenever p∈ W ∩ H^ it holds Σ_p(U,ϕ)∩ W=Σ_p(V,ψ)∩ W. In particular, the germ at p of the Segre variety is well defined. It will bedenote by Σ_p. It contains p if and only if p ∈ H. Recall that, by Proposition <ref>, we have H^ℂ⊂ H^× H^ *.Let π_1 :H^ℂ→ H^ and π_2 :H^ℂ→ H^ * be the canonical projections. For p ∈H^, if we identify H^×{p}≃H^, we have, by (<ref>) and (<ref>),π_2^-1(p̅) = {z ∈ H^; ϕ^(z,p̅) = 0} = Σ_p .Similarly, under the identification {p}× H^*≃H^*, we have thatπ_1^-1(p) = {w ∈ H^*; ϕ^(p,w) = 0} = Σ_p^* . We have the following result: Let W ⊂ U ⊂ℂ^N be an open set andϕ(z,z̅) be a real analytic map in U. Suppose thatL⊂ W is a complex variety such that ϕ(z,z̅) = 0 for all z∈ L. Then, for each fixed p∈ L, we haveϕ(z,p̅)= 0 for all z∈ L. Without loss of generality, we can suppose that W = U and that U is ϕ-reflexive. Let H = {ϕ(z,z̅) = 0 }⊂ U. Our hypothesis is that L ⊂ H. Taking complexifications, we find L × L^*⊂ H^⊂^N×^N*. Given p ∈ L, we haveL ×{p̅}⊂ H^∩(^N×{p̅}) = {ϕ^(z,p̅) = 0 }. This is equivalent to L ⊂{ϕ(z,p̅) = 0 }, which is the desired result.As a consequence, if H is a Levi-flat subset and L_p is the Levi leaf at p∈H_reg, then L_p ⊂Σ_p, which gives codim _ℂ,H^(Σ_p) ≤ codim _ℂ,H^(L_p) = 1. This remark motivates the following definition: Let H be a germ of real analytic Levi-flat subset.The point p∈ H is said to be Segre degenerate or simply S-degenerate ifcodim _ℂ,H^ (Σ_p)=0.When codim _ℂ,H^ (Σ_p) =1, the point p∈ H is called Segre ordinary or S-ordinary. We denote by S_d theset ofS-degenerate pointsof H. For a germ ϕ∈_N and for a ϕ-reflexive neighborhood U, equation (<ref>) gives that, whenever (p,q̅) ∈ U × U^*,ϕ^(q,p̅) = 0 ⇔ϕ^(q,p̅) = 0⇔ϕ^(p,q̅) = 0.This applied to the components of a generating map ϕ of a Levi-flat subset H andto a ϕ-reflexive neighborhood U gives the following: We have q∈Σ_p(U,ϕ) if and only if p∈Σ_q(U,ϕ). In particular, if p∈ S_d, then p∈Σ_q for every q∈ U ∩ H^. We have the following proposition: S_d is a complex analytic variety.Following the above notation, we haveS_d={p∈ U ∩ H^; ϕ^(z,p̅)=0∀ z ∈ U ∩ H^}={p∈ U ∩ H^;ϕ^(p,z̅) = 0∀ z ∈ U ∩ H^}, and thenS_d=(⋂_z∈ U ∩ H^{ϕ^(p,z̅) =0})∩ H^.This defines S_d as acomplex analytic set. It is worth commenting that S_d is a proper subset ofH^. Indeed,otherwise, by(<ref>), ϕ^ would vanish over H^× H^ *. This would happen if and only ifϕ^≡ 0, which is impossible. It is a known fact that the set of S-degenerated points of a Levi-flat hypersurfaceform a complex subvariety of codimension at least two contained in H_sing<cit.>. For Levi-flat subsets we can state the following:S_d has codimension at least two inH^.We first suppose thatn= H =1, so thatdim_ℝH=3 and dim_ℂH^=2. By contradiction, suppose that there exists a one-dimensional irreducible component Γ⊂ S_d. We haveΣ_p=H^ for every p∈Γ. As before, let π_2 :H^ℂ⊂ H^× H^ *→ H^ * be the projection in the second coordinate. Then, by (<ref>), we have π_2^-1(p̅)≃Σ_p= H^ for every p∈Γ. Therefore π_2^-1(Γ^*)= H^×Γ^* isa three-dimensional variety. On the other hand, H^ℂ is irreducible and thus π_2^-1(Γ^*)= H^ℂ, which gives Γ^*= π_2(H^ℂ) = H^ *.This is a contradiction, since Γ^* is properly contained in H^ *. The general case n= H >1 follows from the particular one by taking planar sections.Considera complex plane α of codimension n-1 simultaneously transversal to H, H^ and H_sing. The setsH_α=H∩α and H_α^=H^∩α have dimensions dim_ℝH_α=3 and dim_ℂH_α^=2. By the minimality property, we have thatH^_α=(H_α)^ is the -complexification of H_α. Let ϕ be a defining map for H. If (S_α)_d denotes the set of S-degenerated points of H_α, we have(S_α)_d={p∈ H_α^; ϕ|_α(z,p̅)≡ 0onH^_α}⊇{p∈ H^;ϕ(z,p̅)≡0onH^}∩α=S_d∩α.The particular case gives that (S_α)_dis formed by isolated points, which is enough to conclude that codim_ℂ,H^S_d≥ 2.Levi leaves of a real analytic Levi-flat hypersurfaceare closed analytic varieties. The same hods for Levi-flat subsets:The Levi leaves of a germ of Levi-flat subset are closed analytic sets. Indeed, by Proposition <ref>, every Levi leaf contains S-ordinary points. Thus, if p∈ H_reg is S-ordinary and L_p is the corresponding Levi leaf, we have _ℂL_p = _ℂΣ_p. Since L_p ⊂Σ_p, we conclude thatL_p is a component of the analytic set Σ_p.For a germ ofreal analytic Levi-flat subset H at (^N,0),a point p ∈ H_sing is said to be dicritical if it belongs to (the closure of) infinitely many leaves of . The main result in <cit.> states that the notions of dicriticalnessand Segre degeneracy coincide for real analytic Levi-flat subsets.§ LEVI FLAT SUBSETS IN PROJECTIVE SPACESIn this section wepresent some results on real analytic Levi-flat subsets in the complex projective space ℙ^N = ℙ^N_ℂ. If H ⊂ℙ^N is a real analytic variety, then the natural projectionσ: ℂ^N+1∖{0}→ℙ^Nidentifies H with the complex coneH_κ:={z∈ℂ^N+1∖{0};σ(z)∈ H}∪{0}, which is areal analytic subvariety in ℂ^N+1∖{0}.When H is Levi-flat,H_κ naturally inherits the Levi structure of H and H_κ =H + 1. We have that H is algebraic if and only if H_κ is analytic at 0 ∈ℂ^N+1 <cit.>. Thus, in the real algebraic case,some of the local constructions done so far can be repeated forthe germ of H_k at (ℂ^N+1,0). For instance, we can extend the construction ofthe (extrinsic) complexification for a real projective algebraic variety H ⊂ℙ^N. Consider the ideal ℐ(H_κ) in ℂ[z,z̅], wherez = (z_1,…,z_N+1) are coordinates of ℂ^N+1, and take a system of generators ϕ_1,...,ϕ_k, where, for j=1,...,k, each ϕ_jis a bihomogeneous polynomial of bidegree (d_j,d_j) in the variables (z,z̅).Their complexifications definea complex variety H_κ^ℂ in ℂ^N+1×ℂ^N+1, which goes down to an algebraic subvariety H^ℂ⊂ℙ^N×ℙ^N called (extrinsic) projective complexification of H. Note that H^ℂ inherits the properties of the local complexification H_κ^ℂ.We summarize this in the following:Let H⊂ℙ^N be a real algebraic variety. Then H^ℂ⊂ℙ^N ×ℙ^N is a complex algebraic variety, which is irreducible if and only ifH is. We nowexamine the intrinsic complexification H^ of a real analytic Levi-flat subsetH ⊂ℙ^N. In principle, by pasting local -complexifications, we build H^as a complex analytic variety of dimension H+1defined in an open neighborhood of H_reg. When H is algebraic, H^extends to analgebraic subset of ℙ^N, as shown in: Let H⊂ℙ^N be an irreducible real algebraicLevi-flat subset of -dimension n. Then its -complexification H^ extends to an (n+1)-dimensional algebraic variety in ℙ^N. We associate to H itsprojectivecone H_κ, which is analytic and irreducible as a germ at (ℂ^N+1,0). Let H_κ^ℂ denote its complexification at (ℂ^N+1×ℂ^N+1,0). By Proposition <ref>, we have π_1(H_κ^ℂ)=H_κ^, where H^_κ is the -complexification of H_κ. By Proposition <ref>, H^ℂ⊂ℙ^N×ℙ^N is complex algebraic and so is its image π^ℙ_1(H^ℂ)⊂ℙ^Nbythe projection π^ℙ_1: ℙ^N×ℙ^N→ℙ^N in the first coordinate. Note that the cone associated with π^ℙ_1(H^ℂ) is (π^ℙ_1(H^ℂ))_κ=π_1(H_κ^ℂ) = H_κ^. Finally, H_κ^isthe cone of an irreducible algebraic variety in ℙ^Nof dimension n+1 which contains H. The result follows from the uniqueness of the intrinsic complexification as a germ around H. Next we look at Segre varieties of a Levi-flatalgebraic subset H. We identify H with its algebraic cone H_κ at (ℂ^N+1,0) and take a system of bihomogeneous generators ϕ_1,..., ϕ_k ∈ℂ[z,z̅] for the ideal ℐ(H_κ).By Proposition <ref>, the -complexification H^_κ is algebraic. It then follows from Definition <ref> that the Segre varieties of H_κ are algebraic. An arbitrary Levi leaf of H_κ contains S-ordinary points and, at each of these points, it is a component of the corresponding Segre variety. This gives the following:The Levi leaves of areal algebraic Levi-flat subset in ℙ^N are algebraic. As we observed, when a Levi-flat subsetH⊂ℙ^N isreal analytic, its -complexification in principle is defined in a neighborhood of H_reg. However, in certain cases, we can apply extension results of analytic varieties in order to prove that H^ extends to an algebraic variety in ℙ^N. For instance, we can useof the following theorem:(Chow, <cit.>) Let Z⊂ℙ^N be an algebraic set of dimension n and V be a connected neighborhoodof Z in ℙ^N. Then any analytic subvariety of dimension higher than N-n in V that intersects Z extends algebraically to ℙ^N. This allows us to state the following extension result for the -complexification H^:Let H⊂ℙ^N be a real analytic Levi-flat subset of H =n such that N>3 and n>N-12. If the Levi foliation has an algebraic leaf, then H^ extends to an algebraic variety in ℙ^N. We have _ℂ L=n, where L is the Levi leaf which supposed to be algebraic, and _ℂ H^=n+1. Since n>(N-1)/2, we find _ℂ H^ = n+1>N-n.The result then follows from Chow's Theorem.Afoliation of codimension one in ℙ^N tangent to an algebraic Levi-flat hypersurface has arational first integral <cit.>. We can state a version of this result inthe context ofLevi-flat subsets. We consider a real analytic Levi flat subset H ⊂ℙ^N of H = n, invariant by an n-dimensional holomorphic foliation . By Proposition <ref>, H^ is invariant by . We will be mostly concerned with ℱ^:=ℱ|_H^, which is a codimension one foliation on H^, which in principle is asingularvariety. Wemake use of the followingresult on the integrability of foliations in projective manifolds:(X. Gómex-Mont, <cit.>)Let ℱ be a singular holomorphic foliationof codimension q on an irreducible projective manifold M. Assume that every leaf L of ℱ is a quasi-projective subvariety of M. Then there exist a projective manifoldX of dimension q and a rational map f:M→ X such that the leaves of ℱ are contained in the fibers of f. We also need the following generalization of Darboux-Jouanolou Theorem <cit.>: (E. Ghys, <cit.>)Let ℱ be a singular holomorphic foliation of codimension oneon a smooth, compact and connected analytic complex manifold. If ℱ has infinitely many closed leaves, then ℱ has a meromorphic first integral and, therefore, all its leaves are closed. In order to apply the above theorems, we have todesingularize the -complexification H^ usingHironaka's Dessingularization's Theorem <cit.>: there existsa manifold H̃^̃ and a proper bimeromorphic morphism π:H̃^̃→ H^ such that: * π:H̃^̃∖ (π^-1((H^)) →H^∖(H^) is an isomorphism.* π^-1((H^)) is a simple normal crossing divisor. Note that if the real analytic Levi-flat subset H ⊂ℙ^N is tangent to an abient foliation ℱ on ℙ^N, then ℱ^, being the restriction of ℱ to H^, lifts bythe desingularization mapto a foliation ℱ̃^ onH̃^̃.We then have the main result of this section:Let H⊂ℙ^N be a real analytic Levi-flat subset of H = n invariant by a holomorphic foliation ℱ in ℙ^N. Suppose that the -complexification H^ extends to an algebraic variety inℙ^N — which happens, for instance, ifN>3 and n> (N-1)/2. If the Levi foliationhas infinitely many algebraic leaves, thenℱ^ = ℱ|_H^ hasa rational first integral. Let π:H̃^̃→ H^be a desingularization map. H^ is compact and so is H̃^. We lift ℱ^to an n-dimensional foliation ℱ̃^on H̃^̃. Our hypothesis gives that ℱ^ has infinitely many closed leaves and thusthe same holds for ℱ̃^.By Theorem<ref>,ℱ̃^ admits a meromorphic first integral in H̃^̃. So, all leaves of ℱ̃^ are compact. Besides, their π-imagesare compact leaves of ℱ^ in H^. Finally, by Theorem <ref>, there exists a one-dimensional projective manifold Xand a rational map f:H^→ X whose fibers contain the leaves of ℱ^. The rational first integral is obtained by composing f with anon-constant rational map r:X →ℙ^1.When a Levi-flat subset H⊂ℙ^N is algebraic, assembling the conclusions of Propositions <ref> and <ref>, the same argument of the proof of Proposition <ref>givesthe following integrability result: Let H⊂ℙ^N be an algebraic Levi-flat subset invariant by a holomorphic foliation ℱ. Then H^ is algebraic and ℱ^ has a rational first integral. § RATIONAL FUNCTIONS AND LEVI-FLAT SUBSETS Let R be a rational function on ℙ^N and S⊂ℂ be a real algebraic curve. Then R^-1(S) is a Levi-flat hypersurface <cit.>. An equivalent result — with a similar proof — can be statedin the context of this paper: Let X⊂ℙ^N be an irreducible (n+1)-dimensional algebraic variety, R be a rational function on X and S⊂ℂ be a real algebraic curve. Then the set R^-1(S) is an algebraic Levi-flat subset of L-dimension n whose -complexification is X. Our goal in this section isto prove that, with the additional hypothesis that the Levi-flat subset is tangent to a foliation in the ambient space, a reciprocal of this result can be proved by adapting the techniques of <cit.>.Fix the usual notation (F) for the indeterminacy set of a meromorphic function F. We have the following local portrait of Levi-flat subsets tangent to the levels of meromorphic functions: Let H be a germ of irreducible real analytic Levi-flat subsetof H = n at (ℂ^N,0). Suppose that F is a non-constant meromorphic function in H^, such that codim_ℂ (F)≥ 2, which is constant along the Levi leaves. If 0∈ H ∩(F), then there exists an algebraic one-dimensional subset S⊂ℂ such that H⊂F^-1(S). Since the proof goes as that of <cit.>, we just review its main steps and verify that they adapt to our context. It is sufficientto consider the casen=1, for which _ℂH^=2 and 0∈ H^ is an isolated point of indeterminacy of F — the general casen>1 reduces to this particular one by cutting H by an (n-1)-plane α in general position, as we did in Proposition <ref>.Write F=f/g, where f and g are holomorphic functions in H^, without common factors, and consider the mapΦ: z ∈ H^↦(f(z),g(z)) ∈ℂ^2The crucial fact is thatΦ(H) is semianalytic, an open subset of an analytic variety K of the same dimension. In fact, the mapΨ^ℂ: (z,w) ∈ H^× (H^)^* ↦(f(z),g(z), f^*(w),g^*(w)) ∈ℂ^4is finite and thus, by theFinite Map Theorem,Ψ^ℂ(H^ℂ) is an analytic variety. Therefore, consideringΦ̃: z ∈ H^↦ (f(z),g(z),f(z),g(z)) ∈ℂ^4,we have that Φ̃(H) ⊂Ψ^ℂ(H^ℂ)∩Δ is open and thus it is semianalytic. Note thatΨ^ℂ(H^ℂ)∩Δ⊂^4 can be defined by functions that depend only on the two first coordinates. Thus,taking the projection π: ^4→^2, π (z_1,z_2,z_3,z_4)=(z_1,z_2), we have that Φ(H)=π(Φ̃(H))⊂π_1(Ψ^ℂ(H^ℂ)∩Δ) = K is also semianalytic.Note that Φ(H) contains infinitely many complex lines through the origin and thus, if r(z,z̅)=∑_j,k r_jk(z,z̅) is a defining function for K, written inbihomogeneous terms of bidegree (j,k), then r_j,k(z,z̅)≡ 0 for all (j,k), meaningthatK isreal algebraic. Next, project the algebraic set{(z,ξ)∈ℂ^2×ℂ; z∈ Kand ξ z_2=z_1}in theξ-variable. By Tarski-Seidenberg Theorem<cit.>, this projection is semialgebraic, so it lies in a one-dimensionalalgebraic set S⊂ℂ. Thus K⊂G^-1(S). Since ϕ(H)⊂ K andF=G∘ϕ, we concludethat H⊂F^-1(S).Remark that if X ⊂ℙ^N is an algebraic complex variety of _ℂX ≥ 2, then any rational function in X admits points of indeterminacy. This gives us the following: Let ℱ be a holomorphic foliation in ℙ^N tangent to a real analytic Levi-flat subset H of H = n. Suppose that ℱ^ has arational first integral R. Then there exists a real algebraic curve S⊂ℂ such that H⊂R^-1(S).Write H_reg=∪_ℓL_ℓ, where L_ℓ are irreducible complex analytic subvarieties given by the closures of Levi leaves of ℱ, which are levels of the rational function R. Takingp∈(R), then p∈ H, since p∈∩L_ℓ.Applying Proposition <ref> at p, we find a one-dimensional algebraic subset S⊂ℂ such that, locally, H⊂R^-1(S). Since H_reg=∪_ℓL_ℓ and p∈ L_ℓ for every ℓ, then H⊂R^-1(S).With this proposition, we accomplish the proof of Theorem <ref>:By Proposition <ref>,ℱ^=ℱ|_H^ has a rational first integral in H^, say R. The result then follows from Proposition <ref>. In a similar way, the combination of Proposition <ref> and the Corollary <ref> gives:Let H⊂ℙ^N an algebraic Levi-flat subsetinvariant by a foliation ℱ in ℙ^N. Then there exist a rational function R in H^ and a real algebraic curve S⊂ℂ such thatH⊂R^-1(S). § A COMMENT ON BRUNELLA'S INTEGRATION TECHNIQUESIn this section we explain how the techniques of <cit.> can be adapted in order to prove Theorem <ref>. Recall the conditions of its statement: we havea germ of real analytic Levi-flatsubset Hat (ℂ^N,0),of H = n and codim_ (H^) ≥ 2, invariant by a germ of holomorphic foliation ℱ ofdimension n. We start by remarking that, by applyingthe Transversality Lemma (stated a proved in the Appendix) and taking transverse plane sections, we can suppose thatH = 1 and thatH^ has an isolated singularity at 0 ∈ℂ^N. We have the following Lemma: LetH be a real analytic Levi-flat subsetof L-dimension 1 at (ℂ^N,0) invariant by a germ of holomorphic foliationℱ. Then,for each p∈ H^∖{0}, the mirror of Segre variety Σ_p^*⊂ (H^)^* is a non-empty curve invariant by the mirror foliation ℱ^ *. Besides, if p and q are on the same leaf of ℱ^, then Σ_p^* =Σ_q^*.The fact that Σ_p^*⊂ (H^)^* is non-empty for everyp∈ H^ sufficiently near 0 ∈^N follows from Proposition <ref>. Since codim_ℂS_d⩾ 2 and H^ = 2, we can suppose that 0 ∈^N is the only Segre degenerate point, implyingthat Σ^*_p is a curve in H^ * for each p∈ H^∖{0}. Take the two-dimensional foliation ℱ^×ℱ^* in H^×H^ * whose leaf through (p,q^*) ∈ (H^∖{0})× (H^ *∖{0}) is L_p,q^*=L_p× L^*_q^*, where L_p denotes the leaf ofthrough p. Consider the analytic complex set of tangencies between ℱ^×ℱ^ * and H^ℂ⊂ H^×H^ *, denoted by Tang(ℱ^×ℱ^ *,H^ℂ) ⊂ H^ℂ. Since H^Δ⊂ Tang(ℱ×ℱ^*,H^ℂ), the minimality of the complexification implies that (H^Δ)^ℂ=H^ℂ= Tang(ℱ×ℱ^*,H^ℂ).Denote, as before π_1: H^ℂ⊂ H^× H^ *→ H^ the projection in the first coordinate. Then, for each p∈ H^∖{0}, the fiber π_1^-1(p)= Σ_p^* is a one-dimensional analytic set tangent to ℱ×ℱ^*.Thus Σ_p^* is invariant by ℱ^ * and is composed by a finite union of leaves of ℱ^*. It follows that, for a fixed leaf L of ℱ^, the inverse image π_1^-1(L) ⊂ H^ℂ is invariant by ℱ^×ℱ^* and has the formπ_1^-1(L)=L×⋃_λ∈Λ L_λ^*, where the L_λ^*'s are leaves of ℱ^ *and Λ is a finite set. In particularly, if p and q ∈ L, we have Σ_p^*= {p}×⋃_λ∈Λ L_λ^* and Σ_q^*= {q}×⋃_λ∈Λ L_λ^*. Identifying these with ⋃_λ∈Λ L_λ^*, we obtain Σ_p^*= Σ_q^*.Theorem <ref>isa straight consequence of the proposition below, for which the above lemma is a key ingredient. Itrestates Propositions 2 and 4 of <cit.> and its proof follows the very same steps as those in Brunella's paper. The only difference is that here we should also take into account the desingularization divisor of the -complexification H^. The hypothesis on the codimension of (H^) is needed in order to apply Levi's extension theorem for meromorphic functions.Let ℱ be a germ of one-dimensional holomorphic foliation at (ℂ^N,0) tangent to a germ of analytic real Levi-flat subset H of H = 1. Suppose that the -complexification H^ has an isolated singularity at origin and that one of the two following conditions is satisfied:* For every p∈ H^∖{0}, the mirror of Segre varietyΣ_p^* is a proper analytic curve in H^ * passing through the origin;* For everyp∈ H^, the mirror of Segre variety Σ_p^*is a proper analytic curve in H^* passing through the origin when p=0;Then ℱ^ has afirst integral that is purely meromorphic in case (1) and holomorphic in case (2). § EXAMPLESLet Z be a real analytic Levi-flat hypersurfacein acomplex manifold X of _X = n+1. Let ℙT^*X be the cotangent bundle projectivization, which is a ℙ^n-bundle over X whose dimensionis N=2n+1. Denote by ρ the projection ℙT^*X→ X.The regular part Z_reg of Z can be lifted to ℙT^*X, since, for any z∈ Z_reg, T^ℂ_zZ_reg=T_z Z_reg∩J(T_z Z_reg)⊂ T_z Xis acomplex hyperplane. Let H_reg be the lifting of Z_reg in ℙT^*X. Fix y∈H_reg such that ρ(y)=z∈Z_reg. It follows from <cit.> that there exists a neighborhood V⊂ℙT^*X of y and a germof complex variety Y_y at y of dimension n+1 containing H_reg on ℙT^*X.We have that H=H_reg is a germ at y of Levi-flat subset of H = n onM = ℙT^*X. The gluing of the local varieties Y_y produces its-complexification H^. By this procedure, any real analytic Levi-flat hypersurface in a complex manifold X induces a real analytic Levi-flat subset in ℙT^*X.When X=ℙ^(n+1), its projectivized cotangent bundle is isomorphic to the incidence varietyΥ ={(p,α)∈ℙ^n+1×ℙ̌^n+1; p∈α},where ℙ̌^n+1 denotes theparameter space of all hyperplanes in ℙ^n+1 (see<cit.>). Therefore, when considering a real analytic Levi-flat hypersurface Z in ℙ^n+1, what we get is a real analytic Levi-flat subset H in Υ. However Υ is not a complex projective space andour main results on global integrability cannot be applied in this situation.A canonical way to generate Levi-flat subsets is by intersecting Levi-flat hypersurfaces with complex analytic subvarieties. The examples of real analytic Levi subsets we present below are based on this principle.Let H={(z_1,z_2,z_3,z_4)∈ℂ^4;z̅_3z_2-z̅_2z_3=0, z_4=0}. Then H is a real analytic Levi-flat subset in ℂ^4, with degenerate singularities along the z_1-axis. The leaves of the Levi foliation are L_c={z_3=z_2c,z_4=0} for c∈ℝ. Note that the -complexification of H is the hyperplane H^={z_4=0}. On the other hand, since H is a complex cone in ℂ^4∖{0}, we get that H induces a Levi-flat subset in ℙ^3 that satisfies the hypothesis of Theorem <ref>. The foliation ℱ given by the polynomial 1-form ω=z_2dz_3-z_3dz_2 defines a holomorphic foliation on ℙ^3 tangent to H. Moreover, ℱ has a rational first integral R=z_3/z_2, which clearly defines a rational first integral on H^. In ℂ^4 with coordinates(z_1,z_2,z_3,z_4),takeH={z_1^2z̅_3^2-z_1z̅_3|z_2|^2+z_1z_3z̅_2^2-2|z_1|^2|z_3|^2+ z̅_1z̅_3z_2^2-z_3z̅_̅1̅|z_2|^2+z_3^2z̅_̅1̅^2=0, z_4=0}.Then H is a real analytic Levi-flat subsetfoliated by the 2-planesL_c={z_1+cz_2+c^2z_3=0, z_4=0} for c∈ℝ. Again, the -complexification is H^ ={z_4=0}. Naturally, H defines a real analytic Levi-flat subset in ℙ^3 but, in this case, H is not invariant by an ambient holomorphic foliation. Note that, by elimination of c in the system of equations {[z_1+cz_2+c^2z_3=0 ; dz_1+cdz_2+c^2dz_3=0]. we obtain a holomorphic 2-web tangent to H on ℙ^3.Wepresent next a real analytic non-algebraic Levi-flat subset of ℙ^3 of -dimension 1, having an algebraic -complexification and containinginfinitely many algebraic leaves in itsLevi foliation. However, it is not invariant by a global holomorphic foliation on ℙ^3.This shows that, in Theorem <ref>, the assumption of the existence of a global foliation is essential in order togetsemialgebricity. Weadapt anexamplein <cit.>, whose construction is summarized in the following lemma: Let S⊂ℝ^2 be a connected compact real analytic curve without singularities. Let H̃ be the complex cone defined byH̃={(z_0,z_1,z_2)∈ℂ^3; z_0=z_1x+z_2y for(x,y)∈ S}∪{z∈ℂ^3; z_1z̅_2=z̅_1z_2}.Then H̃ is a real analytic Levi-flat hypersurface in ℂ^3∖{0} whose canonical projection σ(H̃) is a real analytic Levi-flat hypersurface in ℙ^2. Besides, if S is not contained in any proper real algebraic curve in ℝ^2, then σ(H̃) is not algebraic.Let us nowtake the projection υ: ℂ^4∖{0}→ℂ^3∖{0} defined byυ(z_0,z_1,z_2,z_3)=(z_0,z_1,z_2) and the real analytic complex conedefined byH'=υ^-1(H̃)∩{(z_0,z_1,z_2,z_3)∈ℂ^4∖{0}; z_0z_3 - z_1z_2=0}. Hence H=σ(H') is a real analytic subvariety in ℙ^3.We have that H⊂ℙ^3 is Levi-flat with H = 1 andits intrinsic complexification is the quadric Q ⊂ℙ^3 defined by z_0z_3 - z_1z_2 =0. Moreover, if wepick S ⊂ℝ^2 real analyticbut non-algebraic, we obtaina real analytic non-algebraic Levi-flat subsetH⊂ℙ^3. Finally, we assertthat H is not tangent to a one-dimensional holomorphic foliationon ℙ^3. In fact, without loss of generality and possibly translatingS, we assume that for all x∈ℝ small enough, there exists at least two distinct points y_1,y_2∈ℝ such that (x,y_1), (x,y_2)∈ S. Given such a x ≠ 0, there are at least two distinct leaves of the Levi-flat subset H passing through [x:1:0:0] ∈ H, corresponding the hyperplanes of equationsz_0 = z_1 x + z_2 y_1 and z_0 = z_1 x + z_2 y_2. Then, around these points, the Levi foliation cannot be tangent to an ambient holomorphic foliation. [scale=.5] fig-artigo.eps (100,145) (x,y_1) (115,130) S (100,45) (x,y_2) (72,24) [x:1:0:0] (-15,30) H § APPENDIXLet M be an N-dimensional complex manifold whose cotangent sheaf is Ω_M = O(T^*M). An n-dimensional holomorphic foliationon M, where 1 ≤ n < N, is the object defined by an analytic coherent subsheaf C of Ω_M of rank N-n satisfying the following properties (see <cit.> for details): * d C_p⊂ (Ω_M∧C)_p for every p ∈ M ∖(C) (integrability condition);* (Ω_M / C) is a set of codimension two. Thisis the singular set of F and denoted by (F). We call C the conormal sheaf of F. Recall that the singular set of a coherent sheaf is the set of points where its stalks failto be free modules over the structural sheaf. Outside (F), the conormal sheaf isthe sheaf of sections of a rank N-n vector subbundleof T^*M, defining an integrable holomorphic distribution of subspaces of dimension N-n onT^*M and, thus, a regular holomorphic foliation of dimension n on M. Then, since codim_ (F) ≥ 2, the foliationis locally induced by holomorphic (N-n)-forms which are locally decomposable outside (F) and satisfy the integrability condition. We emphasize thatour definitiondoes not ask F to be a reduced foliation. By definition, this happens when C is a full sheaf, that is, whenever U ⊂ M isopenand ω is a holomorphic section of Ω_M over U that is also a section ofC over U∖(F), then it is a section of C over U.We finish by proving a transversality lemma that has been used inTheorem <ref>. First a definition. Letbe a germ of singular holomorphic foliation of dimension n at the origin of M = ^N with conormal sheaf C,where 1< n < N. Let α be a germ of hyperplane through0 ∈^N and denote by Ω_αits cotangent sheaf. We say that α is in general position with or transverse toif the singular set of (Ω_M /C)|_α≅Ω_α /(C|_α) has codimension at least two. Thus, C|_α is the conormal sheaf of a foliationof dimension n-1 in (α,0) ≅ (^n-1,0) that will be denoted by |_α. [Transversality] Letbe a germ of singular holomorphic foliation of dimension n at (^N,0). Then the set of hyperplanes through 0 ∈^N transverse toform a generic subset in the Grassmannian Gr_0(N-1,N) ≅ℙ^N-1. We have the following fact: if ωis a germ of holomorphic 1-form at 0 ∈^N (not necessarily integrable) with singular set of codimension at least two, thenthe set of hyperplanes through 0 ∈^N transverse to ω is generic in Gr_0(N-1,N) ≅ℙ^N-1. This is actually a consequence of the proof of <cit.>. The conormal sheaf Cofis coherent and thus, generated by finitely many sectionsat 0 ∈^N, say kholomorphic 1-forms ω_1, …, ω_k. For each i=1,…,k, we can cancel one-codimensional singular components ofω_i, obtainingholomorphic 1-forms ω̃_i such that (ω̃_i) ≥ 2. Note that, since we are notassuming thatis reduced,each ω̃_idoes not necessarily define a section of C,yielding however a sectionoutside (). The set of hyperplanes transverse to each ω̃_i is a generic set Γ_i⊂ Gr_0(N-1,N). Let Γ_0 denote the generic set of hyperplanestransverse to () andconsider the set Γ = ∩_i=0^kΓ_i. ThenΓ⊂ Gr_0(N-1,N) is a generic set formed by hyperplanes transverse to . In fact, fix α∈Γ. Let S_0 = () ∩α and S_i = (ω̃_i|_α) for i=1,…,k. Then S = ∪_i=0^kS_i is a germ of analytic subset in (α,0) ≅ (C^N-1,0) of codimension at least two. We assert that (|_α) ⊂ S. Indeed, if p∈α∖S, then p ∉() and thus there are1-forms ω_i_1, …, ω_i_N-n, all of them non singular at p,such that T_p ={ω_i_1(p) = ⋯ = ω_i_N-n(p) = 0 } .But H is transverse to eachω̃_i_ℓ— and also to ω_i_ℓ — at p, giving that p is not a singular point for|_α.plain Jane Bretas Departamento de Física e Matemática Centro Federal de Educação Tecnológica de Minas Gerais Av. Amazonas, 7675 – Belo Horizonte, [email protected] Fernández-PérezDepartamento de Matemática Universidade Federal de Minas Gerais Av. Antônio Carlos, 6627C.P. 70230123-970– Belo Horizonte – MG, BRAZIL [email protected]érioMolDepartamento de Matemática Universidade Federal de Minas Gerais Av. Antônio Carlos, 6627C.P. 70230123-970– Belo Horizonte – MG, BRAZIL [email protected] | http://arxiv.org/abs/1705.09689v1 | {
"authors": [
"Jane Bretas",
"Arturo Fernández-Pérez",
"Rogério Mol"
],
"categories": [
"math.DS",
"math.CV",
"32S65, 32V40"
],
"primary_category": "math.DS",
"published": "20170526192748",
"title": "Holomorphic foliations tangent to Levi-flat subsets"
} |
http://arxiv.org/abs/1705.11052v1 | {
"authors": [
"Jiaozi Wang",
"Wen-ge Wang"
],
"categories": [
"quant-ph",
"cond-mat.stat-mech",
"nlin.CD"
],
"primary_category": "quant-ph",
"published": "20170527144643",
"title": "Correlations in eigenfunctions of quantum chaotic systems with sparse Hamiltonian matrices"
} |
|
Zero-Shot Learning with Generative Latent Prototype Model Yanan Li, Student Member, IEEE, Donghui Wang, Member, IEEE This work was supported by the National Natural Science Foundation of China under Grants 61473256 and CKCEST project. The authors are with the College of Computer Science, Zhejiang University, Hangzhou, 310027, China e-mail: ([email protected], [email protected]). December 30, 2023 =====================================================================================================================================================================================================================================================================================================================================Zero-shot learning, which studies the problem of object classification for categories for which we have no training examples, is gaining increasing attention from community. Most existing ZSL methods exploit deterministic transfer learning via an in-between semantic embedding space. In this paper, we try to attack this problem from a generative probabilistic modelling perspective. We assume for any category, the observedrepresentation, e.g. images or texts, is developed from a unique prototype in a latent space, in which the semantic relationship among prototypes is encoded via linear reconstruction. Taking advantage of this assumption, virtual instances of unseen classes can be generated from the corresponding prototype, giving rise to a novel ZSL model which can alleviate the domain shift problem existing in the way of direct transfer learning. Extensive experiments on three benchmark datasets show our proposed model can achieve state-of-the-art results. § INTRODUCTIONIt has been estimated that humans can easily distinguish between approximately 30000 basic object categories <cit.> and many more subordinate ones, such as different species of birds. Without seeing them, human beings can even recognize new unseen categories by leveraging other information (e.g. by reading text descriptions about object categories on the internet). In contrast, encumbered with a lack of adequate data, generally machines can only recognize hundreds or thousands categories. To free recognition tasks from exuberant collecting of large labelled image datasets, zero-shot learning (ZSL) is gaining increasing attention in recent years, which aims to recognize instances from the new unseen categories which have no instances during training <cit.>.With the label sets between seen and unseen categories being disjoint, the key in the general methodology of ZSL is to establish the inter-class connections via intermediate semantic representations,either manually defined by human experts annotated attributes <cit.>, or automatically extracted from auxiliary text sources <cit.>. Unseen categories can thus be predicted by transferring information from the training dataset. As a valuable knowledge base given in advance, in theoretical, the semantic representations of unseen categories are encouraged to be leveraged in any stage during ZSL. However, most recent works mainly focus on exploring these representations to construct a more effective classifier during testing. While, how to explore them during training to learn more generalized is equally important but still left far from being solved, since the quality of semantic representation predictor is much more rewarding <cit.>. In addition, due to the disjoint data distribution between seen and unseen classes, direct knowledge transfer will cause the domain shift problem during ZSL, leading to degraded performance. In this letter, we tackle these challenges with ideas from generative learning. We posit that there exists a latent space, as illustrated in Fig.<ref>, where each object category is encoded by a unique data (called prototype) essentially. Any type of object representations, e.g. texts or images, are generated from its corresponding prototypes from different perspectives. This supposition is inspired from the cognitive process of human beings, who have remarkably ability of generating various representations, e.g. images, audios, texts, from high-level category labels<cit.>. For example, it is almost effortless for people to imagine the different picture/audios, given the label 'penguin' and 'sparrow'. To mathematically formulate this institution, we assume that the latent prototypes obey a prior distribution where one can draw samples from. During the data generation process, the category prototype is first generated, from which then different observed representations can be developed. Based on this generation process, we further explore the semantic representations given beforehand to make the training process generalize well across unseen categories. A simple ZSL method encompassing different strategies is proposed to solve the domain shift problem by generating virtual unseen instances. Ahead of time, we further give a simple property about these semantic representations as a basic condition for their application in ZSL. Extensive experiments on real world datasets show our proposed method can achieve state-of-the-art results. § GENERATIVE LATENT PROTOTYPE MODEL§.§ Problem SettingFollowing convention, let _s = {y_s^1, ..., y_s^k} and _t = {y_t^1, ..., y_t^l}, _s ⋂_t = ∅ be disjoint label sets of seen and unseen classes in the source domain _s and target domain _t, respectively. Each category corresponds one-to-one to a unique prototype in the latent space , denoted as _s = {_s^1, ..., _s^k} and _t = {_̨t^1, ..., _̨t^l}. We assume there are two different types of observed category representations in ZSL, i.e. visual featuresand semantic features(defined by attributes/texts). In , all categories in _s and _t are embedded as _s and _t in advance.Given a new test image feature _t, the task of zero-shot learning is to construct a classifier f: max_llog p(y_t^l|_t) by making use of image source dataset {_i, y_i}⊂×_s and all available information in . §.§ ZSL with A Latent Prototypical Space Let us motive our approach from a generative probabilistic modelling perspective.We assume that for each category, there are several different observed representations, e.g. images, texts or audio. Each describes the category from a specific perspective and is developed from the category prototype (an original or first model of something from which other forms are copied or developed[Definition taken from Merriam-Webster.com dictionary.]), which abstracts the common essence about the category from these representations. In the latent prototypical space, we assume the manifold structure of prototypes encodes the underlying semantic relationship between different categories. The similar assumption has been successfully applied in <cit.> .To formulate this process, we use the random variable ∈^m to denote the category prototype in the latent spaceand ∈̨^d to denote the observed representation for clarity. For each observation, the generation process, shown in Fig.1, is as follows. * Choose a category prototype _c ∼ p(_c)=Cat(), where Cat() is the categorical distribution and = [_1, ..., _k+l] contains all the prototypes.* Generate observations of the category prototype _c as ∼̨p(|̨_c). Without loss of generality, we assume p(|̨_c) is a linear Gaussian distribution. Specially, in ZSL, we have two types of representations, i.e. imageand text $̨. Forclassc, we assume∼p(|_c) = (|_x _c, _x)and∼̨p(|̨_c) = (_k _c, _k)forand, respectively._x ∈^d×mand_k ∈^a ×mare the projection matrices,_x ∈^d×d,_k ∈^a×aare their covariance matrices. Becauseand$̨ depict different aspects of the category, we give the conditional independence for them, i.e. p(, |̨_c) = p(|_c) p(|̨_c).Given an instance _t from _t, its class label is predicted asy_t = max_l p(l|), l ∈_t. To aid the prediction in Eq.<ref>, we need to use the textual category representation $̨ as an in-between layer to decouple imagesfrom labell, due to its easy accessability and semantic integrity.Common practice is to assign$̨ on a per-class basis, or a per-image basis. The former is particularly helpful, since it allows the minimum effort of annotating a theoretically unlimited number of unseen categories. For convenience, we consider the former case to give our method, which can easily be extended to the latter.The per-class annotation allows a deterministic labelling of the intermediate semantic layer $̨. Therefore, the label prediction in Eq.<ref> becomes:y_t= max_l p(_̨t^l | ), l ∈_t,§.§ Prerequisite ConditionBefore introducing the proposed method, we first give a discussion about the category representationto assist the ZSL task. Obviously, not any representation has the ability of transcending class boundaries and be used to transfer knowledge for making predictions. It should meet the following property.Basic Property.For ∃_̨t^i ∈_t, if _̨t^i ∉range(_s), where _s = [_̨s^1, ..., _̨s^k] and range(_s)denotes the column space of the matrix _s, then={_s, _t} has no transferability for ZSL.Proof. For_̨t^i, if_̨t^i ∉range(_s),∀α∈^k,_̨t^i ⊥_s α, i.e._̨t^iis not in the subspace spanned by all seen classes. Given_t,∀_̨t^l ∈_t,p(_̨t^l|_t)has the same possibility. Thus Eq.<ref> can't make predictions.This property describes a kind of criterion toevaluate whether a specific category representation is transferrable intuitionally. Similar conclusion about binary attribute representations has been discussed in <cit.>.§.§ Generative Latent Prototype Model (GLaP)Based on the above discussion, we propose the solution for our probabilistic model in Eq.<ref>. §.§.§ Learn directly from _sDue to the absence of target domain instances during training, one natural solution for Eq.<ref> is to learnp(|̨)directly from training data in_sby maximizing its log likelihood, i.e.maxlog__s p(|̨), where p(|̨)∝∫ p(|̨_c)p(|_c)p(_c)d_cAll three distributions are in the exponential family.p(|̨)is actually a linear Gaussian distribution, i.e.(+ ,̱ ), where∈^a ×dand∈̱^aestablish the connection betweenand$̨ and can be solved in closed form <cit.>. Take the simplest case for example. When = , =̱, = ^T(^T)^-1, where = [_̨1, ..., _̨N] and = [_1, ..., _N].However, due to _s ⋂_t = ∅, the underlying data distributions of the object categories differ. Approximating the ideal function p(|̨) for _t using Eq.<ref> suffers from a domains shift problem <cit.>. On one hand, it just optimize the source domain where labelled information of target classes is missing. On the other hand,andmay differ in the semantic relationship among different classes, due to their emphasis in the generation process. Therefore, using Eq.<ref> without any adaptation to the target domain will cause significant performance degradation <cit.>. One natural solution to this problem isloading a small amount of instances for target classes in the training stage to adjust Eq.<ref>. §.§.§ Learn from a virtual _t In the above discussion, we assume the prototypical spaceencodes the essence information of categories and also the true semantic relationship among different categories. Let us denote by _s = [_s^1, ..., _s^k] and _t = [_t^1, ..., _t^l] the prototypes of all classes in _s and _t, respectively. And =[_1, ..., _l] ∈^k× l encodes their semantic relationships. Instead of the graph-based relationship <cit.>,is constrained to be linear in this letter, i.e. ∀_t^i, _t^i = _s _i. Based on our generation process, given the prototype _t^i,in this classis actually a Gaussian distribution, i.e.p(|_t^i) = (_x _t^i, _x) = (_x _s _i, _x)whereencodes its relationship with _s. Thus, to generate unseen instances, we need to estimate the two parameters _x and the invisible _s and _i from training data.First, we estimate _i by means of textual representation , by the following function:_i = min__i ||_̨t^i - _s _i||^2_F + Ω(_i),where Ω is the regularizer of _i, common choice is ℓ_2 or ℓ_1 norm <cit.>. While in the per-image basis, its relationship _i can be obtained using the mean representation.Second, we further simplify Eq.<ref> by estimating _x _s instead of the explicit computation of _s and _x separately. Considering the generation process in Sec.<ref>, we assume different types of representations are produced independently. Given _s, we obtain the prototype for each seen class by maximizing the likelihood of visual representations, i.e.max__x_s^i∏_j=1^N_s^i p(_j), p(_j) = ∫ p(_j|_s^i) p(_s^i) d_s^i,where N_s^i denotes the number of training examples in the class _s^i. Optimizing Eq.<ref> gives rise to _x _s^i = 1/N_s^i∑_j=1^N_s^i_j, _j ∈{|y=i}, which is the mean vector. We denote it as _s^i for clarity. Substituting parameters in Eq.<ref>, we obtain:p(|_t^i) ∼(_s _i, _x)where _s = [_s^1, ..., _s^k] contains all the mean vectors of source classes in _s and _x = σ^2 is a predefined covariance matrix. From this distribution, a bunch of virtual instances for unseen category can be randomly produced, denoted as _t = {(_t^i, _t^i,l̃_i)}. Thus, an alternative strategy for ZSL is to learn directly from _t.§.§.§ Final ObjectiveCombining the above two parts, we need the projectionin Eq.<ref> on the one hand to be optimal for _s, on the other hand to be optimized for unseen categories to solve the domain shift problem. We use a trade-off parameter λ to adjust these two effects. The overall objective function is:max_λlog__s p(|̨) + (1-λ) log__tp(|̨),where log__s p(|̨) means calculating p(|̨) from _s. This objective function gives us 3 strategies for predicting in ZSL. (1). When λ = 1, learn directly from _s; (2). When λ = 0, learn from _t; (3). When 0<λ<1, learn from both _s+_t. Their respective performance is showed in later experiments. § EXPERIMENTS AND RESULT ANALYSISIn order to assess the validity of the statements we made, we conducted a set of experiments on three real world datasets. §.§ Experimental SetupDatasets. We test our work on three benchmark image datasets. Animals with Attributes (AwA) <cit.> consists of 30,475 images of 50 image classes, each paired with 85 human-labelled attributes. We follow the usual procedure <cit.>, 40 classes for training and 10 for testing. Caltech-UCSD Birds-200-2011 (CUB) <cit.> is a fine-grained dataset with 312 attributes annotated for 200 bird classes. It contains 11,788 images in total. Following <cit.>, we use the same 150/50 class split for training and testing. Standford Dogs (Dogs) <cit.> contains 19,501 images of 120 fine-grained dog species, with no attributes annotated. We use 90 classes for training and the rest for testing.Choices forand . We mainly use two different types of observed representations in this letter. We use 3 types of deep features for , extracted from 3 popular CNN architectures, VGG <cit.>, GoogLeNet <cit.> and ResNet <cit.>. We extract respectively 1000D, 1024D and 1000D features from these CNNs, which are denoted as fc8, goog and res_fc. They are both low-dimensional and high-semantic features. For the semantic textual representation , 2 different types are used, continuous human-annotated attributes (denoted as A) and 3 kinds of word vectors learned automatically from Wikipedia (skipgram, cbow <cit.> and glove <cit.>). §.§ Evaluation on the strategy of loading testing instancesIn the first set of experiments, we test the performance of our proposed method under various image featuresand textural descriptions . We consider 3 different kinds of strategies corresponding to diverse λ in Eq.<ref>.They are Baseline (Learn from _s when λ = 1), GLaP #1 (Learn from _t, when λ=0) and GLaP #2 (Learn jointly from _s and _t, when λ = 1/2). In the latter two methods, we generate a small bunch of data from Eq.<ref>. For clarity, we use only two types of , i.e. goog and rec_fc and three different types of , i.e. manual attributes (A), word vectors (W) and a concatenation of attributes and word vectors (A+W). Experimental results are shown in Tab.<ref>.Comparing the results ofGLaP #2 with baseline, we find loading virtual testing instances during training can boostthe ZSL performance greatly, regardless of different category representations.On AwA dataset, it achieves the highest 80.84%, 7% more than our baseline. Similarly, the same degree of improvement can be observed on CUB and Dogs, although the performance is not as good as in AwA. On CUB, the reason of performance degradation is that the much finer granularity can hardly be reached by these general deep features, which lower the discrimination ability. In addition, when adopting just one type of category representations, i.e. either A or W, the ZSL performance is a little decreased than in the case of A+W, with the largest difference being almost 25% in CUB. Indirectly, this phenomenon proves our standpoint in the proposed probabilistic model, i.e. different types of observed representations describe the category from different perspectives. To some extent, they can provide complimentary information about categories to improve recognition performance.Generated virtual instances _talone can be used to solve ZSL problem in Eq.<ref> as well. Comparing with baseline, GLaP #1 usually achieves better results. On AwA, using the textual representation, it even obtain the astonishing highest accuracy, 81.29%. This result shows the potential of unsupervised-learned word vectors in boosting ZSL performance, while refraining from the cumbersome human work of annotating attributes. In contrast with GLaP #2, _t alone performs almost equally, in some cases even better, e.g. on AwA. Impact of the size of _t is also shown in Fig.<ref>. With a small number of generated instances, it performs quite well.Specially, we note that in the generation process of virtual instances, the mean vector of each class, denoted as _s, playsan important role. Therefore, we further conduct experiments on the strategy of learning jointly from _t and _s, denoted as GLaP #3. The experimental results shown in Tab.<ref> demonstrate that this strategy can basically achieve as good performance as GLaP #2. It is worth mentioning that with a little semantic information ofand only a few mean vectors in , the proposed method works well. We assume this will benefit online zero-shot recognition.§.§ Comparison with the State-of-the-artIn the third set of experiments, to better show the zero-shot performance of the proposed methods, we use the concatenation of fc8, goog and res_fc forand compare it with several state-of-the-art ZSL methods. They are SJE <cit.>, HAP<cit.>, ZSLwUA <cit.>, PST <cit.>, TMV<cit.>, AMP <cit.>, UDA <cit.> and UDICA <cit.>.For simple comparison, we use same settings and author-provided results. Results in Table.<ref> testify the effectiveness of generating virtual testing instances in our methods. They achieve the state-of-the-art results. On CUB and Dogs, our methods even exceeds SJE, whoseis constructed with specific word vectors learned from a specific text corpus. One thing should be noted that since our results are obtained based on the baseline method, we expect the result to be further improved when incorporated with other ZSL methods in this table, which is also a work in the future. § CONCLUSIONIn this letter, we proposed a generative latent prototype model for zero-shot learning. We assume observed category descriptions are developed from the category prototype, which is able to encode the true semantic relationship among different categories. Based on this assumption, virtual instances for unseen categories in the target domain can be produced, which give rise to the improved efficiency of our ZSL model. Experiments showed it achieved the state-of-the-art results on three benchmark datasets. IEEEtran | http://arxiv.org/abs/1705.09474v1 | {
"authors": [
"Yanan Li",
"Donghui Wang"
],
"categories": [
"cs.CV"
],
"primary_category": "cs.CV",
"published": "20170526082213",
"title": "Zero-Shot Learning with Generative Latent Prototype Model"
} |
Dynamic degree-corrected blockmodels for social networks:a nonparametric approachLinda S. L. Tan^* and Maria De Iorio^† [3mm] ^* Department of Statistics and Applied Probability, National University of Singapore ^† Department of Statistical Science, University College London A nonparametric approach to the modeling of social networks using degree-corrected stochastic blockmodels is proposed. The model for static network consists of a stochastic blockmodel using a probit regression formulation and popularity parameters are incorporated to account for degree heterogeneity. Dirichlet processes are used to detect community structure as well as induce clustering in the popularity parameters. This approach is flexible yet parsimonious as it allows the appropriate number of communities and popularity clusters to be determined automatically by the data. We further discuss some ways of extending the static model to dynamic networks. We consider a Bayesian approach and derive Gibbs samplers for posterior inference. The models are illustrated using several real-world benchmark social networks. § INTRODUCTIONSocial networks play a central role in the dissemination of information <cit.>, formation of alliances <cit.>, transmission of disease <cit.> and many other areas. It is thus important to understand the underlying structure of a social network and the behavioral patterns in the interactions. A common characteristic of social networks is that they often exhibit community structure, where certain groups of nodes (representing the social actors) are more densely connected within each group than across groups. The community structure may be present due to various factors such as similar interests, social stature or physical locations. Studying the nodal attributes associated with the communities can provide a greater understanding of the network topology, behavior patterns and network dependent processes such as epidemic spreading. However, identifying the community structure in a network can be challenging as the number of communities is typically unknown and the communities can vary in size and rate of interaction. Moreover, results can be distorted if the broad degree distributions often observed in real networks are not taken into account <cit.>. In this article, we propose a non-parametric approach to community detection in social networks by using independent Dirichlet processes <cit.> to capture the blockstructure in the social network and induce clustering in the activity level of nodes.The partitioning of nodes into structurally equivalent groups, such that nodes in the same group relate with other nodes in the exact same way, was first discussed by <cit.>, followed by <cit.>. Building upon the work of <cit.> and <cit.>, <cit.> generalized this deterministic concept and formulated stochastic blockmodels to allow for data variability. In a stochastic blockmodel, the nodes of a network are partitioned into groups and the distribution of ties between the nodes depends only on the group membership of the nodes and the probabilities of interactions between different groups. The stochastic blockmodel is generative and a wide variety of network structures, such as community, hierarchical or core-periphery, can be produced through different choices of the probability matrix. In a priori blockmodeling, exogenous actor attribute data are used to partition the nodes, while the discovery of blockstructures from relational data is referred to as a posteriori blockmodeling <cit.>. <cit.> studied a posteriori blockmodeling for undirected networks when there are only two groups and derived procedures for finding the blockstructure using both maximum likelihood and Bayesian estimation. <cit.> extend their approach to directed valued networks where the number of classes is fixed and address the nonidentifiability problem of the class labels. <cit.> consider a different clustering approach based on latent space models <cit.>, which posits that the probability of a tie is dependent on the positions of the actors in some unobserved space and decreases with distance. The stochastic blockmodel has been extended in many ways. To overcome the restriction that each actor can only belong to one group, <cit.> develop mixed membership stochastic blockmodels (MMSB), where each node is associated with a membership vector describing the probability of the node belonging to each of the groups. Each node can also assume different group membership when interacting with different nodes. <cit.> considers overlapping stochastic blockmodels, where each node can belong simultaneously to multiple groups with independent probabilities. The infinite relational model introduced by <cit.> allows the number of groups to be determined automatically by the data by drawing the membership vector from a Chinese restaurant process <cit.>. A brief review on the CRP is given in Section <ref>.<cit.> note that the stochastic blockmodel often yield poor fits to real-world networks whose degree distributions are much broader than that generated by the stochastic blockmodel. To account for heterogeneity in the degree of nodes, they propose degree-corrected stochastic blockmodels, which modify the probability of a tie between node i in group g_i and node j in group g_j from ω_g_ig_j to θ_i θ_j ω_g_ig_j, where ω_rs denotes the probability of a tie between group r and s while θ_i measures the activity level or “popularity" of node i. Estimates of the parameters are derived using maximum-likelihood and they demonstrate that the degree-corrected blockmodel leads to improved community detection. <cit.> consider a related “assortative MMSB with node popularities" model, that considers a logit link and extends the MMSB to incorporate node popularities. A stochastic variational inference algorithm <cit.> is developed for posterior inference. <cit.> consider degree-corrected stochastic blockmodels using a Bayesian approach and a logistic regression formulation. Posterior inference is obtained via data augmentation with latent Pólya-Gamma variables and a canonically mapped centroid estimator thataddresses label non-identifiability.In this article, we focus on degree-corrected stochastic blockmodels for community detection in undirected social networks using a non-parametric Bayesian approach. The static model is formulated using probit regression and Dirichlet processes are used to capture the communities in the network and induce clustering among the popularity parameters. This approach is highly flexible yet parsimonious as it does not require the number of communities to be fixed in advance and instead allows the appropriate number of communities and popularity clusters to be determined automatically by the data. Our model integrates the approach of <cit.> who uses the CRP to detect community structure and that of <cit.>, who use the DP to induce clustering among the “productivity" and “attractiveness" parameters of a variation of the p_1 model <cit.> and a social relations model <cit.>. While <cit.> implements Bayesian analysis using WinBUGS software <cit.> by considering a truncated DP <cit.>, we derive a Gibbs sampler for posterior inference. We also discuss several ways in which the static model can be extended to dynamic networks and illustrate the applicability of the proposed models using benchmark social networks.This article is organized as follows. We review the Dirichlet process briefly in Section <ref> and describe the proposed models in Section <ref>. We first present a model for static networks and then discuss extensions of this model to dynamic networks. In Section <ref>, we describe how posterior inference for the proposed model can be obtained using Gibbs samplers. In Section <ref>, we use the proposed models to analyze three real-world social networks. We conclude with a discussion of future research directions in Section <ref>. § THE DIRICHLET PROCESS The Dirichlet process <cit.> is widely used in Bayesian nonparametric models, particularly in DP mixture models as a prior over distributions. Let (Θ, ℬ) be a measurable space with G_0 a probability measure on the space and α a positive real number. A random probability measure G is distributed as a DP with base distribution G_0 and concentration parameter α, written G ∼DP(α, G_0), if (G(A_1), …, G(A_r)) ∼Dirichlet (α G_0(A_1), …, α G_0(A_r))for every finite measurable partition A_1, …, A_r, of Θ. The base distribution G_0 is the mean of the DP and α describes the concentration of mass around G_0. The larger α is, the more the DP will concentrate mass around G_0. Suppose the random variables {θ_i|i=1, …, n } are assigned the DP prior G. This is denoted asθ_i|G iid∼ G where G|α, G_0 ∼DP(α, G_0). Note that G is discrete with probability one even when G_0 is continuous. The stick-breaking construction of <cit.> shows that G= ∑_l=1^∞π_l δ_θ^*_l, where δ_θ is a point mass concentrated at θ and θ^*_l iid∼ G_0. The random weights are defined by π_l =V_l ∏_j=1^l-1 (1-V_j) where V_l iid∼Beta(1,α). The construction of the weights {π_l } can be interpreted as starting with a stick of unit length and recursively breaking off a proportion V_l of the remaining stick length. The stick-breaking construction shows clearly that G is a discrete probability distribution. This implies that {θ_i} generated from G will have non-negligible probability of having the same value. Thus, the DP will induce clustering among {θ_i} such that within each cluster, the θ_i'swill assume the same value.Another metaphor on the DP is provided by the Chinese restaurant process <cit.>, which likens θ_i+1 to the (i+1)^th customer entering a Chinese restaurant with infinitely many tables. The customer either selects an occupied table with probability proportional to the number of customers sitting there or a new table with probability proportional to α. The CRP illuminates the “rich gets richer" phenomenon of the DP where larger clusters grow faster. § NON-PARAMETRIC MODELS FOR SOCIAL NETWORKSLet N={1, …, n} be the set of n actors of interest and y=[y_ij] be a n × n adjacency matrix where y_ij is an indicator of a link from actor i to actor j. In this article, we focus on undirected networks without self-links and multiple links. Hence y is symmetric and the diagonal elements of y are zeros. When the network of interest is observed at multiple (discrete) time points, T, we let y_t=[y_t,ij] be the n × n adjacency matrix representing the state of the network at a time t for t=1, …, T.§.§ Static modelFirst we introduce a model for a static network y that aims to detect community structure while incorporating heterogeneity through node-specific popularity parameters.For 1 ≤ i < j ≤ n, we assume thaty_ij|p_ij∼Bernoulli (p_ij), and introduce latent variable ζ_ij | μ_ij∼ N (μ_ij, 1) whereμ_ij = θ_i + θ_j + ∑_k=1^K β_k^* 1{z_i= z_j } ,and y_ij|ζ_ij = 1 if ζ_ij > 0 and 0 if ζ_ij≤ 0. Here we are considering a probit link where Φ^-1(p_ij) = μ_ij and Φ(·) denotes the cumulative distribution function of the standard normal. The parameter θ_i represents the popularity or activity level of actor i, K ≤ n denotes the total number of groups or communities in the network and z_i ∈{1, …, K} represents the group membership of actor i. The coefficient β_k^* measures the rate of interaction in the kth community. Members within a community are assumed to interact with each other at a common rate. A high β_k^* indicates a tight or close-knit community where members interact at a high rate while a low β_k^* indicates a group with little interaction. The third term on the right-hand side of (<ref>) resembles a stochastic blockmodel where non-diagonal entries of the probability matrix are set to a common value (not necessarily zero). In (<ref>), the probability of interaction, p_ij, between actors i and j depends on their individual popularities as well as the interaction rate of their community if they belong to the same community. An interaction between actors from different communities is driven only by their popularities. Thus the presence of a link can be explained by homophily in terms of community membership or popularities and the popularity parameters {θ_i} and community assignments {z_i} are competing to explain the observed network. For model parsimony, a DP is used to induce clustering among the popularity parameters {θ_i}. We assume θ_i|G iid∼ G for i = 1, …, n,G ∼DP(α, G_0),where the base distribution G_0 is N(0,σ_θ^2) and α∼Gamma(a_α, b_α). Let θ^* = [θ_1^*, …, θ_L^*]^T denote the set of unique values among {θ_1, …, θ_n} and let c_i indicate the latent class associated with θ_i so that θ_i = θ_c_i^*. To detect the communities in the network, we consider another DP, H, which is independent of G. We introduce a β_i for each actor i where β_i = β_z_i^* and assume β_i|Hiid∼ H for i = 1, …, n,H∼DP(ν, H_0),where H_0 is N(0, σ_β^2) and ν∼Gamma(a_ν, b_ν). Let β^* = [β_1^*, …, β_K^*]^T be the set of unique values among {β_1, …, β_n}.In this non-parametric approach, the number of clusters L among {θ_i} and the number of communities K are not fixed in advance. Instead, they are random and to be inferred from the data. The prior distribution of L depends on the concentration parameter α, with a larger α implying a larger L a priori. To avoid overfitting and for greater interpretability, α will typically be small relative to n. Uncertainty about L can be expressed by placing a prior on α and we consider a Gamma prior here. The relation between K and ν is similar.Next, we propose some ways of extending the static model to model dynamic networks. Suppose we observe networks y_t = [y_t,ij] for t=1, …, T. For the dynamic models below, we assume that for t=1, …, T, 1 ≤ i < j ≤ n,y_t,ij|p_t,ij∼Bernoulli (p_t,ij). As before we consider the probit link function and introduce the latent variable ζ_t,ij| μ_t,ij∼ N (μ_t,ij, 1) for 1≤ i < j ≤ n, t=1, …, T such that y_t,ij|ζ_t,ij = 1 if ζ_t,ij > 0 and 0 if ζ_t,ij≤ 0. Thus, p_t,ij = Φ(μ_t,ij). §.§ Dynamic model 1Dynamic model I assumes that the community memberships remain unchanged over time but the popularities of the actors can vary with time. This assumption is appropriate for data where the communities arise due to factors that do not or are unlikely to vary drastically over time, for instance, gender, race, physical locations and job positions. In such cases, the changes in ties may be attributed to variations in the activity levels of individual nodes. For 1 ≤ i < j ≤ n and t=1, …, T, letμ_t,ij = θ_it + θ_jt + ∑_k=1^K β_k^* 1{z_i= z_j }.In resemblance of the static model, we assume that the {θ_it} are independent and induce clustering among them using a DP,θ_it|G iid∼ G for i = 1, …, n, t=1, …, T,G∼DP(G_0, α),where G_0 is N(0,σ_θ^2) and α∼Gamma(a_α, b_α). For this model, let θ^* = [θ_1^*, …, θ_L^*]^T denote the set of unique values among {θ_11, …, θ_nT} and c_it indicate the latent class associated with θ_it so that θ_it = θ_c_it^* for i=1, …, n, t=1, …, T. The {β_k^*} and {z_i} are modeled using a DP as described in (<ref>). §.§ Dynamic model IIDynamic model II extends the static model by allowing the tie between nodes i and j at time t to depend on the existence of the tie at the previous time point. It assumes that the popularities and community memberships of the actors remain unchanged over time. For 1 ≤ i < j ≤ n and t=1, …, T, letμ_t,ij =η y_t-1, ij1{t>1}+ θ_i + θ_j + ∑_k=1^K β_k^* 1{z_i= z_j },where η∼ N(0,σ_η^2). The coefficient η can be interpreted as a measure of thepersistence of ties in the network once they are formed. A positive η implies that a tie between two actors is more likely to be present at time t if a tie was present at time t-1 than if it were not, conditional on their popularities and community memberships. On the other hand, a negative η would imply that a tie is more likely to be present at time t if the tie was absent at the previous time point than if it were present. The parameters {θ_i} are modeled as in (<ref>) and {z_i} and {β_k^*} are modeled as in (<ref>). The popularities and communities inferred from this model smooths out the noise in the data and provide an overview of the behavior of actors over time.§ POSTERIOR INFERENCEWe use Gibbs samplers to derive posterior inference for the proposed models. To obtain the updates in the Gibbs sampler, we derive the posterior distribution of each variable conditional on the rest. Detailed derivations are given in the Appendix. Sampling from the DP is performed using the methods described in <cit.> while the concentration parameters α and ν are sampled using the method described in <cit.>. For ease in representation, we introduce the following notations. Let Z_ij be a binary vector of length K where the kth element is 1 if z_i=z_j=k, 0 otherwise, and Z = [Z_12, Z_13, …, Z_(n-1),n]^T be a n(n-1)/2 × K matrix. We also define ζ_ij = ζ_ji and ζ_t,ij = ζ_t,ji for 1 ≤ j < i ≤ n and t=1, …, T. Let 𝒮_m = {(i,j)|i<j, c_i=c_j=m }, 𝒮_t,m = {(i,j)|i<j, c_it=c_jt=m }, 𝒫_m = {(i,j)|j ≠ i, c_i=m, c_j ≠ m }, and 𝒫_t,m = {(i,j)|j ≠ i, c_it=m, c_jt≠ m }. We use (x| μ, σ,a,b) denote the truncated normal distribution with density 1/σϕ(x-μ/σ)/(Φ(b-μ/σ) - Φ(a-μ/σ)), where ϕ(·) denotes the density of the standard normal. In the algorithms presented below, we use K and L to represent the current number of communities and popularity clusters respectively at each iteration and β^* = [β^*_1, …, β_K^*] and θ^*= [θ^*_1, …, θ_L^*] to represent the states currently associated with the clusters. For the static model, the joint distribution p(y,ζ,z, β^*, ν, c, θ^*, α) is given byp(c|α) p(α) p(z|ν) p(ν) p(θ^*) p(β^*) ∏_i<j p(y_ij|ζ_ij) p(ζ_ij|c_i,c_j,θ^*,z_i,z_j,β^*),where ζ = {ζ_11, …, ζ_n-1,n}, c= {c_1, …, c_n} and z={z_1, …, z_n}.. Note that p(c|α) and p(z|ν) are defined as <cit.>: (z_i = k| z_-i, ν) = m_-i,k/n-1+ν for k ∈ z_-i,(z_i ≠ z_jfor allj ≠ i|z_-i, ν) = ν/n-1+ν,(c_i = ℓ| c_-i, α) = n_-i,ℓ/n-1+α for ℓ∈ c_-i,(c_i ≠ c_jfor allj ≠ i|c_-i, α) = α/n-1+α.where z_-i = z\ z_i, c_-i = c\ c_i, m_-i,k = ∑_z_j ∈ z_-i1{z_j = k} and n_-i,ℓ = ∑_c_j ∈ c_-i1{c_j = ℓ}. The Gibbs sampler for the static model is outlined in Algorithm 1. In step 2, suppose that the number of distinct values in z_-i is K'. In the update, z_i can either assume one of these K' distinct values or a new value not assumed by any z_j ∈ z_-i. (<ref>) describe these probabilities and a is a constant that ensures these K'+1 probabilities sum to one. The same idea applies to (<ref>)–(<ref>), [htb!] 0.95 Initialize z, c, θ^* and β^* and cycle through the following updates: * For 1 ≤ i <j ≤ n, draw ζ_ij from (ζ_ij| μ_ij, 1, 0, ∞) if y_ij = 1 and (ζ_ij| μ_ij, 1, -∞, 0) if y_ij = 0, where μ_ij = θ^*_c_i +θ^*_c_j + Z_ij^T β^*.* For i=1, …, n: If m_-i, z_i = 0, remove β_z_i^* from β^*. Draw z_i according to (<ref>):( z_i =k |) = am_-i,kexp{β_k^* ∑_j≠ i:z_j=k(ζ_ij - θ^*_c_i -θ^*_c_j) -m_-i,k/2β_k^*^2} for k ∈ z_-i and (z_i ≠ z_jfor allj ≠ i |) = aν,where a is a normalizing constant that ensures the above probabilities sum to one. If the value of z_i is not in z_-i, draw β_z_i^* ∼(0, σ_β^2) and add it to β^*.* Draw β^* ∼(P^-1∑_i<j(ζ_ij - θ^*_c_i -θ^*_c_j ) Z_ij, P^-1), where P = 1/σ_β^2 I_K + Z^TZ. * Draw γ_1 ∼Beta(α+1, n). Then draw α from the mixture: π_αGamma(a_α+L, b_α-logγ_1) +(1-π_α) Gamma(a_α+L-1, b_α-logγ_1), where π_α/1-π_α = a_α+L-1/n(b_α-logγ_1).* For i=1, …, n: If n_-i, c_i = 0, remove θ_c_i^* from θ^*. Draw c_i according to (<ref>): ( c_i =ℓ |) = bn_-i,ℓexp{θ^*_ℓ∑_j ≠ i(ζ_ij-θ^*_c_j - Z_ij^Tβ^* )-n-1/2θ^*_ℓ^2} for ℓ∈ c_-i and (c_i ≠ c_jfor allj ≠ i |) = bασ_c/σ_θexp{μ_c_i^2/2σ_c^2},where σ_c^2 = ( n-1 +1/σ_θ^2)^-1, μ_c_i = σ_c^2 ∑_j ≠ i(ζ_ij-θ^*_c_j - Z_ij^Tβ^*), and a is a normalizing constants that ensure the above probabilities sum to one. If the value of c_i is not in c_-i, draw θ_c_i^* ∼(μ_c_i, σ_c^2) and add it to θ^*. * For m=1, …, L, draw θ^*_m ∼ N(μ_m,σ_m^2), where σ_m^2 = (1/σ_θ^2 + ∑_𝒮_m 4 +∑_𝒫_m 1)^-1 and μ_m = σ_m^2 [2 ∑_𝒮_m (ζ_ij-Z_ij^Tβ^* )+∑_𝒫_m (ζ_ij-θ^*_c_j -Z_ij^Tβ^* )]. * Draw γ_2 ∼Beta(ν+1, n). Then draw ν from the mixture: π_νGamma(a_ν+K, b_ν-logγ_2) +(1-π_ν) Gamma(a_ν+K-1, b_ν-logγ_2), where π_ν/1-π_ν = a_ν+K-1/n(b_ν-logη). Gibbs sampler for static model. For dynamic model I, the joint distribution p(y, ζ,z, β^*, ν, c, θ^*, α) is given byp(c|α)p(α) p(z|ν)p(ν)p(θ^*)p(β^*) ∏_t=1^T ∏_i<j p(y_t,ij|ζ_t,ij) p(ζ_t,ij|c_it,c_jt,θ^*,z_i,z_j,β^*),whereζ = {ζ_1,11, …, ζ_T, n-1,n}, c= {c_11, …, c_nT} and z={z_1, …, z_n}. Note that p(c|α) is defined as(c_it= ℓ| c_-it, α) = n_-it,ℓ/nT-1+αfor ℓ∈ c_-it, (c_it is not equal to any value inc_-it |c_-it, α) = α/nT-1+α,where c_-it = c\ c_it and n_-it,ℓ be the number of indicators in c_-it that are equal to ℓ. The definition of p(z|ν) remains as in (<ref>). The Gibbs sampler for dynamic model I is outlined in Algorithm 2.[htb!] 0.96 Initialize z, c, θ^* and β^* and cycle through the following updates: * For t=1, …, T, 1 ≤ i <j ≤ n, draw ζ_t,ij from (ζ_t,ij| μ_t,ij, 1, 0, ∞) if y_t,ij = 1 and (ζ_t,ij| μ_t,ij,1, -∞, 0) if y_t,ij = 0, where μ_t,ij = θ^*_c_it +θ^*_c_jt +Z_ij^T β^*. * For i=1, …, n: If m_-i, z_i = 0, remove β_z_i^* from β^*. Draw z_i according to (<ref>):( z_i =k |)= am_-i,kexp{β_k^* ∑_j ≠ i: z_j = k∑_t(ζ_t,ij - θ^*_c_it -θ^*_c_jt )- Tm_-i,k2β_k^*^2 } for k ∈ z_-i and (z_i ≠ z_jfor allj ≠ i|) = aν,where a is a normalizing constant that ensures the above probabilities sum to one. If the value of z_i is not in z_-i, draw β_z_i^* ∼(0, σ_β^2) and add it to β^*.* Draw β^*∼(P^-1∑_i<j Z_ij∑_t(ζ_t,ij - θ^*_c_it -θ^*_c_jt ), P^-1), where P = 1/σ_β^2 I_K + Z^TZ.* As in Step 4 of Algorithm 1. * For t=1, …, T, i=1, …, n: If n_-it, c_it = 0, remove θ_c_it^* from θ^*. Draw c_it according to (<ref>):(c_it=ℓ|) = bn_-it,ℓexp{θ^*_ℓ∑_j ≠ i(ζ_t,ij-θ^*_c_jt -Z_ij^Tβ^*) - n-12θ^*_ℓ^2 } for ℓ∈ c_-i and (c_it≠any value in c_it|) =bασ_c/σ_θexp{μ_c_it^2/2σ_c^2},where σ_c^2 = ( n-1 +1/σ_θ^2)^-1 and μ_c_it = σ_c^2 ∑_j ≠ i(ζ_t,ij-θ^*_c_jt -Z_ij^Tβ^*), and a is a normalizing constants that ensure the above probabilities sum to one. If the value of c_it is not in c_-it, draw θ_c_it^* ∼(μ_c_it, σ_c^2) and add it to θ^*. * For m=1, …, L, draw θ^*_m ∼ N(μ_m,σ_m^2), where σ_m^2 = (1/σ_θ^2 + ∑_t∑_𝒮_t,m4 +∑_t∑_𝒫_t,m 1 )^-1, μ_m = σ_m^2 [2 ∑_t ∑_𝒮_t,m (ζ_t,ij-Z_ij^Tβ^* )+ ∑_t∑_𝒫_t,m (ζ_t,ij-θ^*_c_jt -Z_ij^Tβ^* )]. * As in Step 7 of Algorithm 1. Gibbs sampler for dynamic model I.For dynamic model II, the joint distribution is given byp(y,ζ,z, β^*, ν, c, θ^*, α) =p(c|α)p(α)p(z|ν)p(ν) p(θ^*)p(β^*) p(η) ×∏_i<j{[∏_t≥1p(y_t,ij|ζ_t,ij) ] p(ζ_1,ij|c_i,c_j,θ^*,z_i,z_j,β^*) [ ∏_t≥2 p(ζ_t,ij|c_i,c_j,θ^*,z_i,z_j,β^*, η, y_t-1, ij) ]},where ζ = {ζ_1,11, …, ζ_T, n-1,n}, c= {c_1, …, c_n} and z={z_1, …, z_n}. Note that p(c|α) and p(z|ν) are as defined in (<ref>). The Gibbs sampler for dynamic model II is outlined in Algorithm 3. [htb!] 0.95 Initialize z, c, θ^* and β^* and cycle through the following updates: * For t=1, …, T, 1 ≤ i <j ≤ n, draw ζ_t,ij from (ζ_t,ij| μ_t,ij, 1, 0, ∞) if y_t,ij = 1 and (ζ_t,ij| μ_t,ij,1, -∞, 0) if y_t,ij = 0, where μ_t,ij =η y_t-1,ij1{t > 1} + θ^*_c_i +θ^*_c_j + Z_ij^Tβ^*. * For i=1, …, n: If m_-i, z_i = 0, remove β_z_i^* from β^*. Draw z_i according to (<ref>):( z_i =k |)= am_-i,kexp{β_k^* ∑_t ∑_j≠ i: z_j=k(ζ̃_t,ij - θ^*_c_i -θ^*_c_j), -Tm_-i,k/2β_k^*^2 } for k ∈ z_-i and (z_i ≠ z_jfor allj ≠ i|) = aν,where b is a normalizing constant that ensures the above probabilities sum to 1. If the value of z_i is not in z_-i, draw β_z_i^* ∼(0, σ_β^2) and add it to β^*.* Draw β^*∼(P^-1∑_i<j Z_ij∑_t(ζ̃_t,ij - θ^*_c_i -θ^*_c_j ), P^-1), where P = 1/σ_β^2 I_K + TZ^TZ.* As in Step 4 of Algorithm 1.* For i=1, …, n: If n_-i, c_i = 0, remove θ_c_i^* from θ^*. Draw c_i according to (<ref>): (c_i=ℓ|) = bn_-i,ℓexp{θ^*_ℓ∑_t ∑_j ≠ i( ζ̃_t,ij-θ^*_c_j -Z_ij^Tβ^*)-T(n-1)2θ^*_ℓ^2}.for ℓ∈ c_-i and (c_i ≠ c_jfor allj ≠ i|) = bασ_c/σ_θexp{μ_c_i^2/2σ_c^2},where σ_c^2 = ( T(n-1) +1/σ_θ^2)^-1, μ_c_i = σ_c^2 ∑_t ∑_j ≠ i(ζ̃_t,ij-θ^*_c_j -Z_ij^Tβ^*) and b is a normalizing constant that ensures the above probabilities sum to 1. If the value of c_i is not in c_-i, draw θ_c_i^* ∼(μ_c_i, σ_c^2) and add it to θ^*. * For m=1, …, L, draw θ^*_m ∼(μ_m,σ_m^2), where σ_m^2 = (1/σ_θ^2 +∑_𝒮_m 4T + ∑_𝒫_m T)^-1 and μ_m = σ_m^2 (2 ∑_𝒮_m ( ζ̃_t,ij-Z_ij^Tβ^* )+∑_𝒫_m ( ζ̃_t,ij - θ^*_c_j -Z_ij^Tβ^* )). * As in Step 7 of Algorithm 1 * Draw η∼(μ_η, σ_η,1^2), where σ_η,1^2 = ( 1/σ_η^2 + ∑_t ≥ 2∑_i<j y_t-1,ij^2 )^-1 and μ_η =σ_η,1^2 ∑_t≥ 2∑_i<j y_t-1,ij (ζ_t,ij - θ^*_c_i -θ^*_c_j - Z_ij^Tβ^* ). Gibbs sampler for dynamic model II.We code Algorithms 1–3 in Julia and all experiments are run on an Intel Core i5 CPU @ 3.30GHz, 8.0GB RAM. We note that it is also possible to use software such as OpenBUGS to obtain posterior inference for the proposed models by considering a truncated DP approach <cit.>. However, we observe that the runtime in OpenBUGS is significantly longer than Julia especially as the number of nodes increases. For the examples, we initialize multiple MCMC chains from random starting points and make use of diagnostic plots to check for convergence. §.§ Cluster AnalysisGiven the sample of clusterings from the MCMC output, we can assess clustering by computing the posterior similarity matrix S, which is a n × n symmetric matrix whose (i,j) entry contains the posterior probability that actors i and j belong to the same cluster. This probability is estimated by the proportion of times actors i and j cluster together and it is not affected by the problem of “label-switching" <cit.> or the number of clusters varying across iterations. We can also compute a single (hard) clustering estimate, for instance, by using the maximum a posteriori (MAP) approach or methods based on the posterior similarity matrix or Rand index <cit.>. Here we consider the Binder's loss function <cit.>, which is defined as the total number of disagreements between the estimated and true clustering among all pairs of actors. The R package mcclust provides a function, minbinder, that can be used to find the clustering c^* = [c_1^*, …, c_n^*] that minimizes the posterior expectation of this loss. The posterior expected loss can be written as ∑_i<j | 1_{c_i^* = c_j^*} - S_ij|,where the sum is taken over all possible pairs of actors and S_ij is the (i,j) entry of the posterior similarity matrix.§ APPLICATIONS We investigate the performance of the static and dynamic models on three well-known social network datasets and compare the fitted models with results obtained previously by other authors. The first is a karate club network studied by <cit.> from 1970 to 1972, the second is a dolphins social network <cit.> and the third is a dataset collected by <cit.> at an African clothing factory in Zambia. These datasets are available at the UCI Network Data Repository (<https://networkdata.ics.uci.edu/>).§.§ Karate club networkThis dataset contains 78 undirected friendship links among 24 members, which are constructed based on interactions outside of club activities. Due to disputes over the price of karate lessons, the club was divided informally into two factions, led by the karate instructor “Mr Hi" (actor 1) and the president “John A." (actor 34) respectively (these names are pseudonyms). During the study, the club eventually split into two separate clubs when Mr Hi was fired for trying to raise lesson fees unilaterally and his supporters left to join the new club formed by Mr Hi. All members joined clubs following their own factions except actor 9, who crossed factions to join Mr Hi's club because he was only three weeks away from a test for black belt at the time of the split and he could not bear to give up his rank. We fit the static model to this dataset using Algorithm 1. Three chains were run in parallel, each consisting of 40,000 iterations with the first 30,000 discarded as burn-in. The total runtime is 172 seconds. A thinning factor of 5 was applied and the remaining 6000 samples were used for posterior inference. We set a_ν=b_ν=a_α=b_α =5 and σ_θ^2 = σ_β^2 = 1. Figure <ref> shows the posterior distributions of the number of communities (K), the number of popularity clusters (L), and the DP concentration parameters α and ν.The mode of K is 3 and that of L is 4. The fitted model is quite parsimonious with a relatively small number of clusters for both popularity and community. Figure <ref> shows the posterior similarity matrices for the clusterings according to community (left) and popularity (right). Each element in the matrix represents the proportion of times that the actors concerned belong to the same cluster. Figure <ref> plots the posterior mean of θ_i against the degree for each actor. While the factional leaders, Mr Hi (actor 1) and John A. (actor 34), and a few other actors {2, 3, 33} have high popularity, the rest of the members have much lower activity levels generally.Using the similarity matrices, we compute hard clustering estimates using Binder's loss function. There are three communities, one of which contains a single node {3} and three popularity clusters. Figure <ref> shows plots of the karate club network where nodes of the same color belong to the same cluster and singletons are not colored. We run Algorithm 1 again, fixing z and c to obtain estimates of β^* and θ^* for these clusterings. The conditional posterior mean and standard deviation (in brackets) of these parameters for each cluster are shown in the legend of Figure <ref>.There are three clusters for the popularity parameters {θ_i}, the first contains {Mr Hi, John A., 3}, the second contains {2, 33} and the third contains all remaining members. For the communities, we note that the β_k^* for groups 1 and 3 are strongly positive, indicating a high interaction rate within each group. The posterior mean and standard deviation ofβ_k^* for the singletons necessarily equal that of the prior distribution. Note that group 3 corresponds exactly to the faction led by John A. as concluded in <cit.> while group 1 together with the singleton {3} correspond to the faction led by Mr Hi. From the posterior probability matrix, actor 3 has a posterior probability of about 0.4 of being clustered together with members in group 1 (Mr Hi's faction) and a probability of about 0.05 of being clustered together with members in group 3 (John A.'s faction). It is thus reasonable to combine actor 3 with group 1. Hence, our proposed static model is able to identify members in the factions accurately. Incidentally, if we drop {θ_i} from the static model and consider just the blockmodel, we obtain five clusters, four of which are singletons: {1}, {3}, {33}, {34} and the fifth cluster contains all other members. This result is similar to the phenomenon discussed in <cit.>, who note that the non-degree-corrected blockmodel with K=2 splits the network into high-degree and low-degree nodes instead of by factions, while the degree-corrected version splits it according to factions albeit with one misclassification. In addition, <cit.> observe that the non-degree-corrected blockmodel with K=4 splits the network according to factions correctly after the merging of sub-communities. These observations highlight the importance of accounting for degree variation in blockmodels as well as the difficulties in determining an appropriate number of clusters. Our static model tries to address these issues using a non-parametric approach via the automatic clustering structures induced by the DP. We observed that the clusters identified by the static model can be sensitive to the DP concentration parameters in some cases. For example, if we adopt a more conservative prior, say by setting a_ν=b_ν=a_α=b_α=10, then we obtain three communities, the first corresponds to the faction led by John A., the second contains {5, 6, 7, 11, 17} and the third contains all remaining members. Here, the second cluster emerges as one with a higher interaction rate than the third. However, merging the second and third clusters still yields Mr Hi's faction. While the clustering estimates return hard partitions of the network which are easy to interpret, the posterior similarity matrices reveal finer details regarding the degree of affiliation of actors towards the clusters that they are assigned to in the hard split. For the posterior similarity matrix for popularities,there are two main blocks but the partitioning among actors {1, 34, 3, 33, 2} is not so straightforward. For the posterior similarity matrix for communities, actor 10 is assigned to the cluster led by John A., but he has a somewhat lower posterior probability (∼ 0.5) of being together with the other members in this cluster than the rest, and also has some posterior probability (∼ 0.2) of being in the same cluster as members in Mr Hi's faction. §.§ Dolphins social network<cit.> constructed an undirected social network describing the associations among a community of 62 bottlenose dolphins living off Doubtful Sound, New Zealand after observing them for seven years from 1994–2001. This dataset has been widely studied in community detection, see for instance, <cit.> and <cit.>. In this network, the nodes represent dolphins and the ties represent higher than expected frequency of being sighted together. Of the 62 dolphins, 33 are males, 25 are females and the gender of the remaining 4 are unknown.We apply Algorithm 1 to this network, using 15,000 iterations with a burn-in of 5000 iterations and a thinning factor of 5 in each chain. Three chains were run in parallel and the total runtime is 250 seconds. We set a_ν = b_ν = a_α = b_α =10 and σ_θ^2 = σ_β^2 = 1. The marginal posterior distributions of K, L, ν and α are shown in Figure <ref>. The posterior of K is concentrated on larger values as compared to L and K has a mode of 7 while the mode of L is 2.The posterior similarity matrices in Figure <ref> show the community and popularity clustering structure in this network. Around five communities can be seen in the matrix on the left while the right matrix shows faint outlines of two clusters. Next we use Binder's loss to obtain clustering estimates for the community structure and popularity clusterings based on the MCMC samples. This yields 16 communities and a single popularity cluster. Of the 16 communities, 9 are singletons so there are essentially only 7 communities. We run Algorithm 1 again, fixing c and z to obtain estimates of β^* and θ^* for these clusterings. The estimate of θ^* is -0.92 ± 0.03. Figure <ref> shows the observed dolphins social network where the nodes are labeled with the names of the dolphins, and males, females and dolphins of unknown gender are represented using squares, circles and triangles respectively. Nodes of the same color belong to the same community while the singletons are not colored. From Figure <ref>, most of the singletons can be regarded as peripherals (e.g. Zig, TR82, Quasi, MN23); they have few links and lie at the margins of the network. While some of them can clearly be pushed into certain clusters, others such as SN89 lie at the edge of different groups. The estimate of β_k^* is indicative of the rate of interaction for each group k and this is shown in the legend along with the standard deviation in brackets. Groups 1–3 and 6–7 represent close-knit communities while groups 4–5 have low within-group interaction rates. Previously, <cit.> studied the community structure of this dolphins network by using a clustering algorithm which is based on removing links with high “betweeness" measures to extract the groupings <cit.>. They also investigated the role that gender and age homophily played in the formation of communities. They concluded that there are 2 main communities and 4 sub-communities; the first sub-community matches group 1 exactly, the second matches group 2 together with the singletons (Zig, TR82, Quasi, MN23), the third matches groups 7 and 4 combined and the fourth matches groups 3 and 6 combined plus the singletons TR120, TR88, TSN83, Zipfel and SN89. We note that the posterior similarity matrix does suggests some of these combinations. Thus, the communities detected by Algorithm 1 agree largely with the results of <cit.> and also that of <cit.>. In addition, Figure <ref> also provides some evidence of assortative mixing by sex. For example, group 6 consists almost entirely of females while groups 2 and 7 are composed of mainly males.§.§ Kapferer's tailor shop network<cit.> collected data on the interactions among 39 workers in a tailor shop in Zambia, Southern Africa, from June 1965 to February 1966, and he examined how these social networks relate to major events taking place in the factory. The workers' duties can be classified into eight categories: head tailor (worker number 19), cutter (16), line 1 tailor (1–3, 5–7, 9, 11–14, 21, 24), button machiner (25–26), line 3 tailor (8, 15, 20, 22–23, 27–28), ironer (29, 33, 39), cotton boy (30–32, 34–38) and line 2 tailor (4, 10, 17–18). These positions require different levels of skills and some like the head tailor, cutter, line 1 tailors and button machiners were perceived as having more prestige. Here we focus on the symmetric “sociational" networks (based on convivial interactions) recorded at two time points, the first was before an aborted strike and the second was after a successful strike for higher wages. The network at the second time point (223 edges) is much denser than the first (158 edges) as the workers strive to be more united (thereby expanding their social relations) in their efforts to change the wage system. This dataset has been widely studied, for instance, by <cit.> and <cit.> using block structures and <cit.> using Bayesian exponential random graph models. §.§.§ Dynamic model IFirst, we fit dynamic model I to the data using Algorithm 2. Dynamic model I assumes that the communities remain constant over time and that the emergence or dissolution of ties are due to changes in the activity level of individual actors. The hyperparameters are set as a_ν=b_ν=a_α=b_α=10 and σ_θ^2=σ_β^2=1. We use three parallel chains, each with 15,000 iterations and the first 5000 iterations are discarded as burn-in. The total runtime is 139 seconds. A thinning factor of 5 was applied and posterior inferences are based on the remaining 6000 iterations. The posterior distributions of K, L, ν and α are shown in Figure <ref>. The mode of K is 6 and the mode of L is 4. Next we compute the posterior similarity matrices and use Binder's function to obtain hard clustering estimates. This yields nine communities and three popularity clusters. Of the nine communities, five are singletons so there are essentially only four communities. We run Algorithm 2 again, fixing z and c to obtain estimates of β^* and θ^* for these clusterings. The results are shown in Figure <ref>, and the mean and standard deviation (in brackets) of β^* and θ^* are reported for each group.The first row shows the four communities which are constant across the two time points (the singletons (19–21, 25–26) are not colored). The shapes of the nodes represent the positions of the workers as explained in the legend. The plots indicate a high degree of job homophily in the communities even though these social networks are constructed based on casual interactions<cit.>. In particular, groups 1 and 2 consists of workers with jobs perceived to be of higher prestige: cutter, line 1 and line 2 tailors, group 3 consists of line 3 tailors and group 9 consists of all the ironers and cotton boys. The estimates of β_k^* are strongly positive, indicating a high interaction rate within each group.There are three popularity clusters with increasing means, -1.41 (group 1), -0.46 (group 2) and 0.57 (group 3). Thus, we can consider the three clusters as representing “low", “average" and “high" popularity. Actors 19 (head tailor) and 16 (cutter) are the only two actors with high popularity at t=1, and they maintained high popularity at t=2. This is not surprising since they are regarded by <cit.> to be in “supervisory" positions and play critical roles in the operation of the factory. From the barplot (left) in Figure <ref>, the number of workers with low popularity decreased from t=1 to t=2 while the number with average or high popularity increased. This reflects the efforts of the workers in expanding social ties after the first unsuccessful strike.Examining the results more closely using the barplot (right) in Figure <ref>, the proportion of workers with low and average popularity actually remained unchanged over the two time points for the ironers, cotton boys and line 2 tailors (positions with lower prestige). The changes in popularity arise mainly from line 1 tailors, button machiners and line 3 tailors. In particular, two line 1 tailors, {21, 24} and a button machiner {25}moved from average to high popularity. These observations are consistent with the analysis of <cit.>, who noted that line 1 tailors made a strong attempt to expand their links after the first unsuccessful strike as they stand to benefit the most from the change in wage system. <cit.> also noted that the button machiner, Meshak (actor 25) played a crucial role in the unfolding events at the factory and for the latter part was regarded as a supervisor by the factory owner. §.§.§ Dynamic model IINext, we fit dynamic model II to the data using Algorithm 3. In this model, the parameter η provides an indication of the persistence of ties formed. The probability that a tie is formed at any time point depends on whether a tie exists at the previous time point as well as the community membership of the nodes and their popularities. Using 15,000 iterations, discarding the first 5000 as burn-in and applying a thinning factor of 5 for each of three independent chains, the total runtime is 106 seconds. Figure <ref> shows the posterior distributions of K, ν, L, α and η based on 6000 MCMC samples. The modes of K and L are both 6. The posterior mean of η is 0.58 and its posterior mass is concentrated on positive values. This indicates that a tie is likely to persist at the second time point given that it existed at the first time point. Figure <ref> shows the posterior similarity matrices. We note that the block structures are not clear-cut.Figure <ref> shows the hard clusterings computed using Binder's function and the estimates of β^* and θ^* for these clusterings. The communities detected are largely similar to that of Dynamic model I except for changes to the assignment of individuals {14, 16, 19, 21}. A new “community" consisting of {19,21} appeared. However, the β_k^* estimate for this group is 0.54 with a large standard deviation of 0.8. Thus, this is not truly a “community" in the sense that there is a high interaction rate between the actors. The number of popularity clusters increased from three in dynamic model I to six in model II. In model II, the popularity of an actor summarizes his activity level across all time points. Figure <ref> shows how the mean of θ_i varies with the degree of an actor at each time point. The head tailor and the cutter have significantly higher popularity than the other workers, followed by actor 24 (Ibrahim) and actors in popularity group 2. We note that group 2 includes several individuals who play significant roles in the factory's social relationships <cit.>. These include Lyashi (11), who tried to win followers in support of his view of the factory structure, Hastings (13), who took on many supervisory duties of the cutter at time 2, Meshak (25), who was regarded as a leader by the factory owner, and Mubanga (34), an influential figure among unskilled workers.§ CONCLUSION AND FUTURE WORK We present a non-parametric Bayesian approach for detecting communities in social networks, using degree-corrected stochastic blockmodels. In the proposed static model, the number of communities and popularity clusters does not have to be fixed in advance and is inferred from the data automatically through the use of the DP. For the karate club network and the dolphins social network, we find that the static model returns sensible results although there is some sensitivity to the DP concentration parameters. The inferred popularity clusters also summarizes the popularities of the actors and helps in the identification of key players in the network. We discuss two extensions of thestatic model to dynamic networks. Dynamic model I enables the study of the change in activity level of actors over the entire duration while dynamic model II provides a measure of the persistence of links formed in the network. While the Gibbs samplers are feasible for small networks, they do not scale well to large networks and more efficient methods of estimation, such as variational approximation methods, can be developed.§ ACKNOWLEDGMENTSLinda Tan is supported by the National University of Singapore Overseas Postdoctoral Fellowship. chicago Let 𝒫 = {(i,j)|1 ≤ i<j ≤ n } and 𝒫_s = {(i,j) ∈𝒫|i=sorj=s}. § UPDATES OF STATIC MODEL* For i<j,p(ζ_ij|)∝ p(y_ij| ζ_ij) p(ζ_ij| c_i, c_j, θ^*, z_i, z_j, β^*)∝1{ζ_ij > 0}^y_ij1{ζ_ij≤ 0}^1-y_ijexp{-1/2[ζ_ij^2 -2ζ_ij( θ^*_c_i +θ^*_c_j + Z_ij^Tβ^*) ] }.* For s=1, …, n, p(z_s| ) ∝ p(z_s|z_-s, ν) ∏_(i,j) ∈𝒫_s p(ζ_ij| θ^*_c_i, θ_c_j^*, β_z_s^*).∴( z_s =k |)= a'm_-s,kexp{-1/2∑_(i,j) ∈𝒫_s(ζ_ij - θ^*_c_i -θ^*_c_j - β_k^* 1{z_i=z_j=k})^2} for k ∈ z_-s, (z_s ≠ z_jfor allj ≠ s|) = a'ν∫exp{-1/2∑_(i,j) ∈𝒫_s(ζ_ij - θ^*_c_i -θ^*_c_j - β_k^* 1{z_i=z_j=k} )^2}1/√(2π)σ_βexp{-β_k^*^2/2σ_β^2} dβ_k^*= a'νexp{-1/2∑_(i,j) ∈𝒫_s(ζ_ij - θ^*_c_i -θ^*_c_j )^2 },where a' is a normalizing constant that ensures the probabilities sum to one. Hence we can simplify the expressions to that in (<ref>).* [t] p(β^*| )∝exp{-1/2∑_i<j(ζ_ij - θ^*_c_i -θ^*_c_j - Z_ij^Tβ^* )^2}exp(-β^*^Tβ^*/2σ_β^2) ∝exp{-1/2(β^*^T (Z^TZ + 1/σ_β^2) β^* -2β^*^T ∑_i<j Z_ij(ζ_ij - θ^*_c_i -θ^*_c_j) }.Note that Z^TZ= ∑_i<j Z_ijZ_ij^T is a K × K diagonal matrix where the kth diagonal element counts the number of pairs of (z_i, z_j) that assume a common value k.* For i=1, …,n,(c_i|)∝exp{-1/2∑_i<j (ζ_ij - θ^*_c_i -θ^*_c_j -Z_ij^Tβ^* )^2} p(c|α)∝exp{θ^*_c_i∑_j:j ≠ i(ζ_ij-θ^*_c_j -Z_ij^Tβ^*)-n-1/2θ^*_c_i^2 } p(c|α). ∴ P(c_i ≠ c_jfor allj ≠ i|) ∝α∫exp{θ^*_c_i∑_j:j ≠ i(ζ_ij-θ^*_c_j -Z_ij^Tβ^*) -n-1/2θ^*_c_i^2 }1/√(2π)σ_θexp{-θ^*_c_i^2/2σ_θ^2}d θ^*_c_i = α/σ_θ√(2π)∫exp{θ^*_c_i∑_j:j ≠ i(ζ_ij-θ^*_c_j -Z_ij^Tβ^*)-1/2(n-1 + 1/σ_θ^2)θ^*_c_i^2 }d θ^*_c_i = ασ_c/σ_θexp{μ_c_i^2/2σ_c^2}. * For m=1, …, L, p( θ^*_m|)∝exp{-1/2∑_i<j (ζ_ij - θ^*_c_i -θ^*_c_j -Z_ij^Tβ^* )^2}exp{-θ^*_m^2/2σ_θ^2}∝exp{θ^*_m (2 ∑_𝒮_m (ζ_ij-Z_ij^Tβ^* )+∑_𝒫_m (ζ_ij-θ^*_c_j -Z_ij^Tβ^* )) -θ^*_m^2/2(1/σ_θ^2 +∑_𝒮_m4 +∑_𝒫_m 1 )}.§ UPDATES OF DYNAMIC MODEL I* For t=1, …, T, i<j,p(ζ_t,ij |) ∝1{ζ_t,ij > 0}^y_t,ij1{ζ_t,ij≤ 0}^1-y_t,ijexp{-1/2(ζ_t,ij^2 -2ζ_t,ij( θ^*_c_it +θ^*_c_jt + Z_ij^Tβ^*) ) }.* For s=1, …, n, p(z_s|) ∝ p(z_s|z_-s, ν) ∏_t ∏_(i,j) ∈𝒫_s p(ζ_t,ij| c_it, c_jt, θ^*, β_k^*).∴( z_s =k |)= a'm_-s,kexp{-1/2∑_t ∑_(i,j) ∈𝒫_s(ζ_t,ij - θ^*_c_it -θ^*_c_jt - β_k^* 1{z_i=z_j=k})^2}for k ∈ z_-s and(z_s ≠ z_jfor allj ≠ s|) = a'νexp{-1/2∑_t ∑_(i,j) ∈𝒫_s(ζ_t,ij - θ^*_c_it -θ^*_c_jt )^2}where a' is a normalizing constant to ensure probabilities sum to one. Hence we can simplify the expressions to (<ref>). * [t] p(β^*| )∝exp{-1/2∑_t ∑_i<j(ζ_t,ij - θ^*_c_it -θ^*_c_jt - Z_ij^Tβ^* )^2}exp(-β^*^Tβ^*/2σ_β^2) ∝exp{-1/2(β^*^T T Z^TZ β^* -2β^*^T∑_i<j Z_ij∑_t (ζ_t,ij - θ^*_c_it -θ^*_c_jt) + β^*^Tβ^*/σ_β^2)}. * For i=1, …,n, t=1, …, T,(c_it|)∝exp{-1/2∑_i<j (ζ_t,ij - θ^*_c_it -θ^*_c_jt -Z_ij^Tβ^* )^2} p(c|α).∝exp{-n-1/2θ^*_c_it^2 + θ^*_c_it∑_j:j ≠ i(ζ_t,ij-θ^*_c_jt -Z_ij^Tβ^*) } p(c|α) ∴(c_it≠ c_jt' for allj ≠ iort' ≠ t|)= bα∫exp{-n-1/2θ^*_c_it^2 + θ^*_c_it∑_j:j ≠ i(ζ_t,ij-θ^*_c_jt -Z_ij^Tβ^*) }1/√(2π)σ_θexp{-θ^*_c_it^2/2σ_θ^2}d θ^*_c_it = bα/σ_θ√(2π)∫exp{-1/2(n-1 + 1/σ_θ^2)θ^*_c_it^2 + θ^*_c_it∑_j:j ≠ i(ζ_t,ij-θ^*_c_jt -Z_ij^Tβ^*) }d θ^*_c_it. * For m=1, …, L, p(θ^*_m|) ∝exp{-1/2∑_t ∑_i<j (ζ_t,ij - θ^*_c_it -θ^*_c_jt -Z_ij^Tβ^* )^2}exp{-θ^*_m^2/2σ_θ^2}∝exp{-θ^*_m^2/2(1/σ_θ^2 + ∑_t ∑_𝒮_t,m4 + ∑_t ∑_𝒫_t,m 1)+ θ^*_m (2 ∑_t∑_𝒮_t,m (ζ_t,ij-Z_ij^Tβ^* )+ ∑_t ∑_𝒫_t,m (ζ_t,ij-θ^*_c_jt -Z_ij^Tβ^* ))}§ UPDATES OF DYNAMIC MODEL II * For t=1, …, T, 1 ≤ i <j ≤ n,p(ζ_t,ij|) ∝1{ζ_t,ij > 0}^y_t,ij1{ζ_t,ij≤ 0}^1-y_t,ij exp{-1/2(ζ_t,ij ^2 -2ζ_t,ij ( η y_t-1,ij1{t > 1} + θ^*_c_i +θ^*_c_j + Z_ij^Tβ^*) ) }* [t] p(z_s| )∝ p(z_s|z_-s, ν) ∏_t∏_(i,j) ∈𝒫_s p(ζ_t,ij| θ^*_c_i, θ_c_j^*, η, y, β_z_s^*) ∝ p(z_s|z_-s, ν) exp{-1/2∑_t ∑_(i,j) ∈𝒫_s(ζ̃_t,ij - θ^*_c_i -θ^*_c_j - β_k^* 1{z_i=z_j=k})^2} For k ∈ z_-s, ( z_s =k |) = a'm_-s,kexp{-1/2∑_t ∑_(i,j) ∈𝒫_s(ζ̃_t,ij - θ^*_c_i -θ^*_c_j - β_k^* 1{z_i=z_j=k})^2}.and(z_s ≠ z_jfor allj ≠ s|) = a'νexp{-1/2∑_t ∑_(i,j) ∈𝒫_s(ζ̃_t,ij - θ^*_c_i -θ^*_c_j )^2}where a' is a normalizing constant to ensure probabilities sum to one. Hence we can simplify the expressions to (<ref>)* [t] p(β^*|)∝exp{-1/2∑_t ∑_i<j(ζ̃_t,ij - θ^*_c_it -θ^*_c_jt - Z_ij^Tβ^* )^2}exp(-β^*^Tβ^*/2σ_β^2) ∝exp{-1/2(β^*^T T Z^TZ β^* -2β^*^T∑_i<j Z_ij∑_t (ζ̃_t,ij - θ^*_c_it -θ^*_c_jt) + β^*^Tβ^*/σ_β^2)}. * For i=1, …,n,p(c_i|)∝ p(c_i|c_-i,α) exp{-1/2∑_t ∑_i<j( ζ̃_t,ij - θ^*_c_i -θ^*_c_j -Z_ij^Tβ^* )^2} .∝p(c_i|c_-i,α) exp{θ^*_c_i∑_t ∑_j:j ≠ i( ζ̃_t,ij -θ^*_c_j -Z_ij^Tβ^*) - T(n-1)/2θ^*_c_i^2 } ∴(c_i ≠ c_jfor allj ≠ i|) ∝α∫exp{θ^*_c_i∑_t ∑_j:j ≠ i( ζ̃_t,ij-θ^*_c_j -Z_ij^Tβ^*)-T(n-1)/2θ^*_c_i^2 }1/√(2π)σ_θexp{-θ^*_c_i^2/2σ_θ^2}d θ^*_c_i∝α/σ_θ√(2π)∫exp{θ^*_c_i∑_t ∑_j:j ≠ i( ζ̃_t,ij-θ^*_c_j -Z_ij^Tβ^*) - 1/2(T(n-1) + 1/σ_θ^2)θ^*_c_i^2 }d θ^*_c_i* For m=1, …, L, p(θ^*_m|) ∝exp{-1/2∑_t ∑_i<j (ζ̃_t,ij - θ^*_c_i -θ^*_c_j -Z_ij^Tβ^* )^2}exp{-θ^*_m^2/2σ_θ^2}∝exp{-1/2θ^*_m^2 (1/σ_θ^2 + ∑_𝒮_m 4T + ∑_𝒫_m T) + θ^*_c (2 ∑_t ∑_𝒮_m (ζ̃_t,ij-Z_ij^Tβ^* ) +∑_t ∑_𝒫_m (ζ̃_t,ij -θ^*_c_j -Z_ij^Tβ^* ))}* [t] p(η |)∝exp{ -1/2∑_t=2^T ∑_i<j ( ζ_t,ij - η y_t-1,ij - θ^*_c_i -θ^*_c_j - β^TZ_ij )^2- η^2/2σ_η^2}∝exp{ -η^2/2( 1/σ_η^2 + ∑_t=2^T ∑_i<j y_t-1,ij^2 ) + η∑_t=2^T ∑_i<j y_t-1,ij (ζ_t,ij - θ^*_c_i -θ^*_c_j - β^TZ_ij )}. | http://arxiv.org/abs/1705.09088v1 | {
"authors": [
"Linda S. L. Tan",
"Maria De Iorio"
],
"categories": [
"stat.AP"
],
"primary_category": "stat.AP",
"published": "20170525082059",
"title": "Dynamic degree-corrected blockmodels for social networks: a nonparametric approach"
} |
firstpage–lastpage Four-fermion interactions and the chiral symmetry breaking in an external magnetic fieldYu-xin Liu December 30, 2023 ========================================================================================= In its first four years of operation, the Fermi Large Area Telescope (LAT) detected 3033γ-ray emitting sources. In the Fermi-LAT Third Source Catalogue (3FGL) about 50% of the sources have no clear association with a likely γ-ray emitter. We use an artificial neural network algorithm aimed at distinguishing BL Lacs from FSRQs to investigate the source subclass of 559 3FGL unassociated sources characterised by γ-ray properties verysimilar to those of Active Galactic Nuclei. Based on our method, we can classify 271 objects as BL Lac candidates, 185 as FSRQ candidates, leaving only 103 without a clear classification. we suggest a new zoo for γ-ray objects, where the percentage of sources of uncertain type drops from 52% to less than 10%. The result of this study opens up new considerations on the population of the γ-ray sky, and it will facilitate the planning of significant samples for rigorous analyses and multiwavelength observational campaigns. methods: statistical – galaxies: active – BL Lacertae objects: general – gamma-rays: galaxies§ INTRODUCTION The Fermi Large Area Telescope (LAT) has provided the most comprehensive view of the γ-ray sky in the 100 MeV-300 GeV energy range <cit.>. The most recent catalogue of γ-ray sources detected by the LAT, the third Fermi Large Area Telescope source catalogue <cit.>, isbased on data collected in four years of operation, from 2008 August 4 (MJD 54682) to 2012 July 31 (MJD 56139)[Data are available from the Fermi Science Support Center website:http://fermi.gsfc.nasa.gov/ssc/data/access/lat/4yr_catalog/] and contains 3033 sources. The two largest γ-ray source classes are Active Galactic Nuclei (AGN), with 1745 objects, and pulsars (PSR), with 167 objects. Out of 1745 AGN, 1144 are blazars, subdivided into 660 BL Lacertae (BLL) and 484 Flat Spectrum Radio Quasars (FSRQ). The catalogue includes also 573 blazars of uncertain type (BCU), i.e. γ-ray sources positionally coincident with an object showing distinctive broad-band blazar characteristics but lacking reliable optical spectrum measurements. In addition, 30 per cent of the 3FGL sources, 1010 objects, have not even a tentative association with a likely γ-ray-emitting object and are referred to as unassociated sources.As a result, the nature of about half the γ-ray sources, i.e. BCU and unassociated sources, is still not completely known. Since blazars are the most numerous γ-ray source class, we expect that a large fraction of unassociated sources might belong to one of its subclasses, BLLs or FSRQs. Rigorous determination of whether an unassociated source is a BLL or a FSRQ requires the optical spectrum of the correct counterpart. FSRQs have strong, broad emission lines at optical wavelengths, while BLLs show at most weak emission lines, sometimes display absorption features, and can also be completely featureless <cit.>. For this reason detailed optical spectral observation campaigns to identify the nature of many unassociated sources are in progress <cit.>. Unfortunately, optical observations are demanding and time consuming. An easy screening method to suggest the nature of a γ-ray source counterpart could be very useful for the scientific community in order to plan new focused observational campaigns and research projects. Machine-learning techniques are powerful tools for screening and ranking objects according to their predicted classification. Recently, <cit.> developed a method based on machine-learning techniques to distinguish pulsars from AGN candidates among 3FGL unassociated sources using only γ-ray data. In this work we explore the possibility of applying our Blazar Flaring Pattern (B-FlaP) algorithm, which is based on an artificial neural network technique <cit.>, to provide a preliminary and reliable identification of AGN-like unassociated sources as likely BLL or FSRQ candidates.The paper is organized as follows: in Sectn. <ref> we provide a brief description of the machine-learning technique we employed, in Sectn. <ref> we present results of the algorithm at classifying 3FGL unassociated sources and we test our predictions through optical spectral observations of a number of targets, and we discuss implications of our results in Sectn. <ref>. § MACHINE-LEARNING ANALYSIS The aim of this work is to examine the nature of 3FGL unassociated sources in order to select the best candidate sources, according to their predicted source class, for multiwavelength observations and to estimate the number of new γ-ray sources in each class that we might expect to identify in the future. Machine-learning algorithms are the best techniques for screening and classification of unassociated sources based on γ-ray data only. Such techniques were applied to 3FGL unassociated sources by <cit.> to pinpoint potentially novel source classes, and by <cit.> to classify them as likely AGN or PSR including, for the latter, predictions on the likely type of pulsar.Focusing on the latter approach, they distinguished AGN from PSRusing their γ-ray timing and spectral properties combining results from Random Forest <cit.> and Boosted logistic regression <cit.>. Out of 1010 unassociated sources, 559 were classified as likely AGN and 334 as likely PSR with an overall accuracy of ∼96 per cent. In addition, they used the same approach to classify pulsars into “young” and millisecond, leaving unexplored the distinction of AGN subclasses.Here we want to integrate to the analysis performed in <cit.> a classification of 559 3FGL unassociated sources likely AGN as likely BLL or FSRQ using the B-FlaP method described in <cit.>. B-FlaP uses Empirical Cumulative Distribution Function (ECDF) and Artificial Neural Network (ANN) machine-learning techniques to classify blazars taking advantage of different γ-ray flaring activity for BLLs and FSRQs. We used a two-layer ANN algorithm <cit.> to quantify the blazar flaring including as input the source parameters associated to 10 γ-ray flux values corresponding to the 10%, 20%, ..., 100% fraction of observations below this flux. The output was set up to have two possibilities: FSRQ or BLL, with a likelihood (L) assigned to each so thatL_ BLL=1-L_ FSRQ. The closer to 1 is the value of L, the greater the likelihood that the source is in that specific source class. ANN was optimized using as source sample all 660 BLL and 484 FSRQ in the 3FGL catalogue through a learning method based on a standard back-propagation algorithm. The left-hand panel of Figure <ref> shows the likelihood distribution applied to all 3FGL blazars, which shows a clear separation of the two sub-classes of blazars based on flaring patterns.Defining the precision as the positive association rate, a L_ BLL value greater than 0.566 provides a precision of 90 per cent for recognizing BLLs, while L_ BLL less than 0.230 identifies FSRQs with 90 per cent precision. Thanks to this approach, we have been able to apply B-FlaP to the full sample of 3FGL BCUs <cit.>, obtaining statistical classifications for approximately 85 per cent of the sources[The list of B-FlaP BCU classifications is published online at:]. Comparing the B-FlaP predictions with spectroscopic observations that were subsequently retrieved in literature <cit.>, we note that there is a very good agreement between predictions and spectroscopic classifications. With 55 classified BCUs, 52 turn out to be spectroscopically confirmed, while 3, 2 FSRQ (3FGL J0343.3+3622, 3FGL J0904.3+4240) and one BLL (3FGL J1129.4-4215), are not consistent with the observations. While for 3FGL J0343.3+3622 (L_BLL = 0.583) and 3FGL J0904.3+4240 (L_BLL = 0.673) we have an intermediate likelihood value, meaning that the sources fall in a region characterised by a substantial overlap of the two different source classes, the case of 3FGL J1129.4-4215 is more problematic. With its low L_BLL = 0.062, it should be a high confidence FSRQ, while the observed counterpart definitely shows a BL Lac nature <cit.>. This source, however, has multiple possible counterparts (SUMSS J113014-421414 and SUMSS J113006-421441), all lying several arc minutes away from the signal centroid. In such circumstances, it may happen that the γ-ray signal is affected by contamination or badly associated, leading to the observed contradiction.<cit.> used any type of AGN to classify unassociated sources as likely AGN, including not only blazars, but also radio galaxies, compact steep spectrum quasars, Seyfert galaxies, narrow-line Seyfert 1s and other non-blazar active galaxies as well. These objects produce an overall contamination of ∼1.5 per cent to the blazar sample, slightly changing the value of precision of the ANN given the classification thresholds. As a result, the precision for recognizing BLLs and FSRQ decreases to 87 per cent, introducing a contamination given by non-blazars of ∼2 per cent for the former and ∼ 3 per cent for the latter. § RESULTS AND VALIDATION In this section we discuss the results of our optimized ANN algorithm at classifying BLL and FSRQ candidatesamong 3FGL unassociated sources. Applying our optimized algorithm to the 559unassociated sources classified as likely AGN by <cit.>, we find that 271 are classified as BLL candidates, 185 as FSRQ candidates, and 103 remain likely AGN of uncertain type. The right-hand panel of Figure <ref> shows the likelihood distribution applied to likely AGN, which reflects very well those of known BLL and FSRQ. Interestingly, we find that the ratio of likely BLL to FSRQ obtained by our analysis (∼ 1.4) is very similar to the ratio of known BLL and FSRQ (1.4).Table <ref> shows a portion of individual results of the classification of 3FGL unassociated sources classified as likely AGN, where, for each source, we provide the ANN likelihood (L) to be a BLL or an FSRQ, and the predicted classification according to the defined classification thresholds. The second andthird columns of the list show the Galactic longitude and latitude respectively. The full table is available electronically from the journal.Since we did not include any spectral information in the ANN algorithm, we validate our results comparing the spectra for known BLL and FSRQ with those of likely blazar subclasses. Gamma-ray BLL have average spectra that are flatter than those of FSRQs <cit.>. The best-fitting photon spectral index (in 3FGL named power-law index) distribution has a mean value of 2.02±0.25 for the former and 2.45±0.20 for the latter, where uncertainties are reported at the 1σ confidence level. Figure <ref> shows that the power-law index distribution for likely BLL and FSRQ is consistent with those of known BLL and FSRQs (mean value of 2.10±0.27 for the former and 2.54±0.21 for the latter). Another way to validate the predictions of our method is to compare them with classifications obtained after the release of the 3FL catalogue. Currently, optical spectroscopic observations campaigns to hunt blazars among unassociated γ-ray sources are ongoing <cit.>. These follow-up multiwavelength classification efforts have resulted in 24 new blazar associations, 21 classified as BLL and 3 as FSRQs. Since our algorithm was optimized to select the best targets to observe in other wavelengths, we can evaluate the performance of our method analyzing the positive association rate (precision). Out of 24 new blazars with optical spectra, B-FlaP classifies 22 as BLL, while 2 remain unclassified. For the subset of 22 BLL candidates, our prediction matches in about 90 per cent of the objects with optical spectra, in agreement with our classification thresholds definition, while we cannot assert anything about the precision in the classification of FSRQ. This result give a strong confirmation about the optimal performance of our classification algorithm even if the number of new blazar associations is still small. §.§ Optical spectroscopic observations Encouraged by optimal performance of our classification algorithm, we carried out optical spectral observations at the Asiago Astrophysical Observatory of the best targets within the unassociated source sample classified as likely AGN. The new observations were executed with the 1.82m Copernico telescope[Website: <http://www.oapd.inaf.it/index.php/en>] and the 1.22m Galileo telescope[Website: <http://www.dfa.unipd.it/index.php?id=300>], configured for long-slit optical spectroscopy, with the instrumental configurations reported in Table <ref>. Both telescopes are able to provide moderate spectral resolution data (R ∼ 600) over a wavelength range spanning 3700Å to 7500Å, achieving a continuum signal to noise ratio (SNR) of order ∼ 20 in 2 hours of exposure on targets with an optical magnitude V ∼ 17.Taking into account the observational limitations introduced by the magnitude constraints and the geographical position of the observing site, we performed an observing campaign, selecting the targets for observations from the list of 3FGL unassociated sources. Due to the quite large uncertainties on the positions of γ-ray sources, especially faint ones, the identification of the plausible optical counterparts to be observed was obtained by combining radio and X-ray observations, in a similar way to the method described in <cit.>. The reason for combination lies in the theoretical expectation that a high-energy source, powered by a jet of relativistic charged particles, which produce γ-ray photons, should suffer significant energy losses from synchrotron radiation at radio and X-ray frequencies <cit.>. Since the spatial resolution of detectors operating at such lower energies is far better than in the γ-ray band (down to a few arcseconds), the optical counterparts were associated with objects emitting both radio and X-ray photons, within the γ-ray signal confinement area at the 95 per cent confidence level. Although this technique proved reliable to support the association of BCU targets to optical counterparts, in the case of unassociated sources, further care was required. These sources are generally faint and with quite large positional uncertainties. Consequently, also the expected synchrotron losses are weak and might be missed in standard X-ray and radio surveys. In general we are able to select the low-energy candidate counterpart by matching the NRAO VLA Sky Survey <cit.> with the Swift satellite X-ray catalogue <cit.> in circular regions of 10' in radius, centered on the γ-ray signal centroid. In some cases, however, the sources are too weak to be listed in a catalogue, particularly at X-rays, and a subsequent reanalysis of the observations of the target is the only way to pinpoint the most likely optical counterpart to such sources.When all the observational constraints were satisfied, we observed the targets, collecting a total of 2 hours of exposure time for every target, split in observations lasting from 20 up to 30 minutes each. The exact duration of the single exposures was determined by the best trade-off between the requirement to improve the SNR, the need to track the spectrum on the surface of the detectors, and the contamination from cosmic rays and night sky emission lines. These background contributions, indeed, may lead to saturation effects on the detector, with consequent loss of spectral information, and must therefore be removed. Combining several short exposures to form a longer observation can immediately filter out the cosmic ray background, since it follows a random pattern that is easily identified and masked out from the complete data set. The same process leads to an efficient subtraction of the sky emission lines, because the combined spectrum is not subject to saturation limits andcan collect an arbitrarily high signal, in order to interpolate the sky contribution from regions close to the source and to remove them from the spectrum. All the observations were taken together with comparison FeAr arc lamp spectra to perform wavelength calibration, and they were paired with observations of spectro-photometric standard stars, to provide flux calibration. We proceeded with the standard long slit spectroscopic data reduction procedure, which involves bias subtraction and flat field correction, by means of standard IRAF tasks[Website <http://www.iraf.noao.edu>], arranged in a specific pipeline that is optimized to work with the Asiago telescopes instrumental configuration. In this study, we obtained 5 new spectra,illustrated in Fig. <ref>. We detail in the following the characteristics of these spectra and the resulting insights.§.§.§ 3FGL 0032.5+3912This object has been associated with an optical source that, once observed with the 1.82m telescope, showed the characteristic spectrum of an elliptical galaxy. The identification of absorption lines like the Ca II λλ3933,3969 doublet (detected at 4530Å and 4572Å), together with Mg I λ5175 and Na I λ5893 (detected at 5961Å and 6789Å) places the redshift of this object at z = 0.152. Its optical spectrum is consistent with an elliptical galaxy that may host BLL activity, in agreement with the BLL classification suggested by the machine learning technique.§.§.§ 3FGL 2224.4+0351Observed with the 1.82m telescope, this object shows a featureless continuum spectrum, with a peak close to 7000Å, subsequently decaying towards the short wavelength regions. No clear signs of emission and absorption lines are detected in the spectrum, consistent with the predicted BLL classification of the source.§.§.§ 3FGL 2247.2-0004This source has been observed with the 1.22m telescope and shows a featureless continuum spectrum, decaying towards the short wavelength regime, much like the previous case. The lack of clear emission and absorption lines is consistent with its predicted BLL classification.§.§.§ 3FGL 2300.0+4053The spectrum of this source shows a more prominent power-law continuum that increases towards the short wavelength regime. With the exception of some unidentified spike-like features, very likely descending from increase of noise due to lower detector efficiency, the classification turns out to be consistent with a source of BLL type.§.§.§ 3FGL 2358.5+3827When observed with the 1.22m telescope, this source shows a clear system of emission lines that can be identified as the close [O II] λλ3727,3929 doublet (detected as an unresolved feature at 4477Å) and the strong [O III] λλ4959,5007 doublet (falling at 5956Å and 6013Å), which place this source at redshift z = 0.201. The detection of strong narrow lines, together with an underlying continuum that becomes weaker at short wavelength, are suggestive of an obscured AGN activity. The source has a 1.4GHz flux measured by the NVSS ofF_1.4GHz = 57.4 ± 1.8mJythat, adopting a standard Λ Cold Dark Matter Cosmology with H_0 = 70kms^-1Mpc^-1, Ω_Λ = 0.7 and Ω_M = 0.3, corresponds to a distance of 920.3 Mpc and to an intrinsic luminosityν L_ν = (5.49 ± 0.17) · 10^40ergcm^-2s^-1.This suggests that this object could be classified as a Narrow Line Radio Galaxy (NLRG), in spite of the predicted classification as a BLL, although the detection of such type of objects in γ-rays, at the inferred redshift, is extremely rare.§ DISCUSSION AND CONCLUSIONS One of the main goals of our investigation is to complete the census of blazar subclasses in the 3FGL source catalogue using the ANN technique based on B-FlaP <cit.>. B-FlaP is well suited to perform preliminary and reliable classification of likely blazars when detailed observational or multiwavelength data are not yet available. This is the typical situation for almost all unassociated sources in Fermi-LAT catalogues. Recently, <cit.> applied a number of machine-learning techniques to classify 3FGL unassociated sources as likely pulsar or AGN, focusing only on the former, to identify the most promising unassociated source to target in pulsar search. We applied our algorithm to 559 3FGL unassociated sources classified as likely AGN to investigate their source subclass. These sources can be divided in 271 BLL candidates, 185 FSRQ candidates, leaving only 103 without a clear classification. We validated our predictions comparing their γ-ray spectra with the expected ones. In addition, we compared our results with the source classes inferred by recently published optical spectroscopic observations <cit.>. This comparison results in 29 new blazar associations, out of which 5 are obtained thanks to our new optical observations. For the subset of 27 overlapping sources, our prediction matches in ∼ 90 per cent of the objects as expected. Such excellent agreement confirms the power of our method as a classifier for unidentified sources as well. Our work can help to identify targets both for blazar searches and for follow-up studies of blazars at very-high γ-ray energies with ground-based imaging air Cherenkov telescopes (MAGIC, HESS, VERITAS).<cit.> have recently published a paper aimed at researching blazar candidates among the Fermi-LAT 3FGL catalogue using a combination of boosted classification trees and multilayer perceptron artificial neural networks methods. Their work is divided in two steps. In the first one they applied the combined classifier to separate 3FGL unassociated sources as blazar or pulsar candidates, while in the second one they use the same approach to determine the BLL or FSRQ nature of both blazar candidates and of BCU. In contrast to our approach, they used both spectral and timing γ-ray parameters to separate source classes. Out of 595 blazar candidates among the 3FGL unassociated sources, <cit.> study the blazar subclass nature for 417 sources that have no caution flag as described in <cit.>. Out of these, 371 match with our blazar candidate sample. Applying their classifier to this sample they divide them into 192 BLL and 129 FSRQ candidates. The comparison with our corresponding subset of 223 BLL candidates shows that our prediction is in agreement with <cit.> for 174 objects (about 80 per cent) and in disagreement for 28 (about 12 per cent). We observe that 13 objects in disagreement are characterized by a very low prediction value (L_BLL<0.7), thus the discrepancy between the two approaches decreases significantly when defining a more robust classification threshold. In addition, comparing our subset of 83 FSRQ candidates we observe that our prediction is in agreement for 62 sources (about 75 per cent) and in disagreement for 8 (about 9 per cent). Interestingly, analysing the PowerLaw Index distribution for the sources that are in disagreement with <cit.> we observe that they are located in the region of overlap between BLL and FSRQ (between 2.1 and 2.9) making difficult their classification including even spectral information. Only future optical spectroscopic observations will unveil the real nature of these sources. As a result, although <cit.> applied a very different approach from our work, we find a good overall agreement, indicating that both methods are useful classifiers. We obtained the same result applying our optimised algorithm to all 417 un-flagged sources classified as blazar candidates by <cit.>.Putting together the overall result of this study with the ones obtained in <cit.> and <cit.> we can characterize the entire γ-ray population proposing a new distribution of 3FGL sources, as shown in Figure <ref>, where cells in red represent results obtained in this work. Table <ref> shows the number of γ-ray sources per each class reported in the 3FGL catalogue and after this work. The number of BLL (or candidates) increased by a factor of 1.9, while that of FSRQ of 1.7, raising the ratio of BLL to FSRQ from 1.36 to 1.55. Interestingly, out of 180 blazars of uncertain type, only 20 (11 per cent) are located at low Galactic latitude (|b|<10 deg). We expect that a very small fraction (less than 3 per cent) of non-blazar AGN subclasses (Seyferts, radio galaxies and other AGN) could contaminate the sample of blazar of uncertain type. As an important result, the efforts aimed at classifying 3FGL sources decreased the fraction of uncertain sources (BCU and unassociated sources) from 52 per cent to 10 per cent, discriminating the best targets for futurefollow-up multi-wavelength observations.§ ACKNOWLEDGEMENTSWe thank the anonymous referee for his/her very helpful comments and suggestions to our manuscript. Support for science analysis during the operations phase is gratefully acknowledged from the Fermi-LAT collaboration for making the 3FGL results available in such a useful form, the Institute of Space Astrophysics and Cosmic Physics of Milano -Italy (IASF INAF) and The Goddard Space Flight Center NASA. DS acknowledges support through EXTraS, funded from the European Commission Seventh Framework Programme (FP7/2007-2013) under grant agreement n. 607452. We thank Pablo Saz Parkinson because this paperbuilds directly on his work. This work is based on observations collected at Copernico telescope (Asiago, Italy) of the INAF - Osservatorio Astronomico di Padova and on observations collected with the 1.22m Galileo telescope of the Asiago Astrophysical Observatory, operated by the Department of Physics and Astronomy “G. Galilei” of the University of Padova.999[Abdo et al.(2010)]abd10 Abdo, A., et al., 2010, ApJ, 716, 30[Acero etal.(2015)]3FGL Acero, F., et al. , 2015, ApJS, 218, 23[Ackermann et al.(2015)]ack15 Ackermann, M., et al., 2015, ApJ, 810, 14[Álvarez Crespo et al. (2016a)]AC16a Álvarez Crespo, N., et al., 2016, AJ, 151, 32[Álvarez Crespo et al. (2016b)]AC16b Álvarez Crespo, N., et al., 2016, Ap&SS, 361, 316[Álvarez Crespo et al.(2016c)]alv16 Álvarez Crespo, N.,et al. 2016b AJ, 151, 95[Atwood et al.(2009)]FLAT Atwood, W. B., et al., 2009, ApJ, 697, 1071[Bishop (1995)]bis95 Bishop, C. M., Neural Networks for Pattern Recognition, 1995[Boettcher, Harris & Krawczinski (2012)]Boettcher12 Boettcher, M., Harris, D. E., & Krawczinsk,i H., 2012, Relativistic Jets fom Active Galactic Nuclei. Wiley-VCH Verlag GmbH & Co. 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"authors": [
"David Salvetti",
"Graziano Chiaro",
"Giovanni La Mura",
"David J. Thompson"
],
"categories": [
"astro-ph.HE"
],
"primary_category": "astro-ph.HE",
"published": "20170527154450",
"title": "3FGLzoo. Classifying 3FGL Unassociated Fermi-LAT Gamma-ray Sources by Artificial Neural Networks"
} |
Evolution of Social Power in Social Networks with Dynamic Topology Mengbin Ye, Student Member, IEEE Ji Liu, Member, IEEEBrian D.O. Anderson, Life Fellow, IEEE Changbin Yu, Senior Member, IEEE Tamer Başar, Life Fellow, IEEEM. Ye, B.D.O. Anderson and C. Yu are with the Research School of Engineering, Australian National University . B.D.O. Anderson is also with Hangzhou Dianzi University, Hangzhou, China, and with Data61-CSIRO (formerly NICTA Ltd.) in Canberra, A.C.T., Australia. J. Liu and T. Başar are with the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign . December 30, 2023 ================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================= The recently proposed DeGroot-Friedkin model describes the dynamical evolution of individual social power in a social network that holds opinion discussions on a sequence of different issues. This paper revisits that model, and uses nonlinear contraction analysis, among other tools, to establish several novel results. First, we show that for a social network with constant topology, each individual's social power converges to its equilibrium value exponentially fast, whereas previous results only concluded asymptotic convergence. Second, when the network topology is dynamic (i.e., the relative interaction matrix may change between any two successive issues), we show that each individual exponentially forgets its initial social power. Specifically, individual social power is dependent only on the dynamic network topology, and initial (or perceived) social power is forgotten as a result of sequential opinion discussion. Last, we provide an explicit upper bound on an individual's social power as the number of issues discussed tends to infinity; this bound depends only on the network topology. Simulations are provided to illustrate our results. opinion dynamics, social networks, influence networks, social power, dynamic topology, nonlinear contraction analysis, discrete-time systems§ INTRODUCTION Social network analysis is the study of a group of social actors (individuals or organisations) who interact in some way according to a social connection or relationship. The study of social networks has spanned several decades <cit.> and across several scientific communities. In the past few years, perhaps in part due to lessons learned and tools developed from extensive research on coordination of autonomous multi-agent systems <cit.>, the systems and control community has taken an interest in social network analysis.Of particular interest in this context is the problem of “opinion dynamics”, which is the study of how individuals in a social network interact and exchange their opinions on an issue or topic. A critical aspect is to develop models which simultaneously capture observed social phenomena and are simple enough to be analysed, particularly from a system-theoretic point of view. The seminal works of <cit.> proposed a discrete-time opinion pooling/updating rule, now known as the French-DeGroot (or simply DeGroot) model. A continuous-time counterpart, known as the Abelson model, was proposed in <cit.>. These opinion updating rules are closely related to consensus algorithms for coordinating autonomous multi-agent systems <cit.>. The Friedkin-Johnsen model <cit.> extended the French-DeGroot model by introducing the concept of a “stubborn individual”, i.e., an individual who remains attached to its initial opinion. This helped to model social cleavage<cit.>, a phenomenon where opinions tend towards separate clusters. Other models which attempt to explain social cleavage include the Altafini model with negative/antagonistic interactions <cit.> and the Hegelsmann-Krause bounded confidence model <cit.>. Simultaneous opinion discussion on multiple, logically interdependent topics was studied with a multidimensional Friedkin-Johnsen model <cit.>. The concept of social power or social influence has been integral throughout the development of these models. Indeed, French Jr's seminal paper <cit.> was an attempt to quantitatively study an individual's social power in a group discussion. Broadly speaking, in the context of opinion dynamics, individual social power is the amount of influence an individual has on the overall opinion discussion. Individuals which maximise the spread of an idea or rumour in diffusion models were identified in <cit.>. The social power of an individual in a group can change over time as group members interact and are influenced by each other. Recently, the DeGroot-Friedkin model was proposed in <cit.> to study the dynamic evolution of an individual's social power as a social network discusses opinions on a sequence of issues. In this paper, we present several major, novel results on the DeGroot-Friedkin model. In Section <ref>, we shall provide a precise mathematical formulation of the model, but here we provide a brief description to better motivate the study, and elucidate the contributions of the paper.The discrete-time DeGroot-Friedkin model <cit.> is a two-stage model.In the first stage, individuals update their opinions on a particular issue, and in the second stage, each individual's level of self-confidence for the next issue is updated. For a given issue, the social network discusses opinions using the DeGroot opinion updating model, which has been empirically shown to outperform Bayesian learning methods in the modelling of social learning processes <cit.>. The row-stochastic opinion update matrixused in the DeGroot model is parametrised by two sets of variables. The first is individual social powers, which are the diagonal entries of the opinion update matrix (i.e. the weight an individual places on its own opinion). The second is the relative interaction matrix, which is used to scale the off-diagonal entries of the opinion update matrix to ensure that, for any given values of individual social powers, the opinion update matrix remains row-stochastic. In the original model <cit.>, the relative interaction matrix was assumed to be constant over all issues, and constant throughout the opinion discussion on any given issue. Under some mild conditions on the entries of the relative interaction matrix, the opinions reach a consensus on every issue.At the end of the period of discussion of an issue, i.e., when opinions have effectively reached a consensus, each individual undergoes a sociological process of self-appraisal (detailed in the seminal work <cit.>) to determine its impact or influence on the final consensus value of opinion. Such a mechanism is well accepted as a hypothesis <cit.> and has been empirically validated <cit.>. Immediately before discussion on the next issue, each individual self-appraises and updates its individual social power (the weight an individual places on its own opinion) according to the impact or influence it had on discussion of the previous issue. In updating its individual social power, an individual also updates the weight it accords its neighbours' opinions, by scaling using the relative interaction matrix, to ensure that the opinion updating matrix for the next issue remains row-stochastic. This process is repeated as issues are discussed in sequence. The primary objective of the DeGroot-Friedkin model is to study the dynamical evolution of the individual social powers over the sequence of discussed issues.The model is centralised in the sense that individuals are able to observe and detect their impact relative to every other individual in the opinion discussions process, which indicates that the DeGroot-Friedkin model is best suited for networks of small or moderate size. Such networks are found in many decision making groups such as boards of directors, government cabinets or jury panels. Distributed models of self-appraisal have been studied in continuous time <cit.> as well as discrete time <cit.> to extend the original DeGroot-Friedkin model. Dynamic topology, but restricted to doubly-stochastic relative interaction matrices, was studied in <cit.>. §.§ Contributions of This PaperThis paper significantly expands on the original DeGroot-Friedkin model in several different respects. In the original paper <cit.>, LaSalle's Invariance Principle was used to arrive at an asymptotic stability result. Exponential convergence was conjectured but not proved. In this paper, a novel approach based on nonlinear contraction analysis <cit.> is used to conclude an exponential convergence property for non-autocratic social power configurations. Autocratic social power configurations are shown to be unstable, or asymptotically stable, but not exponentially so. Additional insights are also developed; an upper bound on the individual social power at equilibrium is established, dependent only on the relative interaction matrix. The ordering of individuals' equilibrium social powers can be determined <cit.>, but numerical values for nongeneric network topologies cannot be determined.The paper is also the first to provide a complete proof of convergence for the DeGroot-Friedkin model with dynamic topology. Dynamic topology for the DeGroot-Friedkin model was studied in <cit.> and a stability result was conjectured based on extensive simulation. By dynamic topology, we mean relative interaction matrices which are different between issues, but remain constant during the period of discussion for any given issue. Relative interaction matrices encode trust or relationship strength between individuals in a network. A network discussing sometimes sports and sometimes politics will have different interaction matrices; some individuals are experts on sports and others on politics. These factors can influence the trust or relationship strength between individuals. This gives rise to the concept of issue-driven topology change. In addition, allowing for dynamic relative interaction matrices is a natural way of describing network structural changes over time. For many reasons, new relationships may form and others may die out. For example, an individual may attempt to, after each issue, form new relationships, disrupt other relationships, and adjust relationship strengths in order to maximise its individual social power. This gives rise to the concept of individual-driven topology change. The idea that an individual intentionally modifies topology to gain its social power was studied in <cit.> by assuming constant topology, but this can be more naturally modelled using dynamic topology.A conference paper <cit.> by the authors studied the special case of periodically varying topology and proved the existence of periodic trajectories, but did not provide a convergence proof. In this paper, we show that for relative interaction matrices which vary arbitrarily across issues, the individual social powers converge exponentially fast to a unique trajectory (as opposed to unique stationary values for constant interactions). Specifically, every individual forgets its initial social power estimate (initial condition) for each issue exponentially fast. For any given issue, and as the number of issues discussed tends to infinity, individuals' social powers are determined only by the network interactions on the previous issue. This paper therefore concludes that a social network described by the DeGroot-Friedkin model is self-regulating in the sense that, even on dynamic topologies, sequential discussion combined with reflected self-appraisal removes perceived social power (initial estimates of social power). True social power is determined by topology. Periodically varying topologies are presented as a special case. §.§ Structure of the Rest of the PaperSection <ref> introduces mathematical notations, nonlinear contraction analysis and the DeGroot-Friedkin model. Section <ref> uses nonlinear contraction analysis to study the original DeGroot-Friedkin model. Dynamic topologies are studied in Section <ref>. Simulations are presented in Section <ref>, and concluding remarks are given in Section <ref>.§ BACKGROUND AND PROBLEM STATEMENTWe begin by introducing some mathematical notations used in the paper. Let 1_n and 0_n denote, respectively, the n× 1 column vectors of all ones and all zeros. For a vector x∈ℝ^n, 0≼ x and 0 ≺ x indicate component-wise inequalities, i.e., for all i∈{1,…,n}, 0≤ x_i and 0<x_i, respectively. The n-simplex is . The canonical basis of ℝ^n is given by 𝐞_1, …, 𝐞_n. Define Δ_n = Δ_n \{𝐞_1, …, 𝐞_n } and . The 1-norm and infinity-norm of a vector, and their induced matrix norms, are denoted by ‖·‖_1 and ‖·‖_∞, respectively. For the rest of the paper, we shall use the terms “node”, “agent”, and “individual” interchangeably. We shall also interchangeably use the words “self-weight”, “social power”, and “individual social power”. An n× n matrix with all entries nonnegative is called a row-stochastic matrix (respectively doubly stochastic) if its row sums all equal 1 (respectively if its row and column sums all equal 1). We now provide a result on eigenvalues of a matrix product, to be used later. Let A, B∈ℝ^n× n be symmetric. If A is positive definite, then AB is diagonalizable and has real eigenvalues. If, in addition, B is positive definite or positive semidefinite, then the eigenvalues of AB are all strictly positive or nonnegative, respectively.§.§ Graph TheoryThe interaction between individuals in a social network is modelled using a weighted directed graph, denoted as 𝒢 = (𝒱, ℰ, C). Each individual corresponds to a node in the finite, nonempty set of nodes 𝒱 = {v_1, …, v_n}. The set of ordered edges is ℰ⊆𝒱×𝒱. We denote an ordered edge as e_ij = (v_i, v_j) ∈ℰ, and because the graph is directed, in general, e_ij and e_ji may not both exist. An edge e_ij is said to be outgoing with respect to v_i and incoming with respect to v_j. The presence of an edge e_ij connotes that individual j learns of, and takes into account, the opinion value of individual iwhen updating its own opinion. The incoming and outgoing neighbour sets of v_i are respectively defined as 𝒩_i^+ = {v_j ∈𝒱 : e_ji∈ℰ} and 𝒩_i^- = {v_j ∈𝒱 : e_ij∈ℰ}. The relative interaction matrix C∈ℝ^n× n is associated with 𝒢, the relevance of which is explained below. The matrix C has nonnegative entries c_ij, termed “relative interpersonal weights” in <cit.>.The entries of C have properties such that 0 < c_ij≤ 1 ⇔ e_ji∈ℰ and c_ij = 0 otherwise. It is assumed that c_ii = 0 (i.e., there are no self-loops), and we impose the restriction that ∑_j∈𝒩_i^+ c_ij = 1 (i.e., C is a row-stochastic matrix). The word “relative” therefore refers to the fact that c_ij can be considered as a percentage of the total weight or trust individual i places on individual j compared to all of individual i's incoming neighbours.A directed path is a sequence of edges of the form (v_p_1, v_p_2), (v_p_2, v_p_3), … where v_p_i∈𝒱, e_ij∈ℰ. Node i is reachable from node j if there exists a directed path from v_j to v_i. A graph is said to be strongly connected if every node is reachable from every other node. The relative interaction matrix C is irreducible if and only if the associated graph 𝒢 is strongly connected. If C is irreducible, then it has a unique left eigenvector γ^⊤≻ 0 satisfying γ^⊤1_n = 1, associated with the eigenvalue 1 (Perron-Frobenius Theorem, see <cit.>). Henceforth, we call γ^⊤ the dominant left eigenvector of C. §.§ The DeGroot-Friedkin ModelWe define 𝒮 = {0, 1, 2, …} to be the set of indices of sequential issues which are being discussed by the social network. For a given issue s ∈𝒮, the social network discusses it using the discrete-time DeGroot consensus model (with constant weights throughout the discussion of the issue). At the end of the discussion (i.e. when the DeGroot model has effectively reached steady state), each individual undergoes reflected self-appraisal, with “reflection” referring to the fact that self-appraisal occurs following the completion of discussion on the particular issue s. Each individual then updates its own self-weight, and discussion begins on the next issue s+1 (using the DeGroot model but now with adjusted weights). The DeGroot-Friedkin model assumes the opinion dynamics process operates on a different time-scale than that of the reflected appraisal process. This allows for a simplification in the modelling and is reasonable if we consider that having separate time-scales merely implies that the social network reaches a consensus on opinions on one issue before beginning discussion on the next issue. If this assumption is removed, i.e., the time-scales are comparable, then the distributed DeGroot-Friedkin model is used <cit.>. However, at this point the analysis of the distributed model is much more involved, and has not yet reached the same level of understanding as the original model.We next explain the mathematical modelling of the opinion dynamics for an issue and the updating of self-weights from one issue to the next. §.§.§ DeGroot Consensus of OpinionsFor each issue s ∈𝒮, individual i updates its opinion y_i(s,·) ∈ℝ at time t+1 asy_i(s, t+1) = w_ii(s) y_i(s, t) + ∑_j≠ i^n w_ij(s) y_j(s, t)where w_ii(s) is the self-weight individual i places on its own opinion and w_ij(s) is the weight placed by individual i on the opinion of its neighbour individual j. Note that ∀ i,j, w_ij(s) ∈ [0,1] is constant for any given s. As will be made apparent below, ∑_j = 1^n w_ij = 1, which implies that individual i's new opinion value y_i(s, t+1) is a convex combination of its own opinion and the opinions of its neighbours at the current time instant. The opinion dynamics for the entire social network can be expressed as y(s, t+1) =W(s)y(s, t)where y(s, t) = [y_1(s, t), ,y_n(s, t)]^⊤ is the vector of opinions of the n individuals in the network at time instant t. This model was studied in <cit.> with 𝒮 = {0} (i.e., only one issue was discussed), and with individuals who remember their initial opinions y_i(s,0) <cit.>.Let the self-weight (individual social power) of individual i be denoted by x_i(s) = w_ii(s) ∈ [0,1] (the i^th diagonal entry of W(s)) <cit.>, with the individual social power vector given as x(s) = [x_1, , x_n]^⊤. For a given issue s, the influence matrix W(s) is defined as W(s) =X(s) + ( I_n -X(s)) Cwhere C is the relative interaction matrix associated with the graph 𝒢, and the matrix X(s) ≐ diag[ x(s)]. From the fact that C is row-stochastic with zero diagonal entries, (<ref>) implies that W(s) is a row-stochastic matrix. It has been shown in <cit.> that W(s) defined as in (<ref>) ensures that for any given s, there holds lim_t →∞ y(s,t) = (ζ(s)^⊤ y(s,0))1_n. Here, ζ(s)^⊤ is the unique nonnegative left eigenvector of W(s) associated with the eigenvalue 1, normalised such that 1_n^⊤ζ(s) = 1. That is, the opinions converge to a constant consensus value.Next, we describe the model for the updating of W(s) (specifically w_ii(s) via a reflected self-appraisal mechanism).Kronecker products may be used if each individual has simultaneous opinions on p unrelated topics, y_i ∈ℝ^p, p ≥ 2. Simultaneous discussion of p logically interdependent topics is treated in <cit.> under the assumption that 𝒮 = {0}. §.§.§ Friedkin's Self-Appraisal Model for Determining Self-Weight The Friedkin component of the model proposes a method for updating the individual self-weights, x(s). We assume the starting self-weights x_i(0)≥ 0 satisfy ∑_i x_i(0) = 1.[The assumption that ∑_i x_i(0) = 1 is not strictly required, as we will prove in Section <ref> that if 0≤ x_i(0) < 1,∀ i and ∃ j : x_j(0) > 0, then the system will remain inside the simplex Δ_n for all s ≥ 1.] At the end of the discussion of issue s, the self-weight vector updates asx(s+1) = ζ(s)Note that ζ(s)^⊤1_n = 1 implies that x(s) ∈Δ_n, i.e., ∑_i=1^n x_i(s) = 1 for all s. From (<ref>), and because C is row-stochastic, it is apparent that by adjusting w_ii(s+1) = ζ_i(s), individual i also scales w_ij(s+1), j ≠ i using c_ij to be (1-w_ii(s+1))c_ij to ensure that W(s) remains row-stochastic.The precise motivation behind using (<ref>) as the updating model for x(s) is detailed in <cit.>, but we provide a brief overview here in the interest of making this paper self-contained. As discussed in Subsection <ref>, for any given s, there holds lim_t →∞y(s,t) = (ζ(s)^⊤ y(s,0))1_n. In other words, for any given issue s, the opinions of every individual in the social network reaches a consensus value ζ(s)^⊤ y(s,0) equal to a convex combination of their initial opinion values y(s,0). The elements of ζ(s)^⊤ are the convex combination coefficients. For a given issue s, ζ_i(s) is therefore a precise manifestation of individual i's social power or influence in the social network, as it is a measure of the ability of individual i to control the outcome of a discussion <cit.>. The reflected self-appraisal mechanism therefore describes an individual 1) observing how much power it had on the discussion of issue s (the nonnegative quantity ζ_i(s)), and 2) for the next issue s+1, adjusting its self-weight to be equal to this power, i.e., x_i(s+1) = w_ii(s+1) = ζ_i(s).Lemma 2.2 of <cit.> showed that the system (<ref>) is equivalent to the discrete-time systemx(s+1) =F( x(s))where the nonlinear map F( x(s)) is defined asF(x(s) ) = 𝐞_iifx(s) = 𝐞_ifor anyi α ( x(s)) [ γ_1/1-x_1(s);⋮; γ_n/1-x_n(s) ] otherwisewith α( x(s)) = 1/∑_i=1^n γ_i/1- x_i(s) whereγ = [γ_1, γ_2, , γ_n]^⊤ is the dominant left eigenvector of C. Note that ∑_i F_i = 1, where F_i is the i^th entry of F. We now introduce an assumption which will be invoked throughout the paper. The matrix C∈ℝ^n× n, with n ≥ 3, is irreducible, row-stochastic, and has zero diagonal entries. Irreducibility of C implies, and is implied by, the strongly connectedness of the graph 𝒢 associated with C.This assumption was in place in <cit.> by and large throughout its development. Dynamic topology involving reducible C is a planned future work of the authors. A special topology studied in <cit.> is termed “star topology”, the definition and relevance of which follow.A strongly connected graph[While it is possible to have a star graph that is not strongly connected, this paper, similarly to <cit.>, deals only with strongly connected graphs.] 𝒢 is said to have star topology if ∃ a node v_i, called the centre node, such that every edge of 𝒢 is either to or from v_i.The irreducibility of C implies that a star 𝒢 must include edges in both directions between the centre node v_i and every other node v_j, j ≠ i. We now provide a lemma and a theorem (the key result of <cit.>) regarding the convergence of F(x(s)) as s→∞, and a fact useful for analysis throughout the paper.Suppose that n ≥ 3, and suppose further that 𝒢 has star topology, which without loss of generality has centre node v_1. Let C satisfy Assumption <ref>. Then, ∀ x(0) ∈Δ_n, lim_s→∞ x(s) = 𝐞_1. This implies that ∀ x(0) ∈Δ_n, a network with star topology converges to an “autocratic configuration” where centre individual 1 holds all of the social power. <cit.> Suppose that n≥ 3 and let γ^⊤, with entries γ_i, be the dominant left eigenvector of C∈ℝ^n× n, satisfying Assumption <ref>. Then, ‖γ‖_∞ = 0.5 if and only if C is associated with a star topology graph, and in this case γ_i = 0.5 where i is the centre node; otherwise,‖γ‖_∞ < 0.5. For n≥ 3, consider the DeGroot-Friedkin dynamical system (<ref>) with C satisfying Assumption <ref>. Assume further that the digraph 𝒢 associated with C does not have star topology. Then, *For all initial conditions x(0) ∈Δ_n, the self-weights x(s) converge to x^* as s→∞, where x^* ∈int(Δ_n) is the unique fixed point satisfying x^* =F( x^*). *There holds x^*_i < x^*_j if and only if γ_i < γ_j for any i,j, where γ_i is the i^th entry of the dominant left eigenvector γ. There holds x^*_i = x^*_j if and only if γ_i = γ_j. *The unique fixed point x^* is determined only by γ, and is independent of the initial conditions. §.§ Quantitative Aspects of the Dynamic Topology ProblemIn the introduction, we discussed in qualitative terms that we are seeking to study the evolution, and in particular the convergence properties, of social power in dynamically changing social networks. Now, we provide quantitative details on the problem of interest. Specifically, we will consider dynamic relative interaction matrices C(s) which are issue-driven or individual-driven. As we have now properly introduced the DeGroot-Friedkin model, it is appropriate for us to expand on this motivation, using the following two examples.Example 1 [Issue-driven]: Consider a government cabinet that meets to discuss the issues of defence, economic growth, social security programs and foreign policy. Each minister (individual in the cabinet) has a specialist portfolio (e.g. defence) and perhaps a secondary portfolio (e.g. foreign policy). While every minister will partake in the discussion of each issue, the weights c_ij(s) will change. For example, if minister i's portfolio is on defence, then c_ji(s_defence) will be high as other ministers j place more trust on minister i's opinion. On the other hand, c_ji(s_security) will be low. It is then apparent that C(s_defence) ≠C(s_security) in general. This motivates the incorporation of issue-dependent or issue-driven topology into the DeGroot-Friedkin model.Example 2 [Individual-driven]: Consider individual i and individual j in a network, and suppose that c_ij(s) = 0 for s = 0. However, after several discussions (say 5 issues), individual i has observed that individual j consistently has a high impact on discussions, i.e., ζ_j(s) is large. Then, individual i may form an interpersonal relationship such that c_ij(s) > 0 for s ≥ 6 (which implies that individual i begins to take into consideration the opinion of individual j). The two examples above are different from each other, but both equally provide motivation for dynamic topology. We assume that ∀ s, C(s) satisfies Assumption <ref>. Given that C(s) is dynamic, the opinion dynamics for each issue is then given by y(s,t+1) = W(s)y(s,t) where W(s) = X(s) + (I_n - X(s))C(s)which records the fact that C(s) is dynamic, in distinction to (<ref>). Precise details of the adjustments to the model arising from dynamic C are left for Section <ref>. We can thus formulate the key objective of this paper at this point as follows.To study the dynamic evolution (including convergence) of x(s) over a sequence of discussed issues by using the DeGroot model (<ref>) for opinion discussion, where W(s) is given in (<ref>), with the reflected self-appraisal mechanism (<ref>) used to update x(s). §.§ Contraction Analysis for Nonlinear SystemsIn this subsection, we present results on nonlinear contraction analysis in <cit.>, specifically results on discrete-time systems from Section 5 of <cit.>. This analysis will be used to obtain a fundamental convergence result for the original DeGroot-Friedkin model. The analysis framework that we build will enable an extension to the study of dynamic C.Consider a deterministic discrete-time system of the formx(k+1) = f_k(x(k), k)with n× 1 state vector x and n× 1 vector-valued function f. It is assumed that f is smooth, by which we mean that any required derivative or partial derivative exists, and is continuous. The associated virtual[The term “virtual” is taken from <cit.>; δx is a virtual, i.e. infinitesimal, displacement.] dynamics is δx(k+1) = ∂f_k/∂x(k)δx(k)Define the transformationδz(k) = Θ_k(x(k), k) δx(k)where Θ_k(x(k), k) ∈ℝ^n× n is uniformly nonsingular. More specifically, uniform nonsingularity means that there exist a real number κ > 0 and a matrix norm ‖·‖^' such that κ < ‖Θ_k(x(k), k) ‖^' < κ^-1 holds for all x and k. If the uniformly nonsingular condition holds, then exponential convergence of δz to 0_n implies, and is implied by, exponential convergence of δx to 0_n. The transformed virtual dynamics can be computed asδz(k+1) = F(k) δz(k)where F(k) =Θ_k+1(x(k+1), k+1) ∂f_k/∂x(k)Θ_k(x(k), k)^-1 is the transformed Jacobian. Given the discrete-time system (<ref>), a region of the state space is called a generalised contraction region with respect to the metric ‖x‖_Θ, 1 = ‖Θ_k(x(k), k)x(k) ‖_1 if in that region, ‖F(k) ‖_1 < 1 - η holds for all k, where η > 0 is an arbitrarily small constant. Note that here we are in fact working with the 1-norm metric in the variable space δz which in turn leads to a weighted 1-norm in the variable space δx. Here, the weighting matrix is Θ_k(x(k),k) and the weighted 1-norm is well defined over the entire state space because Θ is required to be uniformly nonsingular. Given the system (<ref>), consider a tube of constant radius with respect to the metric ‖x‖_Θ, 1, centred at a given trajectory of (<ref>). Any trajectory, which starts in this tube and is contained at all times in a generalised contraction region, remains in that tube and converges exponentially fast to the given trajectory as k →∞.Furthermore, global exponential convergence to the given trajectory is guaranteed if the whole state space is a generalised contraction region with respect to the metric ‖x‖_Θ, 1. Detailed proof of the theorem can be found in the seminal paper <cit.>, but with a focus on contraction in the Euclidean metric ‖x‖_Θ, 2 = ‖Θ_k(x(k), k)x(k) ‖_2, as opposed to the absolute sum metric. However, norms other than the Euclidean norm can be studied because the solutions of (<ref>) can be superimposed. This is because (<ref>) around a specific trajectory x(k) represents a linear time-varying system in δz coordinates (Section 3.7,<cit.>). In the paper, we require use of the 1-norm metric because the 2-norm metric does not deliver a convergence result. We provide a sketch of the proof here, modified for the 1-norm metric, and refer the reader to <cit.> for precise details. In a generalised contraction region, there holds‖δz(k+1) ‖_1 = ‖F(k) δz(k) ‖_1‖δz(k+1)‖_1 < (1-η) ‖δz(k)‖_1since ‖F(k)‖_1 < 1 - η holds for all k inside the generalised contraction region[We need η > 0 to eliminate the possibility that lim_k→∞‖F(k) ‖_1 = 1, which would not result in exponential convergence.]. This implies that lim_k→∞δz(k) = 0_n exponentially fast, which in turn implies that lim_k→∞δx(k) = 0_n exponentially fast due to uniform nonsingularity of Θ_k(x(k),k). The definition of δx then implies that any two infinitesimally close trajectories of (<ref>) converge to each other exponentially fast.The distance between two points, P_1 and P_2, with respect to the metric ‖·‖_Θ,1 is defined as the shortest path length between P_1 and P_2, i.e., the smallest path integral ∫_P_1^P_2‖δz‖_1 = ∫_P_1^P_2‖δx‖_Θ,1. A tube centred about a trajectory x_1(k) and with radius R is then defined as the set of all points whose distances to x_1(k) with respect to ‖·‖_Θ,1 are strictly less than R.Let x_2(k) ≠x_1(k) be any trajectory that starts inside this tube, separated from x_1(k) by a finite distance with respect to the metric ‖·‖_Θ,1. Suppose that the tube is contained at all times in a generalised contraction region. The fact that lim_k→∞‖δx(k) ‖_Θ,1 = 0 then implies that lim_k→∞∫_x_1(k)^x_2(k)‖δx (k) ‖_Θ,1 = 0 exponentially fast. That is, given the trajectories x_2(k) and x_1(k), separated by a finite distance with respect to the metric ‖·‖_Θ,1, x_2(k) converges to x_1(k) exponentially fast. Global convergence is obtained by setting R = ∞.If the contraction region is convex, then all trajectories converge exponentially fast to a unique trajectory.This immediately follows because any finite distance between two trajectories shrinks exponentially in the convex region. § CONTRACTION ANALYSIS FOR CONSTANT C In this section, before we address dynamic topology in Section <ref>, we derive a convergence result for the constant DeGroot-Friedkin model (<ref>) (i.e., C is constant for all s∈𝒮) using nonlinear contraction analysis methods as detailed in Section <ref>. The framework built using nonlinear contraction analysis is then applied in the next section to the DeGroot-Friedkin Model with dynamic topology.In order to obtain a convergence result, we make use of two properties of F( x(s)) established in <cit.>, but it must be noted that beyond these two properties, the analysis method is novel. The map F(x(s)) is continuous on Δ_n. If 𝒢 does not have star topology, then the following contraction-like property holds [pp. 390, Appendix F, <cit.>].Define the set 𝒜 = {x∈Δ_n : 1-r ≥ x_i ≥ 0, ∀ i ∈{1, , n}}, where r ≪ 1 is a small strictly positive scalar. Then, there exists a sufficiently small r such that x_i(s) ≤ 1-r implies x_i(s+1) < 1-r, for all i.By choosing r sufficiently small, it follows that x(s) ∈𝒜, ∀s > 0. In other words, F(𝒜) ⊂𝒜. We term this a contraction-like property so as not to confuse the reader with our main result; this property establishes a contraction only near the boundary of the simplex Δ_n.As a consequence of the above two properties, one can easily show, using Brouwer's Fixed Point Theorem (as shown in <cit.>), that there exists at least one fixed point x^* = F(x^*) in the convex compact set 𝒜. In <cit.>, a method involving multiple inequalities is used to show that the fixed point x^* is unique. This is done separately to the convergence proof. In the following proof, we are able to establish exponential convergence to a fixed point, and as a consequence of the method used, immediately prove that it is unique. Lastly, we present a third, easily verifiable property. If x(s_1) ∈Δ_n for some s_1 < ∞, then x(s) ∈int(Δ)_n for all s > s_1. Since x(s_1) ∈Δ_n, ∃ j : x_j(s_1) > 0. In addition, γ_i > 0, ∀ i because C is irreducible. It then follows that α(x(s_1)) > 0, and thus x_i(s_1+1) > 0,∀ i. Thus, x(s) ∈int(Δ_n) for all s > s_1.§.§ Fundamental Contraction Analysis We now state a fundamental convergence result of the system (<ref>). In the original work <cit.>, LaSalle's Invariance Principle for discrete-time systems was used to prove an asymptotic convergence result. The result in this paper strengthens this by establishing exponential convergence. In the following proof, when we say a property holds uniformly, we mean that the property holds for all x(s) ∈𝒜. Suppose that n≥ 3 and suppose further that C satisfies Assumption <ref> and the associated 𝒢 does not have star topology. The system (<ref>), with initial conditions x(0) ∈Δ_n, converges exponentially fast to a unique equilibrium point x^* ∈int(Δ_n).Consider any given initial condition x(0) ∈Δ_n. According to Property <ref>, x(s) ∈𝒜, ∀s > 0 for a sufficiently small r. It remains for us to study the system (<ref>) for x(s) ∈𝒜. Therefore, in the following analysis, we assume that s > 0. The proof heavily utilises the concepts and terminology of Section <ref>.Define the Jacobian of F(x(s)) at the s^th issue as J_F(x(s)) = {∂ F_i/∂ x_j (x(s))}. We obtain, for j = i,∂ F_i/∂ x_i(x(s)) = γ_i α( x(s)) /(1 - x_i(s))^2 - γ_i^2 α( x(s))^2 /(1-x_i(s))^3 = x_i(s+1) 1 - x_i(s+1)/1 - x_i(s)Similarly, we obtain, for j ≠ i, ∂ F_i/∂ x_j(x(s)) = - γ_i γ_j α( x(s))^2 /(1-x_i(s))(1 - x_j(s))^2 = - x_i(s+1) x_j(s+1) /1 - x_j(s)Accordingly, we have the following virtual dynamics δx(s+1) = J_F(x(s)) δx(s)Note that J_F(x(s)) is uniformly well defined and continuous because x_i(s) < 1-r, ∀i,s, thus enabling nonlinear contraction analysis to be used.Because there are scenarios where |λ_max(J_F(x(s)))| > 1 (as observed in our simulations), this implies that it is not always possible to find a matrix norm such that ‖J_F(x(s)) ‖ < 1 uniformly. We are therefore motivated to seek a contraction result via a coordinate transform. However, rather than study a transformation of x(s), we will study a transformation of the virtual displacement δx(s) as detailed in Section <ref>. Specifically, consider the following transformed virtual displacementδz(s) = Θ(x(s),s) δx(s)where Θ(x(s),s) = diag[1/(1-x_i(s))], i.e., Θ is a diagonal matrix with the i^th diagonal element being 1/(1-x_i(s)). It should be noted here that Θ(x(s),s) in this proof explicitly depends only on the argument x(s), unlike the general result presented in Section <ref>, and so we shall write it henceforth as Θ(x(s)).The contraction-like Property <ref> establishes that 1 > 1-x_i(s) > r > 0, which in turn implies that Θ(x(s)) is uniformly nonsingular, with λ_min(Θ(x(s))) > 1 and λ_max(Θ(x(s))) < 1/r. In other words, κ < ‖Θ(x(s)) ‖_1 < κ^-1 for some κ > 0, ∀ x(s) ∈𝒜, as required in Section <ref>. The transformed virtual dynamics is given byδz(s+1) = Θ(x(s+1)) J_F(x(s)) Θ(x(s))^-1δz(s)= H̅(x(s)) δz(s)where H̅(x(s)) = Θ(F(x(s))) J_F(x(s)) Θ(x(s))^-1 is the Jacobian associated with the transformed virtual dynamics. By denoting Φ̅(x(s)) = J_F(x(s)) Θ(x(s))^-1, one can write H̅(x(s)) = Θ(F(x(s))) Φ̅(x(s)).The matrix Φ̅(x(s)) is computed in (<ref>) below, and note that it can be considered as being solely dependent on x(s+1) = F(x(s)). Therefore, we let Φ(x(s+1)) = Φ̅(x(s)). For brevity, we drop the argument x(s+1) where there is no ambiguity and write simply Φ.Note that for each row i, ϕ_ii = x_i(s+1) (1-x_i(s+1)) andwhere ϕ_ij is the (i,j)^th element of Φ. From the fact that 0 < x_i(s) < 1-r, ∀i, it follows that all diagonal entries of Φ are uniformly strictly positive and all off-diagonal entries of Φ are uniformly strictly negative. Notice that Φ = Φ^⊤. Lastly, for any row i, there holds∑_j = 1^n ϕ_ij= x_i(s+1) [ 1 - x_i(s+1) - ∑_j = 1, j≠ i^n x_j(s+1) ] = 0because x_i(s+1) + ∑_j = 1, j≠ i^n x_j(s+1) = 1. In other words, Φ has row and column sums equal to 0. We thus conclude that Φ is the weighted Laplacian associated with an undirected, completely connected[By completely connected, we mean that there is an edge going from every node i to every other node j.] graph with edge weights which vary with x(s+1). The edge weights, -ϕ_ij, are uniformly lower bounded away from zero and upper bounded away from 1. This implies that 0 = λ_1(Φ) < λ_2(Φ) ≤≤λ_n(Φ) < ∞ <cit.>, i.e., Φ is uniformly positive semidefinite with a single eigenvalue at 0, with the associated eigenvector 1_n.Since Φ̅(x(s)) = Φ(x(s+1)) andΘ(x(s+1)) = Θ(F(x(s))), we note that H̅(x(s)) can be considered as depending solely on x(s+1). Letting H(x(s+1)) = H̅(x(s)), we complete the calculation H(x(s+1)) = Θ(x(s+1)) Φ(x(s+1)) to obtain that, for any i∈{1, …, n}, h_ii(x(s+1)) = x_i(s+1)h_ij(x(s+1)) = -x_i(s+1) x_j(s+1)/1-x_i(s+1),j ≠ iwhere h_ij(x(s+1)) is the (i,j)^th element of H(x(s+1)). For brevity, and when there is no risk of ambiguity, we drop the argument x(s+1) and simply write H. We note that the diagonal entries and off-diagonal entries of H(x(s+1)) are uniformly strictly positive and uniformly strictly negative, respectively. Notice that Φ1_n = 0_n ⇒H1_n = Θ(x(s+1)) Φ(x(s+1)) 1_n = 0_n. In other words, each row of H sums to zero. It follows that H is the weighted Laplacian matrix associated with a directed, completely connected graph with edge weights which vary with x(s+1). The edge weights, -h_ij, are uniformly upper bounded away from infinity and lower bounded away from zero. It is well known that if a directed graph contains a directed spanning tree, the associated Laplacian matrix has a single eigenvalue at 0, and all other eigenvalues have positive real parts <cit.>. With A = Θ(x(s+1)) uniformly positive definite and B = Φ(x(s+1)) uniformly positive semidefinite, it follows from Lemma <ref> that H = AB has a single zero eigenvalue and all other eigenvalues are strictly positive and real. By observing that trace(H) = ∑_i=1^n x_i(s+1) = 1 = ∑_i=1^n λ_i(H), we conclude that max_i (λ_i(H)) < 1 uniformly, since n≥ 3. We now establish the stronger result that ‖H‖_1 < 1 uniformly, which is required to obtain our stability result. See Remark <ref> below for more insight. Observe that ‖H‖_1 < 1 if and only if, for all i ∈{1, , n}, there holds ∑_j=1^n | h_ji| < 1, or equivalently,x_i + ∑_j= 1, j ≠ i^n ( x_i/1-x_j) x_j < 1and notice that we have dropped the time argument s+1 for brevity. From the fact that x_i > 0, ∀i (recall α(x(s)) > 0), and n ≥ 3, we obtain x_i + x_j < 1 ⇒ x_i/(1-x_j) < 1 for all j ≠ i. Combining this with the fact that x_i + ∑_j=1, j≠ i^n x_j = 1, we immediately verify that (<ref>) holds for all i.Because 𝒜 is bounded, this implies that ‖H‖_1 < 1 - η for some η > 0 and all x(s) ∈𝒜. Recalling the transformed virtual dynamics in (<ref>), we conclude that‖δz(s+1) ‖_1 = ‖H(x(s+1))δz(s) ‖_1 < (1-η) ‖δz(s) ‖_1We thus conclude that the transformed virtual displacement δz converges to zero exponentially fast. Recall the definition of δz(s) in (<ref>), and the fact that Θ(x(s)) is uniformly nonsingular. It then follows that δx(s) →0_n exponentially, ∀ x(s) ∈𝒜. We have thus established that 𝒜 is a generalised contraction region in accordance with Definition <ref>. Because 𝒜 is compact and convex, we conclude from Theorem <ref> and Corollary <ref> that all trajectories of x(s+1) = F(x(s)) with x(0) ∈Δ_n, converge exponentially to a single trajectory. According to Brouwer's Fixed Point Theorem, there is at least one fixed point x^* = F(x^*) ∈int(Δ_n), which is a trajectory of x(s+1) = F(x(s)). It then immediately follows that all trajectories of x(s+1) = F(x(s)) converge exponentially to a unique fixed point x^*∈int(Δ_n) (recall Property <ref>). The fixed point 𝐞_i of the map F(x) is unstable if γ_i < 1/2. If γ_i = 1/2, i.e., v_i is the centre node of a star graph, then the fixed point 𝐞_i is asymptotically stable, but is not exponentially stable. Without loss of generality, consider 𝐞_1. One can avoid F(x) in (<ref>) (and its Jacobian) misbehaving as x→𝐞_1 by multiplying α(x) by 1/(1-x_1) and by multiplying each entry γ_i/(1-x_i) by 1-x_1. One can then differentiate and obtain J_F(x) and evaluate it at x = 𝐞_1. Specifically, we obtain ∂ F_1/∂ x_1 = (1-γ_1)/γ_1, ∂ F_i/∂ x_1 = -γ_i/γ_1, ∂ F_i/∂ x_j = 0 for all i,j ≠ 1. Note that this immediately proves that F(x) is continuous at each vertex of the simplex Δ_n, greatly simplifying the proof in Lemma 2.2 of <cit.>.It follows that J_F(x) has a single eigenvalue at (1-γ_1)/γ_1 and all other eigenvalues are 0. If γ_1 < 1/2, then (1-γ_1)/γ_1 > 1 and the fixed point 𝐞_1 is unstable. If γ_1 = 1/2, then J_F(x) has a single eigenvalue at 1. A discrete-time counterpart to Theorem 4.15 in <cit.> (converse Lyapunov theorem) then rules out e_1 as an exponentially stable fixed point of F(x) (asymptotic stability was established in Lemma <ref>). We omit the proof of the discrete-time counterpart to Theorem 4.15 of <cit.> due to space limitations.When we first analyse H, we establish that ∀ i, λ_i(H) is real, nonnegative and less than 1. This tells us that the trajectories of (<ref>) about x^* are not oscillatory in nature. It also follows that the spectral radius of H, given by ρ(H), is strictly less than 1. In other words, H is Schur stable, and according to <cit.>, there exists a submultiplicative matrix norm ‖·‖^' such that ‖ H ‖^' < 1. However, we must recall that H(x(s+1)) is in fact a nonconstant matrix which changes over the trajectory of the system (<ref>). It is not immediately obvious, and in fact is not a consequence of the eigenvalue property, that a single submultiplicative matrix norm ‖·‖^'' exists such that ‖H‖^'' < 1 for all x∈𝒜. Existence of such a norm ‖·‖^'' would establish the desired stability property.In fact, the system δz(s+1) = H(x(s+1))δz(s), with H∈ℳ, ℳ = {H(x(s+1)) : x(s+1) ∈𝒜}, can be considered as a discrete-time linear switching system with state δz, and thus under arbitrary switching, the system is stable if and only if the joint spectral radius is less than 1, that is ρ(ℳ) = lim_k→∞max_i {‖H_i_1H_i_k‖^1/k : H_i ∈ℳ} < 1 <cit.>. This is of course a more restrictive condition than simply requiring that ρ(H_i) < 1. It is known that even when ℳ is finite, computing the joint spectral radius is NP-hard <cit.> and the question “ρ(ℳ) ≤ 1?" is an undecidable problem <cit.>. The problem is made even more difficult because in this paper, the set ℳ is not finite. We were therefore motivated to prove the stronger, and nontrivial, result that ‖H‖_1 < 1,∀ x∈𝒜 in order to bypass this issue.For the given definition of δz in (<ref>), we are able to obtain z_i(s+1) = - ln (1-x_i(s+1)) where z_i is the i^th element of z(x(s)). However, we did not present the above convergence arguments by firstly defining z(x(s)) and then seeking to study z(s+1) = G(z(s)). This is because our proof arose from considering x(s+1) = F(x(s)) using the nonlinear contraction ideas developed in <cit.>, which studied stability via differential concepts. It was through (<ref>) that we were able to integrate[Note that in general, the entries of Θ may have expressions which do not have analytic antiderivatives, and thus an analytic z(x(s),s) cannot always be found, but δz(s) can always be defined.] and obtain z_i = -ln(1-x_i). Moreover, it will be observed in the sequel that by conducting analysis on the transformed Jacobian using nonlinear contraction theory, we are able to straightforwardly deal with dynamic relative interaction matrices. It should be noted that <cit.> specifically discusses contraction in the Euclidean metric ‖δz‖_2 = ‖Θδx‖_2. A contraction region in the Euclidean metric requires λ_max(H(x(s))^⊤H(x(s))) < 1 to hold uniformly. This guarantees that δz(s)^⊤δz(s) = δx(s)^⊤M(x(s),s) δx(s) shrinks to zero exponentially fast, where M = Θ^⊤Θ. However, our simulations showed that λ_max(H(x(s))^⊤H(x(s))) was frequently and significantly greater than 1, which indicated that δz(s) defined in (<ref>) is not necessarily contracting in the Euclidean metric. This motivated us to consider contraction of δz(s) in the absolute sum metric, with appropriate adjustments to the proof presented in Section <ref>. Such an approach is alluded to in Section 3.7 of <cit.>. §.§ Extending the Contraction-like AnalysisIn this subsection, we provide a result which significantly expands Property <ref> by providing an explicit value for r and introduces a stronger contraction-like result, which is also applicable to social networks with star topology, unlike Property <ref> established in <cit.>.Suppose that n≥ 3, x(0) ∈Δ_n, and 𝒢 is strongly connected. Definer_j = 1-2γ_j/1-γ_jwhere γ_j is the j^th entry of γ^⊤.If 𝒢 does not have star topology, which implies, from Fact <ref>, that r_j > 0, then for any 0 < r ≤ r_j, there holdsx_j ≤ 1 - r ⇒ F_j(x) < 1-rwhere F_j(x) is the j^th entry of F(x). If 𝒢 has star topology with centre node j, which implies r_j = 0 in accordance with Fact <ref>, then ∄ r > 0 : r ≤ r_j, and thus the contraction-like property in (<ref>) does not hold. It has already been shown that for x(0) ∈Δ_n, there holds x(s) ∈int(Δ_n), i.e., x_i(s) > 0 for all i and s> 0. Consider then s > 0. Suppose that x_j ≤ 1 - r. Then, with r ≤ r_j, there holdsF_j(x) = α(x)γ_j/1 - x_j = 1/γ_j/1-x_j(1+∑_k≠ j^n γ_k/(1-x_k)/γ_j/(1-x_j))γ_j/1-x_j = 1/1+∑_k≠ j^n γ_k/(1-x_k)/γ_j/(1-x_j)≤1/1+∑_k≠ j^n r/γ_jγ_k/(1-x_k)because r ≤ 1 - x_j. From the fact that 1-x_k < 1, we obtain γ_k/(1-x_k) > γ_k, which in turn implies that the right hand side of (<ref>) obeys 1/1+∑_k≠ j^n r/γ_jγ_k/(1-x_k)< 1/1+∑_k≠ j^n γ_k r/γ_j = 1/1+(1-γ_j)r/γ_j = γ_j/γ_j+(1-γ_j)rwith the first equality obtained by noting that ∑_k≠ j^n γ_k = 1-γ_j according to the definition of γ. It follows from (<ref>) and (<ref>) that1-r- F_j(x) > 1 - r - γ_j/γ_j+(1-γ_j)r = γ_j + (1-γ_j)r - rγ_j - (1-γ_j)r^2 - γ_j/γ_j +(1-γ_j)r= r(1-2γ_j) - r^2(1-γ_j)/γ_j +(1-γ_j)r =r(1-γ_j)[1-2γ_j/1-γ_j - r] /γ_j +(1-γ_j)rSubstituting in r_j from (<ref>) then yields1-r- F_j(x) >r(1-γ_j)(r_j - r) /γ_j +(1-γ_j)r≥ 0because r_j ≥ r. In other words, 1-r > F_j(x), which completes the proof. This contraction-like result is now used to establish an upper bound on the social power of an individual at equilibrium. We stress here that, it appears that no general result exists for analytical computation of the vector x^* given γ^⊤. Results exist for some special cases, though, such as for doubly stochastic C and for 𝒢 with star topology<cit.>. While we do not provide an explicit equality relating x_i^* to γ_i, we do provide an explicit inequality. Suppose that n≥ 3 and x(0) ∈Δ_n. Suppose further that 𝒢 is strongly connected, and is not a star graph. Then, x_i^* < γ_i/(1-γ_i).Lemma <ref> establishes that, for any j∈{1, , n}, if x_j ≥ 1-r_j, then the map will always contract in that F_j(x(s)) < x_j. This is proved as follows. Suppose that x_j ≥ 1-r_j. Define r = 1-x_j, which satisfies r ≤ r_j as in Lemma <ref>. Then, we have F_j (x) < 1 - r = x_j. It is then straightforward to conclude that the map F(x) continues to contract towards the centre of the simplex Δ_n until x_i(s) < 1- r_i, ∀ i, where r_i is given by (<ref>). Suppose that x_j^* ≥ 1 - r_j = γ_j/(1-γ_j). According to the arguments in the paragraph above, we have F_j (x^*) < 1 - r_j ≤ x_j^*. On the other hand, the definition of x^* as a fixed point of F implies that x_j^* = F_j(x^*), which leads to a contradiction. Therefore, x_j^* < 1 - r_j = γ_j/(1-γ_j) as claimed. Note that this result is separate from the result of Theorem <ref>, which concluded exponential convergence to a unique fixed point, x^*. Here, we established an upper bound for the values of the entries of the unique fixed point x^*, i.e., the social power at equilibrium, given γ.We mention two specific conclusions following from Corollary <ref>. Firstly, suppose that 𝒢 has star topology with centre node v_1. Then, γ_1 = 0.5 according to Fact <ref>, and thus x_i does not contract. This is consistent with the findings in <cit.>, i.e., Lemma <ref>. Secondly, suppose that 𝒢 is strongly connected and that γ_i < 1/3, ∀i∈{1, , n}. Then, no individual in the social network will have more than half of the total social power at equilibrium, i.e., x_i^* < 1/2, ∀ i ∈{1, , n}. This second result is relevant as it provides a sufficient condition on the social network topology to ensure that no individual has a dominating presence in the opinion discussion. [Tightness of the Bound] The tightness of the bound x_i^* < γ_i/(1-γ_i) increases as γ_kdecreases ∀ k≠ i. This is in the sense that the ratio x_i^*(1-γ_i)/γ_i approaches 1 from below as γ_k decreases ∀ k≠ i. We draw this conclusion by noting that in order to obtain (<ref>), we make use of the inequality 1 - x_k < 1. From the fact that 1-x_k approaches 1 as x_k → 0, and because the contraction-like property of Lemma <ref> holds for x_k ≥γ_k/(1-γ_k), we conclude that the tightness of the bound x_i^* < γ_i/(1-γ_i) increases as γ_k decreases ∀ k ≠ i. If there is a single individual i with γ_i ≫γ_k,∀ k≠ i, we are in fact able to accurately estimate x_i^*. If γ_i ≥ 1/3, and n is large, then we are able to say, with reasonable confidence, that individual i will hold more than half of the total social power at equilibrium, i.e., x_i^* ≥ 0.5 is highly likely. §.§ Convergence Rate for a Set of C Matrices We now present a result on the convergence rate for a constant C which is in a subset of all possible C matrices. Suppose that C∈ℒ, where ℒ = {C∈ℝ^n× n : γ_i < 1/3, ∀ i, n ≥ 3}[According to Fact <ref>, ℒ does not contain any C whose associated graph has a star topology.] and γ_i is the i^th entry of the dominant left eigenvector γ^⊤ associated with C. Then, for the system (<ref>), with x(0) ∈Δ_n, there exists a finite s_1 such that, for all s ≥ s_1, there holds ‖J_F(x(s))‖_1 ≤ 2β - ϵ < 1 - η, where β = max_i γ_i/(1-γ_i) < 1/2 and ϵ, η are arbitrarily small positive constants. For s ≥ s_1, the system (<ref>) contracts to its unique equilibrium point x^* with a convergence rate obeying‖x^* - x(s+1) ‖_1 ≤ (2β - ϵ) ‖x^* - x(s) ‖_1From Corollary <ref>, we conclude that x_i^* < β_i where β_i = γ_i/(1-γ_i) < 1/2. Defining β = max_i β_i, we conclude that x_i^* ≤β - ϵ_1 for all i, where ϵ_1 is an arbitrarily small positive constant. Note that we already established an exponential convergence result in Theorem <ref> and an asymptotic result in Lemma <ref>, but that does not imply that x_i(s) ≤β - ϵ_1 for some finite s. However, we are able to conclude that there exists a strictly positive ϵ satisfyingϵ/2 < ϵ_1 and s_1 < ∞ such that x_i(s) ≤β - ϵ/2 for all s≥ s_1. The Jacobian J_F(x(s)) has column sum equal to 1. We obtain this fact by observing that, for any i,∂ F_i/∂ x_i+ ∑_j = 1, j≠ i^n ∂ F_j/∂ x_i = x_i(s+1) 1 - x_i(s+1)/1 - x_i(s) - ∑_j = 1, j≠ i^n x_i(s+1) x_j(s+1) /1 - x_i(s) = x_i(s+1)/1-x_i(s)[ 1 - x_i(s+1) - ∑_j = 1, j≠ i^n x_j(s+1) ] = 0because x_i(s+1) + ∑_j = 1, j≠ i^n x_j(s+1) = 1 by definition. Note also that the diagonal entries of the Jacobian are strictly positive and for s ≥ s_1, there holds ∂ F_i/∂ x_i ≤β - ϵ/2, ∀ i. This is because x_i(1-x_i) ≤ (β-ϵ/2)(1-β+ϵ/2)for x_i ≤β-ϵ/2 < 0.5 and 1/(1-x_i) ≤ 1/(1-β+ϵ/2). Combining the column sum property and the fact that the off-diagonal entries of the Jacobian are strictly negative, we conclude that for s ≥ s_1, there holds ‖J_F(x(s))‖_1 = 2 max_i ∂ F_i/∂ x_i ≤ 2β - ϵ < 1-η where η is an arbitrarily small positive constant.The quantity 2β - ϵ, which is a Lipschitz constant associated with the iteration, upper bounds the 1-norm of the untransformed Jacobian, and therefore is a lower bound on the convergence rate of the system. In fact, under the special assumption that γ_i < 1/3, ∀ i, we are able to work directly with the Jacobian J_F, as opposed to the transformed Jacobian H. It is in general much more difficult to compute an upper bound on ‖H‖_1 using γ and Corollary <ref> when ∃ i : γ_i ≥ 1/3.Note that ℒ includes many of the topologies likely to be encountered in social networks. Topologies for which γ_i ≥ 1/3 for some i will have an individual who holds more than half the social power at equilibrium. Such topologies are more reflective of autocracy-like or dictatorship-like networks, as opposed to a group of equal peers discussing their opinions.§ DYNAMIC RELATIVE INTERACTION TOPOLOGY In this section, we will explore the evolution of individual social power when the relative interaction topology is issue- or individual-driven, i.e., C(s) is a function of s. Motivations for dynamic C(s) have been discussed in detail in Sections <ref> and <ref>. This section will establish a theoretical result on the problem of dynamic C(s), conjectured and studied extensively with simulations in <cit.> but without any proofs. In our earlier work <cit.>, we provided analysis on the special case of periodically varying C(s), showing the existence of a periodic trajectory. This section provides complete analysis for general switching C(s) and extends the periodic result in <cit.> as a special case.Suppose that for a given social network with n ≥ 3 individuals, there is a finite set 𝒞 of P possible relative interaction matrices, defined as 𝒞 = {C_p ∈ℝ^n× n : p ∈𝒫} where 𝒫 = {1, 2, …, P}. We assume that Assumption <ref> holds for all C_p,p ∈𝒫. For simplicity, we assume that ∄ p such that the graph 𝒢_p associated with C_p has star topology. Let σ(s) : [0,∞) →𝒫 be a piecewise constant switching signal, determining the dynamic switching as C(s) = C_σ(s). Then, the DeGroot-Friedkin model with dynamic relative interaction matrices is given byx(s+1) =F_σ(s)( x(s))where the nonlinear map F_p( x(s)) for p ∈𝒫, is defined asF_p(x(s) ) = 𝐞_iifx(s) = 𝐞_i for anyi α_p ( x(s)) [ γ_p, 1/1-x_1(s); ⋮; γ_p, n/1-x_n(s) ] otherwisewhere α_p( x(s)) = 1/∑_i=1^n γ_p,i/1- x_i(s) and γ_p,i is the i^th entry of the dominant left eigenvector of C_p, . Note that the derivation for (<ref>) is a straightforward extension of the derivation (<ref>) using Lemma 2.2 in <cit.>, from constant C to C(s) = C_σ(s). We therefore omit this step.The system (<ref>) is a nonlinear discrete-time switching system, which makes analysis using the usual techniques for switched systems difficult. For arbitrary switching, one might typically seek to find a common Lyapunov function, i.e., one which would establish convergence for any fixed value of p ∈𝒫. This, however, appears to be difficult (if not impossible) for (<ref>). In the constant C case studied in <cit.>, the convergence result relied on 1) a Lyapunov function which was dependent on the unique equilibrium point x^*, and 2) LaSalle's Invariance Principle for discrete-time systems. Both 1) and 2) are invalid when analysing (<ref>). In the case of 1), the system (<ref>) does not have a unique equilibrium point x^* but rather a unique trajectory x^*(s) (as will be made clear in the sequel). In the case of 2), LaSalle's Invariance Principle is not applicable to general non-autonomous systems.§.§ Convergence for Arbitrary Switching We now state the main result of this section, the proof of which turns out to be fairly straightforward. This is a consequence of the analysis framework arising from the techniques used in the proof of Theorem <ref>. Note that in the theorem statement immediately below, a relaxation of the initial conditions is made; we no longer require ∑_i x_i(0) = 1. A social interpretation of this is given in Remark <ref> just following the theorem. Suppose that ∄ p such that C_p ∈𝒞 is associated with a star topology graph. Then, system (<ref>), with initial conditions 0≤ x_i(0) < 1,∀ i and ∃ j : x_j(0) > 0, converges exponentially fast to a unique trajectory x^*(s) ∈int(Δ_n). In other words, each individual i forgets its initial estimate of its own social power, x_i(0), at an exponential rate. For any given s, x^*(s+1) is determined solely by γ_σ(s). If x(0) = 𝐞_i for some i, then x(s) = 𝐞_i for all s. It is straightforward to conclude that Property <ref>, as stated at the beginning of Section <ref>, holds for each map F_p. With initial conditions x_i(0) < 1, the map F_σ(0)(x(s)) ≠𝐞_i for any i. We also easily verify that with these initial conditions, the matrix W(0) is row-stochastic, irreducible and aperiodic, which implies that the opinions converge for s =0 as in the constant C case. Because C(0) is irreducible, this implies that γ_σ(0),i > 0 for all i, and we conclude that α_σ(0)(x(0)) > 0 because ∃ j : x_j(0) > 0. We thus conclude that x(1) = F_σ(0)(x(0)) ≻ 0, i.e., for issue s = 1, every individual's social power/self-weight is strictly positive, and the sum of the weights is 1. Moreover, because C_p is irreducible ∀ p, this implies that for any p, there holds γ_p,i > 0 for all i. It follows that for s≥ 1, α_σ(s)(x(s)) > 0, which in turn guarantees that x(s+1) = F_σ(s)(x(s)) ≻ 0, i.e., x(s) ∈int(Δ_n) for all s > 0. This satisfies the requirements <cit.> on x(s) which ensures that ∀ s, W(s) is row-stochastic, irreducible, and aperiodic, which implies that opinions converge for every issue. If x(0) = 𝐞_i for some i, then (<ref>) leads to the conclusion that x(s) = 𝐞_i for all s. Denote the i^th entry of F_p by F_p,i. Regarding Property <ref>, stated at the beginning of Section <ref>, for each map F_p, define the set 𝒜_p(r_p) = {x∈Δ_n : 1-r_p ≥ x_i ≥ 0, ∀ i ∈{1, , n}}, where 0 < r_p ≪ 1 is sufficiently small such that x_i(s) ≤ 1-r_p for all i, which implies that F_p,i(x(s)) = x_i(s+1) < 1-r_p. Define 𝒜̅ = {x∈Δ_n : 1-r̅≥ x_i ≥ 0, ∀ i ∈{1, , n}} where r̅ = min_p r_p. Because F_p(𝒜̅) ⊂𝒜̅, it follows that ∪_p=1^P 𝒜_p ⊂𝒜̅, and that for the system (<ref>), for all s> 0, x(s) ∈𝒜̅.Denoting the Jacobian for the system (<ref>) at issue s as J_F_σ(s) = {∂ F_σ(s),i/∂ x_j}, we obtain∂ F_σ(s),i/∂ x_i(x(s)) = γ_σ(s),iα_σ(s)( x(s)) /(1 - x_i(s))^2 - [γ_σ(s),iα_σ(s)( x(s))]^2 /(1-x_i(s))^3 = x_i(s+1) 1 - x_i(s+1)/1 - x_i(s)Similarly, we obtain, for j ≠ i,∂ F_σ(s),i/∂ x_j(x(s)) = - γ_σ(s),iγ_σ(s),j[α_σ(s)( x(s))]^2 /(1-x_i(s))(1 - x_j(s))^2 = - x_i(s+1) x_j(s+1) /1 - x_j(s)Comparing to (<ref>) and (<ref>), we note that the Jacobian of the non-autonomous system (<ref>) with map (<ref>) is expressible in the same form as the Jacobian of the original system (<ref>) with map (<ref>). More precisely, it can be expressed in a form which is dependent on the trajectory of the system, and not explicitly dependent on s. Using the same transformation of δz given in (<ref>) with the same Θ(x(s)), we obtain the exact same transformed virtual dynamics (<ref>), expressed as δz(s+1) = H(x(s+1))δz(s)and it was shown in the proof of Theorem <ref> that, for some arbitrarily small η > 0, there holds ‖H‖_1 < 1 - η for all x(s) ∈𝒜̅, independent of p∈𝒫. It follows that δx(s) →0_n exponentially fast for all x(s) ∈𝒜̅. We thus conclude that 𝒜̅ is a generalised contraction region. Again, because 𝒜̅ is compact and convex, it follows from Theorem <ref> and Corollary <ref> that all trajectories of x(s+1) = F_σ(s)(x(s)) converge exponentially to a single trajectory, which we denote x^*(s). We established earlier that x^*(s) ∈int(Δ_n). Exponential convergence to a single unique trajectory can be considered from another point of view as the system (<ref>) forgetting its initial conditions at an exponential rate. Note also that in one sense, F_σ(s) in (<ref>) is parametrised by γ_σ(s). We conclude from these two points that the unique trajectory x^*(s) is such that x^*(s+1) depends only on γ_σ(s).Finally, following the same analysis as in [pp.393, <cit.>], one can show that lim_s→∞ζ(s) = x^*(s) and lim_s→∞W(x(s)) =X^*(s) + (I_n -X^*(s))C(s) =W(x^*(s)).The above result implies that the system (<ref>), with initial conditions satisfying 0≤ x_i(0) < 1,∀ i and ∃ j : x_j(0) > 0, converges to a unique trajectory x^*(s) as s→∞. For convenience in future discussions and presentation of results, we shall call this the unique limiting trajectory of (<ref>). This is a limiting trajectory in the sense that lim_s→∞x(s) = x^*(s). Theorem <ref> contains a mild relaxation of the initial conditions of the original DeGroot-Friedkin model, and provides a more reasonable interpretation from a social context. One can consider x_i(0) as individual i's estimate of its individual social power (or perceived social power) in the group when the social network is first formed and before discussion begins on issue s = 0. The original DeGroot-Friedkin model requires x(0) ∈Δ_n to avoid an autocratic system (an autocratic system is where x(s) = 𝐞_i for some i, i.e., an individual holds all the social power). However, this is unrealistic because one cannot expect individuals to have estimates such that ∑_i x_i(0) = 1. On the other hand, we do show that the unique limiting trajectory satisfies further, as already commented, ∑ x_i(1)=1, and then easily ∑ x_i(k)=1,∀ k > 1 and x^*(s) ∈int(Δ_n), i.e., x_i^*(s) > 0,∀ i and ∑_i x_i^*(s) = 1,∀ s. We therefore show that, as long as no individual i estimates its social power to be autocratic (x_i(0) = 1) and at least one individual estimates its social power to be strictly positive (∃ j : x_j(0) > 0), then by sequential discussion of issues, every individual forgets its initial estimate of its individual social power at an exponential rate. This occurs even for dynamic relative interaction topologies.§.§ Contraction-Like Property with Arbitrary SwitchingWe now extend Lemma <ref>, Corollary <ref> and Lemma <ref> to the case of dynamic relative interaction matrices.For the system (<ref>), with initial conditions 0≤ x_i(0) < 1,∀ i and for at least one k, x_k(0) > 0, define r̅_j = 1 - 2γ̅_j/1 - γ̅_j, j ∈{1, …, n}where γ̅_j = max_p∈𝒫γ_p,j and γ_p,j is the j^th entry of γ_p. Then, for any 0 < r ≤r̅_j and p ∈𝒫, there holdsx_j ≤ 1 - r ⇒ F_p,j(x) < 1-rwhere F_p,j(x) is the j^th entry of F_p(x). The lemma is proved by straightforwardly checking that, for the given definition of r̅_j, the result in Lemma <ref> holds separately for every map F_p,p ∈𝒫. In other words, for all i,p, x_i(s) ≤ 1 - r ⇒ F_p,i(x(s)) < 1 - r ,∀ r≤r̅_i.For the system (<ref>), with initial conditions 0≤ x_i(0) < 1,∀ i and for at least one j, x_j(0) > 0, there holds x_i^*(s) ≤γ̅_i/(1-γ̅_i),∀ s, where γ̅_j = max_p∈𝒫γ_p,j and x_i^*(s) is the i^th entry of the unique limiting trajectory x^*(s). The proof is a straightforward extension of the proof of Corollary <ref>, and is therefore not included here.For all p∈𝒫, suppose that C_p∈ℒ where ℒ = {C_p ∈ℝ^n× n : γ_p,i < 1/3, ∀ i} and γ_p,i is the i^th entry of the dominant left eigenvector γ_p associated with C_p. Then, there exists a finite s_1 such that, for all s ≥ s_1, there holds ‖J_F_σ(s)(x(s))‖_1 ≤ 2β̅ - ϵ < 1 - η, where β̅ = max_p max_i γ_p,i/(1-γ_p,i) < 1/2 and ϵ, η are arbitrarily small positive constants. For s ≥ s_1, the system (<ref>) contracts to its unique limiting trajectory x^*(s) with a convergence rate obeying ‖x^*(s) - x(s+1) ‖_1 ≤ (2β̅ - ϵ) ‖x^*(s) - x(s) ‖_1Again, the proof is a straightforward extension of the proof of Lemma <ref>, by recalling from the proof of Theorem <ref> that the Jacobian takes on the same form. We thus omit the minor details.The exponential forgetting of initial conditions is a powerful notion. It implies that sequential discussion of topics combined with reflected self-appraisal is a method of “self-regulation" for social networks, even in the presence of dynamic topology. Consider an individual i who is extremely arrogant, e.g. x_i(0) = 0.99. However, individual i is not likeable and others tend to not trust its opinions on any issue, e.g. c_ji(s) ≪ 1 ,∀ j,s. Then, γ_i(s) ≪ 1 because γ(s)^⊤ = γ(s)^⊤C(s) implies γ_i(s) = ∑_j≠ iγ_j(s) c_ji(s). Then, according to Corollary <ref>, x_i^*(s) ≪ 1, and individual i exponentially loses its social power. An interesting future extension would be to expand on the reflected self-appraisal by modelling individual personality. For example, we can consider x_i(s+1) = ϕ_i(ζ_i(s)) where ϕ_i(·) may capture arrogance or humility. We also conclude that, for large s, any individual wanting to have an impact on the discussion of topic s+1 should focus on ensuring it has a large impact on discussion of the prior topic s. This concept can be applied to e.g. <cit.>. §.§ Periodically Varying TopologyIn this subsection, we investigate an interesting, special case of issue-dependent topology, that of periodically varying C(s) which satisfies Assumption <ref> for all s. Preliminary analysis and results were presented in <cit.> without convergence proofs. We now provide a complete analysis by utilising Theorem <ref>. Motivation for Periodic Variations: Consider Example 1 in Section <ref> of a government cabinet that meets to discuss the issues of defence, economic growth, social security programs and foreign policy. Since these issues are vital to the smooth running of the country, we expect the issues to be discussed regularly and repeatedly. Regular meetings on the same set of issues for decision making/governance/management of a country or company then points to periodically varying C(s), i.e., social networks with periodic topology.The system (<ref>), with periodically switching C(s), can be described by a switching signal σ(s) of the form σ(0) = P, and for s≥ 1, σ(Pq + p) = p,[Note that any given s∈𝒮 can be uniquely expressed by a given fixed positive integer P, a nonnegative integer q, and positive p∈𝒫, as shown.] where P < ∞ is the period length, p ∈𝒫 = {1, 2, , P} and q ∈ℤ_≥ 0 is any nonnegative integer. Note that in general, C_i ≠C_j, ∀ i,j ∈𝒫 and i≠ j. Theorem <ref> immediately allows us to conclude that system (<ref>) with periodic switching converges exponentially fast to its unique limiting trajectory x^*(s). This subsection's key contribution is to use a transformation to obtain additional, useful information on the limiting trajectory.For simplicity, we shall begin analysis by assuming that 𝒫 = {1,2}, i.e., there are two different C matrices, and the switching is of period 2. It will become apparent in the sequel that analysis for 𝒫 = {1, 2, , P}, with arbitrarily large but finite P, is a simple recursive extension on the analysis for 𝒫 = {1,2}. For the two matrices case, we obtainx(s+1) =F_1( x(s))ifsis oddF_2( x(s))ifsis evenWe now seek to transform the periodic system into a time-invariant system. Define a new state y ∈ℝ^2n (note that this is not the opinion state given in Section <ref>) asy(2q) =[ y_1(2q); y_2(2q) ] = [ x(2q); x(2q+1) ]and study the evolution of y (2q) for q ∈{0, 1, 2, }. Note thaty(2(q+1)) =[ y_1(2(q+1)); y_2(2(q+1)) ] = [ x(2(q+1)); x(2(q+1)+1) ] In view of the fact that x(2(q+1)) =F_1( x(2q+1)) and x(2(q+1)+1) =F_2( x(2q+2)) for any q ∈{0,1,2,}, we obtainy(2(q+1)) =[ F_1( x(2q+1)); F_2( x(2q+2)) ]Similarly, notice that x(2q+1) =F_2( x(2q)) and x(2q+2) =F_1( x(2q+1)) for any q ∈{0,1,2,}. From this, for q ∈{0,1,2,}, we obtain thaty(2(q+1)) = [ F_1( F_2( y_1(2q))); F_2( F_1(y_2(2q) )) ]= [ G_1 ( y_1(2q)); G_2 ( y_2(2q)) ]for the time-invariant nonlinear composition functions G_1 = F_1 ∘F_2 and G_2 = F_2 ∘F_1. We can thus express the periodic system (<ref>) as the nonlinear time-invariant systemy(2q+2) = G̅ ( y(2q))where G̅ = [G_1^⊤, G_2^⊤]^⊤. The system (<ref>), with initial conditions 0≤ x_i(0) < 1,∀ i and ∃ j : x_j(0) > 0, converges exponentially fast to a unique limiting trajectory x^*(s) ∈int(Δ_n). This trajectory is a periodic sequence, which obeys x^*(s) =y_1^*ifsis oddy_2^*ifsis evenwhere y_1^* ∈int(Δ_n) and y_2^* ∈int(Δ_n) are the unique fixed points of G_1 and G_2, respectively. As mentioned above, one can immediately apply Theorem <ref> to show lim_s→∞x(s) = x^*(s). This proof therefore focuses on using the time-invariant transformation to show that x^*(s) has the properties described in the theorem statement.Part 1: In this part, we prove that the map G_i, i = 1,2 has at least one fixed point. Firstly, we proved in Theorem <ref> that the system (<ref>), with initial conditions 0≤ x_i(0) < 1,∀ i and for at least one j, x_j(0) > 0, will have x(s) ∈int(Δ_n) for all s > 0, which implies that x^*(s) ∈int(Δ_n). Let p ∈{1,2}. The fact that F_p : Δ_n →Δ_n is continuous on Δ_n is straightforward since F_p is an analytic function in Δ_n. Lemma 2.2 in <cit.> shows that F_p is Lipschitz continuous about 𝐞_i with Lipschitz constant 2√(2)/γ_i,p. It is then straightforward to verify that the composition of two continuous functions, G_1 =F_1 ∘ F_2 : Δ_n →Δ_n is continuous. Similarly, G_2 =F_2 ∘ F_1 : Δ_n →Δ_n is also continuous.The proof of Theorem <ref> also showed that for all p, F_p ∈𝒜̅ where 𝒜̅ = {x∈Δ_n : 1-r̅≥ x_i ≥ 0, ∀ i ∈{1, , n}} and r̅ is some small strictly positive constant. For the system (<ref>) with p = 1,2, it follows that F_1(𝒜̅) ⊂𝒜̅⇒ F_2( F_1(𝒜̅)) ⊂𝒜̅, which implies that G_1(𝒜̅) ⊂𝒜̅. Similarly, G_2(𝒜̅)⊂𝒜̅. Brouwer's fixed-point theorem then implies that there exists at least one fixed point y_1^* ∈𝒜̅ such that y_1^* =G_1( y_1^*) (respectively y_2^* ∈𝒜̅ such that y_2^* =G_2( y_2^*)) because G_1 (respectively G_2) is a continuous function on the compact, convex set 𝒜. The arguments in Part 1 appeared in <cit.>, but proofs were omitted due to space limitations.Part 2: In this part, we prove that the unique limiting trajectory of (<ref>) obeys (<ref>). Let y_1^* be a fixed point of G_1. We will show below that y_1^* is in fact unique. Observe that y_1^* =F_2(F_1 ( y_1^*) ). Define y_2^* =F_1( y_1^*). We thus have y_1^* =F_2( y_2^*). Observe that F_1 ( y_1^*)=F_1 ( F_2 ( y_2^*)), which implies that y_2^* =F_1 ( F_2( y_2^*)) =G_2 ( y_2^*). In other words, y_2^* is a fixed point of G_2 (but at this stage we have not yet proved its uniqueness). We now prove uniqueness. Theorem <ref> allows us to conclude that all trajectories of (<ref>) converge exponentially fast to a unique limiting trajectory x^*(s) ∈int(Δ_n). It follows, from (<ref>) and the definition of y(2q), that for all s≥ 0, (<ref>) is a trajectory of the system (<ref>); the critical point here is that (<ref>) holds for all s. Combining these arguments, it is clear that (<ref>) is precisely the unique limiting trajectory.Lastly, we show that y_1^* and y_2^* are the unique fixed point of G_1 and G_2, respectively. To this end, suppose that, to the contrary, at least one of y_1^* and y_2^* is not unique. Without loss of generality, suppose in particular that y_1^'≠y_1^* is any other fixed point of G_1. Then, y_2^'=F_1(y_1^') is a fixed point of G_2, andx(s) =y_1^' ifsis oddy_2^' ifsis evenis a trajectory of (<ref>) that holds for all s≥ 0, and is different from the trajectory (<ref>) because y_1^'≠y_1^*. On the other hand, Theorem <ref> implies that all trajectories of (<ref>) converge exponentially fast to a unique limiting trajectory, which is a contradiction. Thus, y_1^* and y_2^* are the unique fixed point of G_1 and G_2, respectively, and (<ref>) converges exponentially fast to the unique limiting trajectory (<ref>). We now provide the generalisation to periodically switching topology C(s) = C_σ(s), where σ(s) is of the form σ(0) = P, and for s≥ 1, σ(Pq + p) = p. Here, 2 ≤ P < ∞, p ∈𝒫 = {1, 2, , P} and q ∈ℤ_≥ 0. The periodic DeGroot-Friedkin model is described by x(s+1) =F_P( x(s))fors = 0 F_p( x(s = Pq + p )) for alls ≥ 1 A transformation of (<ref>) to a time-invariant system can be achieved by following a procedure similar to the one detailed for the case p=2. A new state variable y ∈ℝ^Pn is defined as y(Pq) =[ y_1(Pq); y_2(Pq); ⋮; y_P(Pq) ] = [ x(Pq);x(P q +1 ); ⋮; x(P q + P -1) ]and we study the evolution of y(Pq) for q ∈{0,1, }. It follows that y_p(P (q+1)) =x( P(q +1) + p - 1 ) ,∀p ∈𝒫Following the logic in the 2 period case, but with the precise steps omitted, we obtainy(P (q+1))= [ F_P-1 (F_P-2 ( ( F_P ( y_1 ( Pq ) ) ) ) ); F_P (F_P-1 ( ( F_1 ( y_2 ( Pq ) ) ) ) ); ⋮; F_P-2 (F_P-1 ( ( F_P ( y_P-1 ( Pq ) ) ) ) ) ] = G̅( y(Pq))where G̅( y)= [ G_1 ( y_1 ), G_2 (y_2 ),, G_P (y_P)]^⊤. This leads to the following generalisation of Theorem <ref>. The system (<ref>), with initial conditions 0≤ x_i(0) < 1,∀ i and for at least one j, x_j(0) > 0, converges exponentially fast to a unique limiting trajectory x^*(s) ∈int(Δ_n). This trajectory is a periodic sequence, which for any q ∈ℤ_≥ 0, obeys x^*(Pq + p - 1) =y_p^*,for allp∈{1, 2, …, P}where y_p^* ∈int(Δ_n) is the unique fixed point of G_p. The proof is obtained by recursively applying the same techniques used in the proof of Theorem <ref>. We therefore omit the details. Note that Lemmas <ref> and <ref> and Corollary <ref> are all applicable to the periodic system (<ref>) because (<ref>) is just a special case of the general switching system (<ref>). §.§ Convergence to a Single Point We conclude Section <ref> by showing that if the set 𝒞 of possible switching matrices has a special property, then the unique limiting trajectory x^*(s) ∈int(Δ_n) is in fact a stationary point.Define 𝒦(γ) = {C_p ∈ℝ^n× n : γ_p = γ, ∀ p ∈𝒫 = {1, 2, , P}} where P is finite. In other words, 𝒦(γ) is a set of C matrices which all have the same dominant left eigenvector γ^⊤. Perhaps the most well-known set is 𝒦(1_n/n), i.e., the set of n× n doubly-stochastic C matrices.Suppose that C(s) = C_σ(s)∈𝒦(γ). Then, the system (<ref>), with initial conditions 0≤ x_i(0) < 1,∀ i and for at least one j, x_j(0) > 0, converges exponentially fast to a unique point x^* ∈int(Δ_n). There holds x^*_i < x^*_j if and only if γ_i < γ_j, for any i,j, where γ_i and x^*_i are the i^th entry of the dominant left eigenvector γ and x^*, respectively. There holds x^*_i = x^*_j if and only if γ_i = γ_j. The map F_σ(s) is parametrised simply by the vector γ_σ(s). Under the stated condition of C(s) = C_σ(s)∈𝒦(γ), the map F_σ(s) is time-invariant. The result in Theorem <ref> is then used to complete the proof. § SIMULATIONSIn this section, we provide a short simulation for a network with 6 individuals to illustrate our key results. The set of topologies is given as 𝒞 = {C_1, , C_5}, i.e., 𝒫 = {1, 2, , 5}. The switching signal σ(s) is generated such that for any given s, there is equal probability that σ(s) = p,∀ p ∈𝒫. The precise numerical forms of C_p given in the appendix.Figure <ref> shows the evolution of individual social power over a sequence of issues for the system as described in the above paragraph, initialised from a set of initial conditions, x(0).Figure <ref> shows the system with a different set of initial conditions x(0) ≠x(0). Notice that individuals 1,2,3 have large perceived social power x_i(0) = 0.95, while individuals 4,5,6 have x_i(0) = 0. In the other set of initial conditions, x_i(0) is large for i = 4,6. Through sequential discussion and reflected self-appraisal, it is clear that the initial conditions are exponentially forgotten and both plots show convergence to the same unique limiting trajectory x^*(s) by about s = 10. This is shown in Fig. <ref>, which displays the individual social powers of selected individuals 1,3 and 6. The solid lines correspond to initial condition set x(0) while the dotted lines correspond to initial condition set x(0). Figure <ref> shows the exponential convergence of the dotted and solid trajectories. Note that for individual 4, its social power is always strictly positive, although for several issues, x_4(s) is close to 0.For each individual, with γ̅_i = max_p∈𝒫γ_p,i, we computed γ̅_1 = 0.4737, γ̅_2 = 0.2371, γ̅_3 = 0.2439, γ̅_4 = 0.2439, γ̅_5 = 0.2439, γ̅_6 = 0.2392. Note that ∑_i γ̅_i ≠ 1 in general due to the definition of γ̅_i. According to Corollary <ref>, we have x^*(s) ≼ [0.9, 0.3108, 0.3226, 0.3226, 0.3226,0.3144]. This is precisely what is shown in Figs. <ref> and <ref>. Since only γ̅_1 > 1/3, we observe that after the first 10 or so issues, only x_1^*(s) > 0.5, i.e., only individual 1 can hold more than half the social power in the limit, under arbitrary switching. Simulations for periodically-varying topology are available in <cit.>. § CONCLUSIONIn this paper, we have presented several novel results on the DeGroot-Friedkin model. For the original model, convergence to the unique equilibrium point has been shown to be exponentially fast. The nonlinear contraction analysis framework allowed for a straightforward extension to dynamic topologies. The key conclusion of this paper is that, according to the DeGroot-Friedkin model, sequential opinion discussion, combined with reflected self-appraisal between any two successive issues, removes perceived (initial) individual social power at an exponential rate. True social power in the limit is determined by the network topology, i.e., interpersonal relationships and their strengths. An upper bound on each individual's limiting social power is computable, depending only on the network topology. A number of questions remain. Firstly, we aim to relax the graph topology assumption from strongly connected (i.e., the relative interaction matrix is irreducible) to containing a directed spanning tree (i.e., the relative interaction matrix is reducible). Moreover, one may consider a graph whose union over a set of issues is strongly connected, but for each issue, the graph is not strongly connected. Stubborn individuals (i.e., the Friedkin-Johnsen model) should be incorporated; only partial results are currently available <cit.>. Effects of noise and other external inputs should be studied, as well as the concept of personality affecting the reflected self-appraisal mechanism (as mentioned in Remark <ref>).The relative interaction matrices used in the simulation are given byC_1= [ 0 0 0 0 0 1; 1 0 0 0 0 0; 0 1 0 0 0 0; 0 0 1 0 0 0; 0 0 0 1 0 0; 0 0 0 0 1 0; ] C_2= [ 0 0 0 0 1 0; 0.8 0 0 0 0 0.2; 0 0.1 0 0 0 0.9; 0 0 1 0 0 0; 0 0 0 1 0 0; 0 0 0 0 1 0; ] C_3= [0000.200.8;0.300.7000;00010.50;010000; 0.7500 0.2500;000010;] C_4= [0000 0.85 0.15;100000;00.700.300;000.500.50;000.9000.1;010000;] C_5= [ 0 0.5 0 0 0 0.5; 0.9 0 0.1 0 0 0; 0.9 0 0 0 0 0.1; 0.9 0.1 0 0 0 0; 0.9 0 0 0.1 0 0; 0.9 0 0 0 0.1 0; ]§ ACKNOWLEDGEMENTThe work of Ye, Anderson, and Yu was supported by the Australian Research Council (ARC) under grantsand , and by Data61-CSIRO. The work of Liu and Başar was supported in part by Office of Naval Research (ONR) MURI Grant N00014-16-1-2710, and in part by NSF under grant CCF 11-11342. IEEEtran [ < g r a p h i c s > ]Mengbin Ye was born in Guangzhou, China. He received the B.E. degree (with First Class Honours) in mechanical engineering from the University of Auckland, Auckland, New Zealand. He is currently pursuing the Ph.D. degree in control engineering at the Australian National University, Canberra, Australia.His current research interests include opinion dynamics and social networks, consensus and synchronisation of Euler-Lagrange systems, and localisation using bearing measurements.[ < g r a p h i c s > ]Ji Liu received the B.S. degree in information engineering from Shanghai Jiao Tong University, Shanghai, China, in 2006, and the Ph.D. degree in electrical engineering from Yale University,New Haven, CT, USA, in 2013. He is currently a Postdoctoral Research Associate at the Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, Urbana, IL, USA. His current research interests include distributed control and computation, multi-agent systems, social networks, epidemic networks, and power networks. [ < g r a p h i c s > ]Brian D.O. Anderson (M'66-SM'74-F'75-LF'07) was born in Sydney, Australia. He received the B.Sc. degree in pure mathematics in 1962, and B.E. in electrical engineering in 1964, from the Sydney University, Sydney, Australia, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, USA, in 1966.He is an Emeritus Professor at the Australian National University, and a Distinguished Researcher in Data61-CSIRO (previously NICTA) and a Distinguished Professor at Hangzhou Dianzi University. His awards include the IEEE Control Systems Award of 1997, the 2001 IEEE James H Mulligan, Jr Education Medal, and the Bode Prize of the IEEE Control System Society in 1992, as well as several IEEE and other best paper prizes. He is a Fellow of the Australian Academy of Science, the Australian Academy of Technological Sciences and Engineering, the Royal Society, and a foreign member of the US National Academy of Engineering. He holds honorary doctorates from a number of universities, including Université Catholique de Louvain, Belgium, and ETH, Zürich. He is a past president of the International Federation of Automatic Control and the Australian Academy of Science. His current research interests are in distributed control, sensor networks and econometric modelling. [ < g r a p h i c s > ]Changbin Yu received the B.Eng (Hon 1) degree from Nanyang Technological University, Singapore in 2004 and the Ph.D. degree from the Australian National University, Australia, in 2008. Since then he has been a faculty member at the Australian National University and subsequently holding various positions including a specially appointed professorship at Hangzhou Dianzi University.He had won a competitive Australian Post-doctoral Fellowship (APD) in 2007 and a prestigious ARC Queen Elizabeth II Fellowship (QEII) in 2010. He was also a recipient of Australian Government Endeavour Asia Award (2005) and Endeavour Executive Award (2015), Chinese Government Outstanding Overseas Students Award (2006), Asian Journal of Control Best Paper Award (2006–2009), etc. His current research interests include control of autonomous aerial vehicles, multi-agent systems and human–robot interactions. He is a Fellow of Institute of Engineers Australia, a Senior Member of IEEE and a member of IFAC Technical Committee on Networked Systems. He served as a subject editor for International Journal of Robust and Nonlinear Control and was an associate editor for System & Control Letters and IET Control Theory & Applications.[ < g r a p h i c s > ]Tamer Başar (S'71-M'73-SM'79-F'83-LF'13) is with the University of Illinois at Urbana-Champaign (UIUC), where he holds the academic positions ofSwanlund Endowed Chair; Center for Advanced Study Professor ofElectrical and Computer Engineering;Research Professor at the Coordinated Science Laboratory; and Research Professorat the Information Trust Institute.He is also the Director of the Center for Advanced Study. He received B.S.E.E. from Robert College, Istanbul, and M.S., M.Phil, and Ph.D. from Yale University. He is a member of the US National Academy of Engineering,the European Academy of Sciences, and Fellow of IEEE, IFAC and SIAM, and has served as president of IEEE CSS, ISDG, and AACC. He has received several awards and recognitions over the years, includingthe IEEE Control Systems Award,the highest awards of IEEE CSS, IFAC, AACC, and ISDG, and a number of international honorary doctorates and professorships. He has over 800 publications in systems, control, communications, networks, and dynamic games, including books on non-cooperative dynamic game theory, robust control, network security, wireless and communication networks, and stochastic networked control. He was the Editor-in-Chief of Automatica between 2004 and 2014, and is currentlyeditor of several book series. His current research interests include stochastic teams, games, and networks; security; and cyber-physical systems. | http://arxiv.org/abs/1705.09756v2 | {
"authors": [
"Mengbin Ye",
"Ji Liu",
"Brian D. O. Anderson",
"Changbin Yu",
"Tamer Başar"
],
"categories": [
"cs.SI",
"cs.MA",
"cs.SY",
"math.DS"
],
"primary_category": "cs.SI",
"published": "20170527022623",
"title": "Evolution of Social Power in Social Networks with Dynamic Topology"
} |
Deciphering the nonlocal entanglement entropy of fracton topological orders Yuan-Ming Lu December 30, 2023 =========================================================================== Most real-world document collections involve various types of metadata, such as author, source, and date, and yet the most commonly-used approaches to modeling text corpora ignore this information. While specialized models have been developed for particular applications, few are widely used in practice, as customization typically requires derivation of a custom inference algorithm. In this paper,we build on recent advances in variational inference methods and propose a general neural framework, based on topic models, to enable flexible incorporation of metadata and allow for rapid exploration of alternative models. Our approach achieves strong performance, with a manageable tradeoff between perplexity, coherence, and sparsity.Finally, we demonstrate the potential of our framework through an exploration of a corpus of articles about US immigration.§ INTRODUCTION Topic models comprise a family of methods for uncovering latent structure in text corpora, and are widely used tools in the digital humanities, political science, and other related fields <cit.>. Latent Dirichlet allocation (LDA; ) is often used when there is no prior knowledge about a corpus. In the real world, however, most documents have non-textual attributes such as author <cit.>, timestamp <cit.>, rating <cit.>, or ideology <cit.>, which we refer to as metadata. Many customizations of LDA have been developed to incorporate document metadata. Two models of note are supervised LDA (SLDA; ), which jointly models words and labels (e.g., ratings) as being generated from a latent representation, and sparse additive generative models (SAGE; ), which assumes that observed covariates (e.g., author ideology) have a sparse effect onthe relative probabilities of words given topics. The structural topic model (STM; ), which adds correlations between topics to SAGE, is also widely used, but like SAGE it is limited in the types of metadata it can efficiently make use of, and how that metadata is used.Note that in this work we will distinguish labels (metadata that are generated jointly with words from latent topic representations)fromcovariates (observed metadata that influence the distribution of labels and words). The ability to create variations of LDA such as those listed above hasbeen limited by the expertise needed to develop custom inference algorithms for each model. As a result, it is rare to see such variations being widely used in practice.In this work, we take advantage of recent advances in variational methods<cit.> to facilitate approximate Bayesian inference without requiring model-specific derivations, andpropose a general neural framework for topic models with metadata, .[Sparse Contextual Hidden and Observed Language AutoencodeR. ]combines the abilities of SAGE and SLDA, and allows for easy exploration of the following options for customization: * Covariates: as in SAGE and STM, we incorporate explicit deviations for observed covariates, as well as effects for interactions with topics.* Supervision: as in SLDA, we can use metadata as labels to help infer topics that are relevant in predicting those labels.* Rich encoder network: we use the encoding network of a variational autoencoder (VAE) to incorporate additional prior knowledge in the form of word embeddings, and/or to provide interpretable embeddings of covariates.* Sparsity: as in SAGE, a sparsity-inducing prior can be used to encourage more interpretable topics, represented as sparse deviations from a background log-frequency. We begin with the necessary background and motivation (<ref>), and then describe our basic framework and its extensions (<ref>), followed by a series of experiments (<ref>). In an unsupervised setting, we can customize the model to trade off between perplexity, coherence, and sparsity, with improved coherence through the introduction of word vectors.Alternatively, by incorporating metadata we can either learn topics that are more predictive of labels than SLDA, or learn explicit deviations for particular parts of the metadata. Finally, by combining all parts of our model we can meaningfully incorporate metadata in multiple ways, which we demonstrate through an exploration of a corpus of news articles about US immigration.In presenting this particular model, we emphasize not only its ability to adapt to the characteristics of the data, but the extent to which the VAE approach to inference provides a powerful framework for latent variable modeling that suggests the possibility of many further extensions.Our implementation is available at <https://github.com/dallascard/scholar>.§ BACKGROUND AND MOTIVATIONLDA can be understood as a non-negative Bayesian matrix factorization model: the observed document-word frequency matrix, 𝐗∈ℤ^D × V (D is the number of documents, V is the vocabulary size) is factored into two low-rank matrices, Θ^D × K and 𝐁^K × V, where each row of Θ, θ_i ∈Δ^K is a latent variable representing a distribution over topics in document i, and each row of 𝐁, β_k∈Δ^V, represents a single topic, i.e., a distribution over words in the vocabulary.[ℤ denotes nonnegative integers, and Δ^K denotes the set of K-length nonnegative vectors that sum to one. For a proper probabilistic interpretation, the matrix to be factored is actually the matrix of latent mean parameters of the assumed data generating process, 𝐗_ij∼Poisson(Λ_ij). See <cit.> or<cit.> for details.] While it is possible to factor the count data into unconstrained matrices,the particular priors assumed by LDA are important for interpretability <cit.>. For example, the neural variational document model (NVDM; ) allows θ_i ∈ℝ^K and achieves normalization by taking the softmax of θ_i^⊤𝐁. However, the experiments in <cit.> found the performance of the NVDM to be slightly worse than LDA in terms of perplexity, and dramatically worse in terms of topic coherence.The topics discovered by LDA tend to be parsimonious and coherent groupings of words which are readily identifiable to humans as being related to each other <cit.>, and the resulting mode of the matrix Θ provides a representation of each document which can be treated as a measurement for downstream tasks, such as classification or answering social scientific questions <cit.>. LDA does not require — and cannot make use of — additional prior knowledge. As such, the topics that are discovered may bear little connection to metadata of a corpus that is of interest to a researcher, such as sentiment, ideology, or time.In this paper, we take inspiration from two models which have sought to alleviate this problem. The first, supervised LDA (SLDA; ), assumes that documents have labels y which are generated conditional on the corresponding latent representation, i.e., y_i ∼ p(y|θ_i).[Technically, the model conditions on the mean of the per-word latent variables, but we elide this detail in the interest of concision.] By incorporating labels into the model, it is forced to learn topics which allow documents to be represented in a way that is useful for the classification task. Such models can be used inductively as text classifiers <cit.>.SAGE <cit.>, by contrast, is an exponential-family model,where the key innovationwas to replace topics with sparse deviations from the background log-frequency of words (d),i.e., p(word|softmax(d + θ_i^⊤𝐁)). SAGE can also incorporate deviations for observed covariates, as well as interactions between topics and covariates, by including additional terms inside the softmax. In principle, this allows for inferring, for example, the effect on an author's ideology on their choice of words, as well as ideological variations on each underlying topic. Unlike the NVDM, SAGE still constrains θ_i to lie on the simplex, as in LDA.SLDA and SAGE provide two different ways that users might wish to incorporate prior knowledge as a way of guiding the discovery of topics in a corpus: SLDA incorporates labels through a distribution conditional on topics; SAGEincludes explicit sparse deviations for each unique valueof a covariate, in addition to topics.[A third way of incorporating metadata is the approach used by various “upstream” models, such as Dirichlet-multinomial regression <cit.>, which uses observed metadata to inform the document prior. We hypothesize that this approach could be productively combined with our framework, but we leave this as future work.]Because of the Dirichlet-multinomial conjugacy in the original model, efficient inference algorithms exist for LDA. Each variation of LDA, however, has required the derivation of a custom inference algorithm, which is a time-consuming and error-prone process. In SLDA, for example, each type of distribution we might assume for p(y|θ) would require a modification of the inference algorithm. SAGE breaks conjugacy, and as such, the authors adopted L-BFGS for optimizing the variational bound. Moreover, in order to maintain computational efficiency, it assumed that covariates were limited to a single categorical label. More recently, the variational autoencoder (VAE) was introduced as a way to perform approximate posterior inference on models with otherwise intractable posteriors <cit.>. This approach has previously been applied to models of text by <cit.> and <cit.>. We build on their work and show how this framework can be adapted to seamlessly incorporate the ideas of both SAGE and SLDA, while allowing for greater flexibility in the use of metadata. Moreover, by exploiting automatic differentiation, we allow for modification of the model without requiring any change to the inference procedure.The result is not only a highly adaptable family of models with scalable inference and efficient prediction; it also points the way to incorporation of many ideas found in the literature, such as a gradual evolution of topics <cit.>, and hierarchical models <cit.>. § : A NEURAL TOPIC MODEL WITH COVARIATES, SUPERVISION, AND SPARSITYWe begin by presenting the generative story for our model, and explain how it generalizes both SLDA and SAGE (<ref>). We then provide a general explanation of inference using VAEs and how it applies to our model (<ref>), as well as how to infer document representations and predict labels at test time (<ref>). Finally, we discuss how we can incorporate additional prior knowledge (<ref>). §.§ Generative Story Considera corpus of D documents, where document i is a list of N_i words, w_i, with V words in the vocabulary. For each document, we may have observed covariates c_i(e.g., year of publication), and/or one or more labels, y_i (e.g., sentiment).Our model builds on the generative story of LDA, but optionally incorporates labels and covariates, and replaces the matrix product of Θ and 𝐁 with a more flexible generative network, f_g, followed by a softmax transform.Instead of using a Dirichlet prior as in LDA, we employ a logistic normal prior on θ as in <cit.> to facilitate inference (<ref>): we draw a latent variable, r,[r is equivalent to z in the original VAE. To avoid confusion with topic assignment of words in the topic modeling literature, we use r instead of z.] from a multivariate normal, and transform it to lie on the simplex using a softmax transform.[Unlike the correlated topic model (CTM; ), which also uses a logistic-normal prior, we fix the parameters of the prior and use a diagonal covariance matrix, rather than trying to infer correlations among topics. However, it would be a straightforward extension of our framework to place a richer prior on the latent document representations, and learn correlations by updating the parameters of this prior after each epoch, analogously to the variational EM approach used for the CTM.] The generative story is shown in Figure <ref> and described in equations below: For each document i of length N_i:# Draw a latent representation on the simplex from a logistic normal prior: r_i ∼𝒩(r|μ_0(α), diag(σ^2_0(α))) θ_i = softmax(r_i) # Generate words, incorporating covariates: η_i = f_g(θ_i , c_i) For each word j in document i:w_ij∼ p(w|softmax(η_i))# Similarly generate labels: y_i ∼ p(y| f_y(θ_i, c_i)),where p(w|softmax(η_i)) is a multinomial distribution and p(y| f_y(θ_i, c_i)) is a distribution appropriate to the data (e.g., multinomial for categorical labels). f_g is a model-specific combination of latent variables and covariates, f_y is a multi-layer neural network, and μ_0(α) and σ_0^2(α) are the mean and diagonal covariance terms of a multivariate normal prior. To approximate a symmetric Dirichlet prior with hyperparameter α, these are given by the Laplace approximation <cit.> to be μ_0,k(α) = 0 and σ_0,k^2 = (K-1)/(α K).If we were to ignore covariates, place a Dirichlet prior on 𝐁, and let η = θ_i^⊤𝐁, this model is equivalent to SLDA with a logistic normal prior. Similarly, we can recover a model that is like SAGE, but lacks sparsity, if we ignore labels, and let η_i =d + θ_i^⊤𝐁+ c_i^⊤𝐁^cov+ (θ_i ⊗c_i)^⊤𝐁^int,where d is the V-dimensional background term (representing the log of the overall word frequency), θ_i ⊗c_i is a vector of interactions between topics and covariates, and 𝐁^cov and 𝐁^int are additional weight (deviation) matrices. The background is included to account for common words with approximately the same frequency across documents, meaning that the 𝐁^∗ weights now represent both positive and negative deviations from this background. This is the form of f_g which we will use in our experiments.To recover the full SAGE model, we can place a sparsity-inducing prior on each 𝐁^∗. As in <cit.>, we make use of the compound normal-exponential prior for each element of the weight matrices, 𝐁_m,n^∗, with hyperparameter γ,[To avoid having to tune γ, we employ an improper Jeffery's prior, p(τ_m,n) ∝ 1/τ_m,n, as in SAGE. Although this causes difficulties in posterior inference for the variance terms, τ, in practice, we resort to a variational EM approach, with MAP-estimation for the weights, 𝐁, and thus alternate between computing expectations of the τ parameters, and updating all other parameters using some variant of stochastic gradient descent. For this, we only require the expectation of each τ_mn for each E-step, which is given by 1/𝐁_m,n^2. We refer the reader to <cit.> for additional details.]τ_m,n ∼Exponential(γ),𝐁_m,n^∗ ∼𝒩(0, τ_m,n). We can choose to ignore various parts of this model, if, for example, we don't have any labels or observed covariates, or we don't wish to use interactions or sparsity.[We could also ignore latent topics, in which case we would get a naïve Bayes-like model of text with deviations for each covariate p(w_ij|c_i) ∝exp (d + c_i^⊤𝐁^cov). ]Other generator networks could also be considered, with additional layers to represent more complex interactions, although this might involve some loss of interpretability.In the absence of metadata, and without sparsity, our model is equivalent to the ProdLDA model of <cit.> with an explicit background term, and ProdLDA is, in turn, a special case of SAGE, without background log-frequencies, sparsity, covariates, or labels. In the next section we generalize the inference method used for ProdLDA; in our experiments we validate its performance and explore the effects of regularization and word-vector initialization (<ref>). The NVDM <cit.>uses the same approach to inference, butdoes not not restrict document representations to the simplex.§.§ Learning and InferenceAs in past work, each document i is assumed to have a latent representation r_i, which can be interpreted as its relativemembership in each topic (after exponentiating and normalizing). In order to infer an approximate posterior distribution over r_i, we adopt the sampling-based VAEframework developed in previous work<cit.>. As in conventional variational inference, we assume a variational approximation to the posterior, q_Φ(r_i |w_i, c_i, y_i), and seek to minimize the KL divergence between it and the true posterior, p(r_i |w_i, c_i, y_i), where Φ is the set of variational parameters to be defined below.After some manipulations (given in supplementary materials), we obtain the evidence lower bound (ELBO) for a single document,ℒ(w_i) = 𝔼_q_Φ(r_i|w_i, c_i, y_i) [ ∑_j=1^N_ilog p(w_ij|r_i, c_i) ] + 𝔼_q_Φ(r_i|w_i, c_i, y_i)[ log p(y_i |r_i, c_i) ] - D_KL[q_Φ(r_i |w_i, c_i, y_i) || p(r_i |α) ]. As in the original VAE, we will encode the parameters of our variational distributions using a shared multi-layer neural network. Because we have assumed a diagonal normal prior on r, this will take the form of a network which outputs a mean vector, μ_i = f_μ(w_i, c_i, y_i) and diagonal of a covariance matrix, σ^2_i = f_σ(w_i, c_i, y_i), such that q_Φ(r_i |w_i, c_i, y_i) = 𝒩(μ_i, σ_i^2). Incorporating labels and covariates to the inference network used by <cit.> and <cit.>, we use:π_i= f_e([𝐖_x x_i ; 𝐖_c c_i; 𝐖_y y_i]) ,μ_i= 𝐖_μπ_i + b_μ,logσ_i^2= 𝐖_σπ_i + b_σ,where x_i is a V-dimensional vector representing the counts of words in w_i, and f_e is a multilayer perceptron. The full set of encoder parameters, Φ, thus includes the parameters of f_e and all weight matrices and bias vectors in Equations <ref>–<ref> (see Figure <ref>). This approach means that the expectations in Equation <ref> are intractable, but we can approximate them using sampling. In order to maintain differentiability with respect to Φ, even after sampling, we make use of the reparameterization trick <cit.>,[ The Dirichlet distribution cannot be directly reparameterized in this way,whichis why we use the logistic normal prior on θ to approximate the Dirichlet prior used in LDA.]which allows us to reparameterize samples from q_Φ(r|w_i, c_i, y_i) in terms of samples from an independent source of noise, i.e.,ϵ^(s) ∼𝒩(0, 𝐈), r_i^(s) = g_Φ(w_i, c_i, y_i, ϵ^(s))= μ_i + σ_i ·ϵ^(s). We thus replace the bound in Equation <ref> with a Monte Carlo approximation using a single sample[Alternatively, one can average over multiple samples.] of ϵ (and thereby of r):ℒ(w_i) ≈∑_j=1^N_ilog p(w_ij|r_i^(s), c_i)+ log p(y_i |r_i^(s), c_i) - D_KL[q_Φ(r_i |w_i, c_i, y_i) || p(r_i |α) ].We can now optimize this sampling-based approximation of the variational bound with respect to Φ, 𝐁^∗, and all parameters of f_g and f_y using stochastic gradient descent. Moreover, because of this stochastic approach to inference, we are not restricted to covariates with a small number of unique values, which was a limitation of SAGE. Finally, the KL divergence term in Equation <ref> can be computed in closed form (see supplementary materials). §.§ Prediction on Held-out Data In addition to inferring latent topics, our model can both infer latent representations for new documents and predict their labels, the latter of which was the motivation for SLDA. In traditional variational inference, inference at test time requires fixing global parameters (topics), and optimizing the per-document variational parameters for the test set.With the VAE framework,by contrast, the encoder network (Equations <ref>–<ref>) canbe usedto directly estimate the posterior distributionfor eachtest document, using only a forward pass (no iterative optimization or sampling).If not using labels, we can use this approach directly, passing the word counts of new documents through the encoder to get a posterior q_Φ(r_i |w_i, c_i). When we also include labels to be predicted, we can first train a fully-observed model, as above, then fix the decoder, and retrain the encoder without labels. In practice, however, if we train the encoder network usingone-hot encodings of document labels, it is sufficient to provide a vector of all zeros for the labels of test documents; this is what weadopt for our experiments (<ref>), and we still obtain good predictive performance.The label network, f_y, is a flexible component which can be used to predict a wide range of outcomes, from categorical labels (such as star ratings; )to real-valued outputs (such as number of citations or box-office returns; ). For categorical labels, predictions are given byŷ_i = y ∈ 𝒴argmax p (y |r_i, c_i). Alternatively, when dealing with a small set of categorical labels, it is also possible to treat them as observed categorical covariates during training. At test time, we can then consider all possible one-hot vectors, e, in place of c_i, and predict the label that maximizes the probability of the words, i.e.,ŷ_i = y ∈ 𝒴argmax ∑_j=1^N_ilog p(w_ij|r_i, e_y).This approach works well in practice (as we show in <ref>), but does not scale to large numbers of labels, or other types of prediction problems, such as multi-class classification or regression. The choice to include metadata as covariates, labels, or both, depends on the data. The key point is that we can incorporate metadata in two very different ways, depending on what we want from the model. Labels guide the model to infer topics that are relevant to those labels, whereas covariates induce explicit deviations, leaving the latent variables to account for the rest of the content. §.§ Additional Prior Information A final advantage of the VAE framework is that the encoder network provides a way to incorporate additional prior information in the form of word vectors. Although we can learn all parameters starting from a random initialization, it is also possible to initialize and fix the initial embeddings of words in the model, 𝐖_x, in Equation <ref>. This leverages word similarities derived from large amounts of unlabeled data, and may promote greater coherence in inferred topics. The same could also be done for some covariates; for example, we could embed the source of a news article based on its place on the ideological spectrum. Conversely, if we choose to learn these parameters, the learned values (𝐖_y and 𝐖_c) may provide meaningful embeddings of these metadata (see section <ref>).Other variants on topic models have also proposed incorporating word vectors, both as a parallel part of the generative process <cit.>, and as an alternative parameterization of topic distributions <cit.>, but inference is not scalable in either of these models. Because of the generality of the VAE framework, we could also modify the generative story so that word embeddings are emitted (rather than tokens); we leave this for future work. § EXPERIMENTS AND RESULTSTo evaluate and demonstrate the potential of this model, we present a series of experiments below. We first test without observed metadata, and explore the effects of using regularization and/or word vector initialization, compared to LDA, SAGE, and NVDM (<ref>). We then evaluate our model in terms of predictive performance, in comparison to SLDA and an l_2-regularized logistic regression baseline (<ref>). Finally, we demonstrate the ability to incorporate covariates and/or labels in an exploratory data analysis (<ref>). The scores we report are generalization to held-out data, measured in terms of perplexity; coherence, measured in terms of non-negative point-wise mutual information (NPMI; ), and classification accuracy on test data. For coherence we evaluate NPMI using the top 10 words of each topic, both internally (using test data), and externally, using a decade of articles from the English Gigaword dataset <cit.>.Since our model employs variational methods, the reported perplexity is an upper bound based on the ELBO.As datasets we use the familiar 20 newsgroups, the IMDB corpus of 50,000 movie reviews <cit.>, and the UIUC Yahoo answers dataset with 150,000 documents in 15 categories <cit.>. For further exploration, we also make use of a corpus of approximately 4,000 time-stamped news articles about US immigration, each annotated with pro- or anti-immigration tone <cit.>. We use the original author-provided implementations of SAGE[<github.com/jacobeisenstein/SAGE>] and SLDA,[<github.com/blei-lab/class-slda>] while for LDA we use Mallet.[<mallet.cs.umass.edu>]. Our implementation of is in TensorFlow, but we have also provided a preliminary PyTorch implementation of the core of our model.[<github.com/dallascard/scholar>] For additional details about datasets and implementation, please refer to the supplementary material.It is challenging to fairly evaluate the relative computational efficiency of our approach compared to past work (due to the stochastic nature of our approach to inference, choices about hyperparameters such as tolerance, and because of differences in implementation). Nevertheless, in practice, the performance of our approach is highly appealing. For all experiments in this paper, our implementation was much faster than SLDA or SAGE (implemented in C and Matlab, respectively), and competitive with Mallet.§.§ Unsupervised EvaluationAlthough the emphasis of this work is on incorporating observed labels and/or covariates, we briefly report on experiments in the unsupervised setting. Recall that, without metadata, equates to ProdLDA, but with an explicit background term.[Note, however, that a batchnorm layer in ProdLDA may play a similar role to a background term, and there are small differences in implementation; please see supplementary material for more discussion of this.]We therefore use the same experimental setup as <cit.> (learning rate, momentum, batch size, and number of epochs) and find the same general patterns as they reported (see Table <ref> and supplementary material): our model returns more coherent topics than LDA, but at the cost of worse perplexity. SAGE, by contrast, attains very high levels of sparsity, but at the cost of worse perplexity and coherence than LDA. As expected, the NVDM produces relatively low perplexity, but very poor coherence, due to its lack of constraints on θ.Further experimentation revealed that the VAE framework involves a tradeoff among the scores; running for more epochs tends to result in better perplexity on held-out data, but at the cost of worse coherence. Adding regularization to encourage sparse topics has a similar effect as in SAGE, leading to worse perplexity and coherence, but it does create sparse topics. Interestingly, initializing the encoder with pretrained word2vec embeddings, and not updating them returned a model with the best internal coherence of any model we considered for IMDB and Yahoo answers,and the second-best for 20 newsgroups.The background term in our model does not have much effect on perplexity, but plays an important role in producing coherent topics; as in SAGE, the background can account for common words, so they are mostly absent among the most heavily weighted words in the topics. For instance, words like film and movie in the IMDB corpus are relatively unimportant in the topics learned by our model, but would be much more heavily weighted without the background term, as they are in topics learned by LDA. §.§ Text ClassificationWe next consider the utility of our model in the context of categorical labels, and consider them alternately as observed covariates and as labels generated conditional on the latent representation. We use the same setup as above, but tune number of training epochs for our model using a random 20% of training data as a development set, and similarly tune regularization for logistic regression.Table <ref> summarizes the accuracy of various models on three datasets, revealing that our model offers competitive performance, both as a joint model of words and labels (Eq. <ref>), and a model which conditions on covariates (Eq. <ref>). Although is comparable to the logistic regression baseline, our purpose here is not to attain state-of-the-art performance on text classification. Rather, the high accuracies we obtain demonstrate that we are learning low-dimensional representations of documents that are relevant to the label of interest, outperforming SLDA, and have the same attractive properties as topic models. Further, any neural network that is successful for text classification could be incorporated into f_y and trained end-to-end along with topic discovery. §.§ Exploratory StudyWe demonstrate how our model might be used to explore an annotated corpus of articles about immigration, and adapt to different assumptions about the data. We only use a small number of topics in this part (K=8) for compact presentation. Tone as a label. We first consider using the annotations as a label, and train a joint model to infer topics relevant to the tone of the article (pro- or anti-immigration). Figure <ref> shows a set of topics learned in this way, along with the predicted probability of an article being pro-immigration conditioned on the given topic. All topics are coherent, and the predicted probabilities have strong face validity, e.g., “arrested charged charges agents operation” is least associated with pro-immigration. Tone as a covariate. Next we consider using tone as a covariate, and build a model using both tone and tone-topic interactions. Table <ref> shows a set of topics learned from the immigration data, along with the most highly-weighted words in the corresponding tone-topic interaction terms. As can be seen, these interaction terms tend to capture different frames (e.g., “criminal” vs. “detainees”,and “illegals” vs. “newcomers”, etc). Combined model with temporal metadata. Finally,we incorporate both the tone annotations and the year of publication of each article, treating the former as a label and the latter as a covariate. In this model, we also include an embedding matrix, 𝐖_c, to project the one-hot year vectors down to a two-dimensional continuous space, with a learned deviation for each dimension. We omit the topics in the interest of space, butFigure <ref> shows the learned embedding for each year, along with the top terms of the corresponding deviations. As can be seen, the model learns that adjacent years tend to produce similar deviations, even though we have not explicitly encoded this information. The left-right dimension roughly tracks a temporal trend with positive deviations shifting from the years of Clinton and INS on the left, to Obama and ICE on the right.[The Immigration and Naturalization Service (INS) was transformed into Immigration and Customs Enforcement (ICE) and other agencies in 2003.]Meanwhile, the eventsof 9/11 dominate the vertical direction, with the words sept, hijackers, and attacks increasing in probability as we move up in the space. If we wanted to look at each year individually, we could drop the embedding of years, and learn a sparse set of topic-year interactions,similar to tone in Table <ref>. § ADDITIONAL RELATED WORK The literature on topic models is vast; in addition to papers cited throughout, other efforts to incorporate metadata into topic models include Dirichlet-multinomial regression (DMR; ), Labeled LDA <cit.>, and MedLDA <cit.>. A recent paper also extended DMR by using deep neural networks to embed metadata into a richer document prior <cit.>. A separate line of work has pursued parameterizing unsupervised models of documents using neural networks <cit.>, including non-Bayesian approaches <cit.>. More recently, <cit.> proposed a neural language model that incorporated topics, and <cit.> developed a scalable alternative to the correlated topic model by simultaneously learning topic embeddings.Others have attempted to extend the reparameterization trick to the Dirichlet and Gamma distributions, either through transformations<cit.> or a generalization of reparameterization <cit.>. Black-box and VAE-style inference have been implemented in at least two general purpose tools designed to allow rapid exploration and evaluation of models <cit.>. § CONCLUSION We have presented a neural framework for generalized topic models to enableflexible incorporation of metadata with a variety of options. We take advantage of stochastic variational inference to develop a general algorithm for our framework such that variations do not require any model-specific algorithm derivations. Our model demonstrates the tradeoff between perplexity, coherence, and sparsity, and outperforms SLDA in predicting document labels. Furthermore, the flexibility of our model enables intriguing exploration of a text corpus on US immigration. We believe that our model and code will facilitate rapid exploration of document collections with metadata. § ACKNOWLEDGMENTS We would like to thank Charles Sutton, anonymous reviewers, and all members of Noah's ARK for helpful discussions and feedback. This work was made possible by a University of Washington Innovation award and computing resources provided by XSEDE.acl_natbib § SUPPLEMENTARY MATERIAL§.§ Deriving the ELBO The derivation of the ELBO for our model is given below, dropping explicit reference to Φ and α for simplicity. For document i,log p(w_i, y_i |c_i) =log∫_r_ip(w_i, y_i, r_i |c_i) dr_i =log∫_r_ip(w_i, y_i, r_i |, c_i) q(r_i |w_i, c_i, y_i)/q(r_i |w_i, c_i, y_i)dr_i=log( 𝔼_q(r_i |w_i, c_i, y_i)[ p(w_i, y_i, r_i |c_i)/q(r_i |w_i, c_i, y_i)] )≥𝔼_q(r_i |w_i, c_i, y_i)[log p(w_i, y_i, r_i |c_i)] - 𝔼_q(r_i |w_i, c_i, y_i)[log q(r_i |w_i, c_i, y_i) ]= 𝔼_q(r_i |w_i, c_i, y_i) [log p(w_i, y_i |r_i, c_i)] + 𝔼_q(r_i |w_i, c_i, y_i) [log p(r_i)]- 𝔼_q(r_i |w_i, c_i, y_i) [log q(r_i |w_i, c_i, y_i) ]= 𝔼_q(r_i |w_i, c_i, y_i) [ ∑_j=1^N_ilog p(w_ij|r_i, c_i)] + 𝔼_q(r_i |w_i, c_i, y_i) [log p(y_i |r_i, c_i)] - D_KL [ q(r_i |w_i, c_i, y_i) || p(r_i ) ] §.§ Model details The KL divergence term in the variational bound can be computed as D_KL[q_Φ(r_i |w_i, c_i, y_i)p(r_i)] = 1/2(tr(Σ_0^-1Σ_i) + (μ_i - μ_0)^⊤Σ_0^-1 (μ_i - μ_0) - K + log|Σ_0|/| Σ_i |)where Σ_i = diag(σ_i^2(w_i, c_i, y_i)), and Σ_0 = diag(σ_0^2(α)).§.§ Practicalities and Implementation As observed in past work, inference using a VAE can suffer from component collapse, which translates into excessive redundancy in topics (i.e., many topics containing the same set of words). To mitigate this problem, we borrow the approach used by <cit.>, and make use of the Adam optimizer with a high momentum, combined with batchnorm layers to avoid divergence. Specifically, we add batchnorm layers following the computation of μ, logσ^2, and η.This effectively solves the problem of mode collapse, but the batchnorm layer on η introduces an additional problem, not previously reported. At test time, the batchnorm layer will shift the input based on the learned population mean of the training data; this effectively encodes information about the distribution of words in this model that is not captured by the topic weights and background distribution. As such, although reconstruction error will be low, the document representation θ, will not necessarily be a useful representation of the topical content of each document. In order to alleviate this problem, we reconstruct η as a convex combination of two copies of the output of the generator network, one passed through a batchnorm layer, and one not. During training, we then gradually anneal the model from relying entirely on the component passed through the batchnorm layer, to relying entirely on the one that is not. This ensures that the the final weights and document representations will be properly interpretable.Note that although ProdLDA <cit.> does not explicitly include a background term, it is possible that the batchnorm layer applied to η has a similar effect, albeit one that is not as easily interpretable. This annealing process avoids that ambiguity.§.§ DataAll datasets were preprocessed by tokenizing, converting to lower case, removing punctuation, and dropping all tokens that included numbers, all tokens less than 3 characters, and all words on the stopword list from the snowball sampler.[<snowball.tartarus.org/algorithms/english/stop.txt>] The vocabulary was then formed by keeping the words that occurred in the most documents (including train and test), up to the desired size (2000 for 20 newsgroups, 5000 for the others). Note that these small preprocessing decisions can make a large difference to perplexity. We therefore include our preprocessing scripts as part of our implementation so as to facilitate easy future comparison. For the UIUC Yahoo answers dataset, we downloaded the documents from the project webpage.[<cogcomp.org/page/resource_view/89>] However, the file that is available does not completely match the description on the website. We droppedCars and Transportation and Social Science which had less than the expected number of documents, and merged Arts and Arts and Humanities, which appeared to be the same category, producing 15 categories, each with 10,000 documents. §.§ Experimental Details For all experiments we use a model with 300-dimensional embeddings of words, and we take f_e to be only the element-wise softplus nonlinearity (followed by the linear transformations for μ and logσ^2). Similarly, f_y is a linear transformation of θ, followed by a softplus layer, followed by a linear transformation to the size of the output (the number of classes). During training, we set S (the number of samples ofϵ) to be 1; for estimating the ELBO at on test documents, we set S=20.For the unsupervised results, we use the same set up as <cit.>: Adam optimizer with β_1=0.99, learning rate =0.002, batch size of 200, and training for 200 epochs. The setup was the same for all datasets, except we only trained for 150 epochs on Yahoo answers because it is much larger.For LDA, we updated the hyperparameters every 10 epochs.For the external evaluation of NPMI, we use the co-occurrence statistics from all New York Times articles in the English Gigaword published from the start of 2000 to the end of 2009, processed in the same way as our data.For the text classification experiments, we use theimplementation of logistic regression. We give it access to the same input data as our model (using the same vocabulary), and tune the strength of l_2 regularization using cross-validation. For our model, we only tune the number of epochs, evaluating on development data. Our models for this task did not use regularization or word vectors.§.§ Additional Experimental Results In this section we include additional experimental results in the unsupervised setting. Table <ref> shows results on the 20 newsgroups dataset, using 20 topics with a 2,000-word vocabulary. Note that these results are not necessarily directly comparable to previously published results, as preprocessing decisions can have a large impact on perplexity. These results show a similar pattern to those on the IMDB data provided in the main paper, except that word vectors do not improve internal coherence on this dataset, perhaps because of the presence of relatively more names and specialized terminology. Also, although the NVDM still has worse perplexity than LDA, the effects are not as dramatic as reported in <cit.>. Regularization is also more beneficial for this data, with both SAGE and our regularized model obtaining better coherence than LDA. The topics from scholar for this dataset are also shown in Table <ref>.Table <ref> shows the equivalent results for the Yahoo answers dataset, using 250 topics, and a 5,000-word vocabulary. These results closely match those for the IMDB dataset, with our model having higher perplexity but also higher internal coherence than LDA. As with IMDB, the use of word vectors improves coherence, both internally and externally, but again at the cost of worse perplexity. Surprisingly, our model without a background term actually has the best external coherence on this dataset, but as described in the main paper, these tend to give high weight primarily to common words, and are more repetitive as a result. | http://arxiv.org/abs/1705.09296v2 | {
"authors": [
"Dallas Card",
"Chenhao Tan",
"Noah A. Smith"
],
"categories": [
"stat.ML",
"cs.CL"
],
"primary_category": "stat.ML",
"published": "20170525180003",
"title": "Neural Models for Documents with Metadata"
} |
A Tale of Three Cities G. Beccari1 M.G. Petr-Gotzens1 H.M.J. Boffin1M. Romaniello1,12D. Fedele2G. Carraro3G. De Marchi4W.-J. de Wit5J.E. Drew6 V.M. Kalari7C.F. Manara4 E.L. Martin8S. Mieske5N. Panagia9L. Testi1J.S. Vink10J.R. Walsh1N.J. Wright6,11December 30, 2023 =========================================================================================================================================================================================================================================Studying the shortness of longest cycles in maximal planar graphs,we improve the upper bound on the shortness exponent of the class of5/4-tough maximal planar graphs presented byHarant and Owens [Discrete Math. 147 (1995), 301–305].In addition, we present two generalizations of a similar result ofTkáč who considered 1-tough maximal planar graphs [Discrete Math. 154 (1996), 321–328];we remark that one of these generalizations gives a tight upper bound.We fix a problematic argument used in the first paper. § INTRODUCTION We continue the study of non-Hamiltonian graphs with the property that removing an arbitrary set of vertices disconnects the graph into a relatively small number of components (compared to the size of the removed set). In the present paper, we construct families of maximal planar such graphs whose longest cycles are short (compared to the order of the graph).More formally, the properties which we study are the toughness of graphs and the shortness exponent of classes of graphs (both introduced in 1973).We recall that following Chvátal <cit.>, the toughness of a graph G is the minimum, taken over all separating sets X of vertices in G, of the ratio of |X| to c(G - X) where c(G - X) denotes the number of components of the graph G - X. The toughness of a complete graph is defined as being infinite.We say that a graph is t-tough if its toughness is at least t.Along with the definition of toughness, Chvátal <cit.> conjectured that there is a constant t_0 such that every t_0-tough graph (on at least three vertices) is Hamiltonian.As a lower bound on t_0, Bauer et al. <cit.> presented graphs with toughness arbitrarily close to 9/4 which contain no Hamilton path (and thus, they are non-Hamiltonian).While remaining open for general graphs, Chvátal's conjecture was confirmed in several restricted classes of graphs; and also various relations among the toughness of a graph and properties of its cycles are known. We refer the reader to the extensive survey on this topic <cit.>.Clearly, every graph (on at least five vertices) of toughness greater than 3/2 is 4-connected, so every such planar graph is Hamiltonian by the classical result of Tutte <cit.>.On the other hand, Harant <cit.> showed that not every 3/2-tough planar graph isHamiltonian; and furthermore,the shortness exponent of the class of 3/2-tough planar graphs is less than 1.We recall that following Grünbaum and Walther <cit.>, the shortness exponent of a class of graphs Γ is the lim inf, taken over all infinite sequences G_n of non-isomorphic graphs of Γ (for n going to infinity), of the logarithm of the length of a longest cycle in G_n to base equal to the order of G_n.Introducing this notation, Grünbaum and Walther <cit.> also presented upper bounds on the shortness exponent for numerous subclasses of the class of 3-connected planar graphs. Furthermore, they remarked that the upper bound for the class of 3-connected planar graphs itself was presented earlier by Moser and Moon <cit.> who used a slightly different notation.Later, Chen and Yu <cit.> showed that every 3-connected planar graph G contains a cycle of length at least |V(G)|^log_3 2; in combination with the bound of <cit.>, it follows that the shortness exponent of this class equals log_3 2.A number of results considering the shortness exponent and similar parameters are surveyed in <cit.>. Considering the class of maximal planar graphs under a certain toughness restriction, Owens <cit.> presented non-Hamiltonian maximal planar graphs of toughness arbitrarily close to 3/2. Harant and Owens <cit.> argued that the shortness exponent of the class of 5/4-tough maximal planar graphs is at most log_9 8.Improving the bound log_7 6 presented by Dillencourt <cit.>, Tkáč <cit.> showed that it is at most log_6 5 for the class of 1-tough maximal planar graphs.In the present paper, we show the following. Let σ be the shortness exponent of the class of maximal planar graphs under a certain toughness restriction.* If the graphs are 54-tough, then σ is at most log_3022. * If the graphs are 87-tough, then σ is at most log_65. * If the toughness of the graphs is greater than 1, then σ equals log_32. We note that log_98 > log_3022, that is, the statement in item (i) of Theorem <ref> improves the result of <cit.>.Furthermore, items (ii) and (iii) provide two different generalizations of the result of <cit.> since 8/7 > 1 and log_65 > log_32.We remark that we fix a problem in a technical lemma presented in <cit.>. The fixed version of this lemma (see Lemma <ref>) is applied to prove the present results. § STRUCTURE OF THE PROOFIn order to prove Theorem <ref>, we shall construct three families of graphs whose properties are summarized in the following proposition. For every i = 1,2,3 and every non-negative integer n, there exists a maximal planar graph F_i,n on f_i(n) vertices whose longest cycle has c_i(n) vertices where* f_1(n) = 1 + 101(1 + 30 + … + 30^n) and c_1(n) = 1 + 93(1 + 22 + … + 22^n) and F_1,n is 5/4-tough, * f_2(n) = 1 + 14(1 + 6 + … + 6^n) and c_2(n) = 1 + 13(1 + 5 + … + 5^n) and F_2,n is 8/7-tough, * f_3(n) = 4 + 5(1 + 3 + … + 3^n) and c_3(n) = 3 · 2^n+3-9n-15 and the toughness of F_3,n is greater than 1. Before constructing the graphs F_1,n, we point out that the use of Proposition <ref> leads directly to the main results of the present paper.We consider an infinite sequence of non-isomorphic graphs F_1,n given by item (i) of Proposition <ref>; and we recall that they are 5/4-tough maximal planar graphs. Furthermore, we have f_1(n) = 1 + 10129(30^n+1 - 1) and c_1(n) = 1 + 9321(22^n+1 - 1). It follows that lim_n→∞log_f_1(n) c_1(n) = log_3022. Thus, the considered shortness exponent is at most log_3022.Using similar arguments and considering items (ii) and (iii) of Proposition <ref>, we obtain the desired upper bounds.Clearly, if G is a maximal planar graph (on at least four vertices), then it is 3-connected. By a result of <cit.>, G contains a cycle of length at least |V(G)|^log_3 2.In combination with the upper bound obtained due to item (iii) of Proposition <ref>, we obtain that for the class of maximal planar graphs of toughness greater than 1, the shortness exponent equals log_3 2.In the remainder of the present paper, we construct the families of graphs having the properties described in Proposition <ref>.Basically, we proceed in four steps. First, we introduce relatively small graphs F_i, 0 called `building blocks' in Section <ref>, and we observe key properties of their longest cycles. We use these building blocks to construct larger graphs F_i, n in Section <ref>, and we show that their longest cycles are short. In Section <ref>, we study the toughness of the building blocks. The toughness of the graphs F_i, n is shown in Sections <ref> and <ref>.We remark that the graphs F_1, n and F_2, n are obtained using a standard construction for bounding the shortness exponent (see for instance <cit.>, <cit.>, <cit.>, <cit.> or <cit.>); the improvement of the known bounds comes with the choice of suitable building blocks. In addition, we formalize the key ideas of this construction to make them more accessible for further usage.The construction of graphs F_3, n can be viewed as a simple modification of the construction used in <cit.> (yet the toughness and longest cycles of the constructed graphs are different). § BUILDING BLOCKSWe start by considering the graph T depicted in Figure <ref> which plays an important role in the latter constructions.We let o_1, o_2, and o_3 be its vertices of degree 6.A T-region of a graph G is an induced subgraph isomorphic to T with a distinction of vertices referring to o_1, o_2, o_3 as to outer vertices and to the remaining vertices as to inner vertices, and with the property that no inner vertex is adjacent to a vertex of G - T.Similarly, we define an H-region for a given graph H and a given distinction of its vertices.We view the T-regions as replacing triangles of a graph with copies of T in the natural way. The basic idea of the present constructions is that if these triangles share many vertices, then every cycle in the resulting graph misses many vertices. We formalize this idea in Proposition <ref>.We recall that a vertex is called simplicial if its neighbourhood induces a complete graph. Let R be a T-region of a graph G and let C be a cycle containingall three simplicial vertices of R and a vertex of G - R. Then the subgraph of C induced by the vertices of R is a path containing all outer vertices of R; two of them as its ends.Clearly, R has three simplicial vertices none of which is an outer vertex. The statement follows from the fact that every simplicial vertex has only one neighbour which is not an outer vertex.Aiming for the graph F_1,0 (the building block for constructing the graphs F_1,n), we consider a graph with a number, say r, of T-regions which all share one outer vertex and which are otherwise disjoint. By Proposition <ref>, every cycle in this graph contains at most 2r + 2 of the 3r simplicial vertices belonging to these T-regions.(We view the building block used in <cit.> simply as choosing r = 3.) With hindsight, we remark that the used construction leads to the upper bound log_3r(2r+2) on the shortness exponent; so we minimize this function over all integers r ≥ 3, that is, we choose r = 10.We let F_1,0 be the graph depicted in Figure <ref>, and we note that F_1,0 is a maximal planar graph. Clearly, F_1,0 has 30 simplicial vertices, and we colour these vertices white. We recall that every cycle in F_1,0 contains at most 22 white vertices.Furthermore, we let F_2, 0 be the graph depicted in Figure <ref>.Clearly, F_2, 0 is a maximal planar graph having 6 simplicial vertices; and we colour these vertices white. We use Proposition <ref>, and we observe that a cycle in F_2, 0 contains at most 5 white vertices.Lastly, we let F_3,0 be the graph T. For every i = 1,2,3, we define the outer face of F_i,0 as given by the present embedding (see Figures <ref> and <ref>).In the following section, we shall use the blocks F_i,0 and construct the graphs F_i,n. § FAMILIES OF TREE-LIKE STRUCTURED GRAPHSWe recall the standard construction used for bounding the shortness exponent, and we formalize it with the following definition and Lemma <ref>.An arranged block is a 5-tuple (G_0, j, W, O, k) where G_0 is a graph, j is the number of vertices of G_0, and W and O are disjoint sets of vertices of G_0 such that the vertices of W are simplicial and independentand O induces a complete graph and such that every cycle in G_0 contains at most k vertices of W. Let (G_0, j, W, O, k) be an arranged block such that k ≥ 1. For every n ≥ 1, let G_n be a graph obtained from G_n-1 by replacing every vertex of W with a copy of G_0 (which contains W and O), and by adding arbitrary edges which connect the neighbourhood of the replaced vertex with the set O of the copy of G_0 replacing this vertex.Then G_n has 1 + (j - 1)(1 + |W| + … + |W|^n) vertices and its longest cycle has at most 1 + (ℓ - 1)(1 + k + … + k^n) vertices where ℓ = j - |W| + k.We note that G_n contains |W|^n+1 vertices of W. For the sake of induction, we prove the statement with an additional claim that every cycle in G_n contains at most k^n+1 vertices of W. Clearly, the statement and the claim are satisfied for n = 0, and we proceed by induction on n.We note that the difference in the order of G_n and G_n-1 equals (j - 1)|W|^n. Thus, G_n has 1 + (j - 1)(1 + |W| + … + |W|^n) vertices.We let C be a cycle in G_n, and we view this cycle simply as a sequence of vertices. For every newly added copy of G_0, we remove from C all but one vertex of the copy and we replace the remaining vertex (if there is such) by the corresponding replaced vertex of G_n-1; and we let C' denote the resulting sequence.Clearly, if C' has at most two vertices, then C visits at most one of the newly added copies of G_0. If C' has at least three vertices, then C' defines a cycle in G_n-1 (since the neighbourhood of every vertex of W in G_n-1 induces a complete graph); and C' contains at most k^n vertices of W (by the induction hypothesis). Thus, C visits at most k^n of the newly added copies of G_0.Similarly, we choose an arbitrary newly added copy of G_0, and we remove from C all vertices not belonging to this copy. We note that the resulting sequence either contains at most two vertices (belonging to O) or it is a cycle in G_0 (since O induces a complete graph). Thus, a cycle in G_n contains at most k vertices belonging to W of one copy of G_0. Furthermore, a cycle contains at most j - |W| + k vertices of one such copy.Consequently, C contains at most k^n+1 vertices of W. Furthermore, the length of C minus the length of a longest cycle in G_n-1 is at most (j - |W| + k -1)k^n which concludes the proof. For i = 1,2, we consider this construction for the graph F_i,0 playing the role of G_0 and the set of its white vertices playing the role of W and the set of vertices of its outer face playing the role of O. For every added copy of F_i,0, we join the vertices of its outer face to the neighbourhood of the corresponding replaced vertex by adding six edges in such a way that the new edges form a 2-regular bipartite graph (that is, a cycle of length 6). We let F_i, n be the resulting graphs, and we observe that they are maximal planar graphs. For instance, see the graph F_2, 1 depicted in Figure <ref>.We start verifying that the constructed graphs (for i = 1,2) have the desired properties.For every i = 1,2 and every non-negative integer n, the graph F_i,n has f_i(n) vertices and its longest cycle has c_i(n) vertices where* f_1(n) = 1 + 101(1 + 30 + … + 30^n) and c_1(n) = 1 + 93(1 + 22 + … + 22^n), * f_2(n) = 1 + 14(1 + 6 + … + 6^n) and c_2(n) = 1 + 13(1 + 5 + … + 5^n). The order of the graphs and the upper bound on the length of their longest cycles follow from Lemma <ref>.We note that a longest cycle in F_1,0, F_2,0 has 94, 14 vertices, respectively.Furthermore, there is a longest cycle which contains an edge of the outer face. Clearly, by removing this edge from the cycle we obtain a path whose ends are vertices of the outer face. We consider a longest cycle in F_i,n-1 and we extend it to a cycle in F_i,n using these paths. The following observation shows that such an extension is possible. For an arbitrary pair, say A, of neighbours of one replaced vertex and an arbitrary pair, say B, of vertices of the outer face of the corresponding F_i,0 (used for replacing this vertex), the bipartite graph (A, B) has a perfect matching.A simple counting argument gives that F_i,n contains a cycle of length c_i(n), for every i = 1,2 and every n. We use a slightly different construction to get the graphs F_3, n. The building block F_3,0 is the graph T whose simplicial vertices are coloured white. We view a subgraph induced by a white vertex and its neighbourhood as a K_4-region, that is, a subgraph isomorphic to K_4 whose white vertex has degree 3 in the whole graph, and the neighbours of the white vertex are called outer vertices of the K_4-region.For n ≥ 1, we let F_3, n be a graph obtained from F_3, n-1 by replacing every K_4-region of F_3, n-1 with a T-region in the natural way (the outer vertices of the K_4-region are the outer vertices of the T-region); and we note that they are maximal planar graphs. For instance, the graph F_3, 1 is depicted in Figure <ref>.We proceed by the following proposition.For every non-negative integer n, the graph F_3,n has 4 + 5(1 + 3 + … + 3^n) vertices and its longest cycle has 3 · 2^n+3-9n-15 vertices.We verify the order of F_3,n using induction on n. Clearly, F_3,0 has 9 vertices,and we note that the difference in the order of F_3,n and F_3,n-1 equals 5 · 3^n. Thus, F_3,n has 4 + 5(1 + 3 + … + 3^n) vertices.We show the length of a longest cycle using a slightly technical argument. We let s_i(n) denote the length of a longest cycle in F_3,n which contains i edges of the outer face (in the embedding which follows naturally from the construction).For the sake of induction, we prove the following equalities: s_0(n) = 3s_1(n-1) -3 s_1(n) = 2s_2(n-1) + s_1(n-1) -3 s_2(n) = 2s_2(n-1) and s_0(n) = 3 · 2^n+3-9n -15 s_1(n) = 2^n+4 - 3n - 7 s_2(n) = 2^n+3. Clearly, s_0(0) = s_1(0) = 9 and s_2(0) = 8, and (using Proposition <ref>) we note that s_0(1) = 24, s_1(1) = 22 and s_2(1) = 16; so the equalities are satisfied for n = 1.We assume that they are satisfied for n-1 and we prove them for n.For n ≥ 1, we view F_3,n as the graph obtained from F_3,0 by replacing each of its K_4-regions with an F_3,n-1-region; and we view a cycle, say C, of F_3,n as a sequence of vertices. We consider one of the F_3,n-1-regions, say R, and we remove from C all vertices not belonging to R; and we let C' be the resulting sequence. We observe that C' either is a cycle or it has at most two vertices.Furthermore, if C visits a vertex not belonging to R, then C' (if it has at least three vertices) is a cycle in R containing at least one edge of its outer face. Clearly, a longest cycle (in F_3,n) whose all vertices belong to the same F_3,n-1-region has s_0(n-1) vertices, and we observe that a longest cycle visiting more than one of the regions has 3s_1(n-1) -3 vertices.We use (<ref>) for s_0(n-1) and s_1(n-1), and we note that3 · 2^n+2-9(n-1)-15 < 3(2^n+3- 3(n-1) - 7) -3. So we get s_0(n) = 3s_1(n-1) -3. By similar arguments, we get s_1(n) = 2s_2(n-1) + s_1(n-1) -3 and s_2(n) = 2s_2(n-1).Consequently, we can use (<ref>) for s_i(n), and we obtains_0(n) =3s_1(n-1)-3 =3(2^n+3 - 3(n-1) - 7) - 3 = 3 · 2^n+3-9n-15 and s_1(n)= 2s_2(n-1) + s_1(n-1) -3 = 2 · 2^n+2 + 2^n+3 - 3(n-1) - 7 - 3 = 2^n+4 - 3n - 7 and s_2(n) = 2s_2(n-1) = 2 · 2^n+2 = 2^n+3. Thus, the equalities are satisfied for n. In particular, a longest cycle in F_3,n has 3 · 2^n+3-9n-15 vertices. With Corollary <ref> and Proposition <ref> on hand, we shall focus on the toughness of the constructed graphs.§ TOUGHNESS OF THE EXTENDED BLOCKSIn this section, we study the toughness of F_i,0 (for i = 1,2,3) and of its extension F^+_i,0 (for i = 1,2) which is a graph obtained by adding a vertex adjacent to all vertices of the outer face of F_i,0.We shall use the following two propositions.Adding a simplicial vertex to a graph does not increase its toughness.We let x be a simplicial vertex of a graph G^+ and we let G = G^+ - x and we let S be a set of vertices of G. We note that c(G^+-S) ≥c(G-S), and the statement follows.Let R be a T-region of a graph G and let I, O be the set of all inner, outer vertices of R, respectively. Let t ≥ 1 and let S be a set of vertices of G such that c(G - S) > 1/t|S|.If |S ∩ O| ≥ 2, then there is a separating set S' = (SI) ∪ A such that c(G - S') > 1/t|S'| where A is chosen as follows.* If |S ∩ O| = 2, then A consists of the non-simplicial vertex of I which is the common neighbour of the vertices of S ∩ O. * If |S ∩ O| = 3, then A consists of two non-simplicial vertices of I.In both cases, we modify S as suggested; and we let S' be the resulting set. Clearly, S' is a separating set and c(G-S') - c(G-S) ≥ 0, and we observe that c(G-S') - c(G-S) ≥ |S'|- |S|.Since t ≥ 1, we have either 0 > 1/t(|S'|- |S|) or |S'|- |S| ≥1/t(|S'|- |S|). Consequently, we obtain c(G-S') - c(G-S) ≥1t|S'|- 1t|S|.We use c(G - S) > 1/t|S| and we conclude that c(G - S') > 1/t|S'|. We recall that a set D of vertices is dominating a graph G if every vertex of G - D is adjacent to a vertex of D. We note that every pair of vertices of degree 6 is dominating T, so every separating set in T has at least two of these vertices. As a consequence of Proposition <ref>, we note the following. The graph T is 3/2-tough. Furthermore, if a graph contains a T-regionand a vertex not belonging to the T-region, then the toughness of the graph is at most 5/4; and we show that this is the correct value for F^+_1, 0 and F_1, 0. The graphs F^+_1, 0 and F_1, 0 are 5/4-tough.By Proposition <ref>, it suffices to show that F^+_1, 0 is 5/4-tough. To the contrary, we suppose that there is a separating set S of vertices such that c(F^+_1, 0 - S) > 4/5|S|.We consider such S adjusted by using Proposition <ref>, in sequence, for all T-regions of F^+_1, 0.We letdenote the set of all components of F^+_1, 0 - S consisting exclusively of inner vertices of some T-region.We note that the existence of such component implies that at least two outer vertices of the corresponding T-region belong to S (since every pair of outer vertices of T is dominating T).We let r_2, r_3 denote the number of T-regions whose exactly 2, 3 outer vertices belong to S, respectively.We let c denote the common vertex of all T-regions. Considering the inner vertices of the T-regions, we call simplicial such vertices white and the remaining such vertices grey. Except for c, the outer vertices of the T-regions are called black. The vertices adjacent to a black vertex but not belonging to a T-region are called blue. We let c' denote the vertex adjacent to all blue vertices and x denote the vertex of F^+_1, 0 not belonging to F_1, 0.We letdenote the set of all components of F^+_1, 0 - S containing a black vertex or a blue vertex.We shall use a discharging argument to avoid complicated inequalities. We assign charge 5 to every component of F^+_1, 0 - S, and we aim to distribute all assigned charge among the vertices of S, and to show that every vertex of S receives charge at most 4, contradicting the assumption that c(F^+_1, 0 - S) > 4/5|S|. We pre-distribute the charge according to the following rules.* Every grey vertex of S receives 4 of the total charge of the components ofbelonging to the same T-region as the grey vertex. * For every T-region, the remaining charge of all components ofbelonging to this T-region is distributed equally among the black vertices of S belonging to this T-region. * Every blue vertex of S receives as much of the total charge of components outsideas possible (at most 4). We note that after the pre-distribution, the charge of every component ofis 0; and we focus on the remaining charge of the rest of the components.If c(F^+_1, 0 - S) - || ≤ 1, then we have || ≥ 1 and the remaining charge is at most 5. Consequently, r_2 ≥ 1 or r_3 ≥ 1, and in both cases, the vertices of S can still receive charge at least 5, a contradiction.We assume that c(F^+_1, 0 - S) - || ≥ 2. We show that F^+_1, 0 - S has no component containing c and no black vertex. The existence of such component implies that all black vertices belong to S, and these vertices can still receive 7/2· 20. A contradiction follows by counting the maximum possible number of components not belonging to .Consequently, F^+_1, 0 - S has at most one component belonging to neithernor . If || ≤ 1, then there is such component (since c(F^+_1, 0 - S) - || ≥ 2). Clearly, this component contains c' or x, that is,all blue vertices or c' and at least two blue vertices belong to S, and a contradiction follows.We assume that || ≥ 2. On the other hand, considering the graph induced by black and blue vertices, we observe that the size of a maximum independent set of this graph is 13. Thus, || ≤ 13.Since || ≥ 2, we note that at least || black and at least || blue vertices belong to S.We let d denote the number of blue vertices of S minus ||. We recall that F^+_1, 0 - S has at most || + 1 components not belonging to . Thus, the remaining charge, which is yet to be distributed, is at most || + 5 - 4d; and since d ≥ 0, it is at most || + 5.If c does not belong to S, thenthe black vertices of S can still receive at least 7/2||. Clearly, 7/2|| ≥ || + 5 since || ≥ 2, a contradiction.We assume that c belongs to S.Clearly, c can receive 4. We note that the black vertices of S can still receive 3r_2 + r_3; and since r_2 + 2r_3 ≥ ||, they can receive at least 1/2|| + 5/2r_2 (thus, at least 1/2||).If c' does not belong to S, then we consider the graph induced by c' and by the black and blue vertices, and we observe that r_2 + d ≥ || - 1 (since there are at least || - 1 components ofnot containing c'). Consequently, we have 1/2|| + 5/2r_2 + 4 > || + 5 - 4d since || ≥ 2, a contradiction.We assume that c' belongs to S, so c' can receive 4. If d ≥ 1, then the remaining charge is at most || + 1; and we have 1/2|| + 4 + 4 > || + 1 since || ≤ 13, a contradiction.We assume that d = 0. If there is not a component containing x as its only vertex, then the remaining charge is ||. Similarly as above, we have 1/2|| + 4 + 4 > ||, a contradiction.We assume that F^+_1, 0 - S has a component consisting of x, so all neighbours of x belong to S. Since d = 0 and since || ≥ 2, we have r_2 ≥ 1. If || ≤ 11, then we have 1/2|| + 5/2r_2 + 4 + 4 ≥ || + 5, a contradiction. If || = 12, then the parity implies that r_2 ≥ 2 or r_3 ≥ 6, and a contradiction follows.We assume that || = 13. We consider the structure ofand we observe that r_3 ≤ 4, that is, r_2 ≥ 5. Consequently, we get 1/2|| + 5/2r_2 + 4 + 4 > || + 5, and we obtain the desired distribution of the assigned charge, a contradiction.Thus, F^+_1, 0 is 5/4-tough. We continue with the following. The graphs F^+_2, 0 and F_2, 0 are 8/7-tough.By Proposition <ref>, it suffices to show the toughness of F^+_2, 0 which we do via contradiction.We suppose that there is a separating set S of vertices in F^+_2, 0 such that c(F^+_2, 0 - S) > 7/8|S|. Since S is separating, we have |S| ≥ 3. Consequently, we can assume that c(F^+_2, 0 - S) ≥ 3.To specify the structure of S, we consider the graph F_2, 0. We note that F_2, 0contains two T-regions and the T-regions share their outer vertices; and we call these outer vertices black (as well as the corresponding vertices of F^+_2, 0). We observe that every pair of black vertices is dominating F_2, 0.Since c(F^+_2, 0 - S) ≥ 3, we have that at least two black vertices belong to S. We note that F^+_2, 0 contains one T-region (and its outer vertices are black), and we modify S using Proposition <ref>; and we let S' be the resulting set. Considering the possibilities, we observe that c(F^+_2, 0 - S') ≤7/8|S'|, a contradiction.We shall use the toughness of the building blocks F_i, 0 (given by Propositions <ref> and <ref> and by Corollary <ref>) to show the toughness of the constructed graphs F_i, n. § GLUING TOUGH GRAPHSWe shall use the following lemma as the main tool for showing the toughness of graphs which are obtained by the standard construction for bounding the shortness exponent.For i = 1,2, let G^+_i and G_i be t-tough graphs such that G_i is obtained by removing vertex v_i from G^+_i. Let U be a graph obtained from the disjoint union of G_1 and G_2 by adding new edgessuch that the minimum degree of the bipartite graph (N(v_1), N(v_2)) is at least t. Then U is t-tough.We assume that t > 0 and that there exists a separating set of vertices in U. We let X be such a set and we let X_i = X ∩ V(G_i) for i = 1,2. Clearly, 2 ≤ c(U-X) ≤ c(G_1-X_1) + c(G_2-X_2), and we use it to show that c(U-X) ≤1/t|X|.If X_i is a separating set in G_i, then the toughness of G_i implies that c(G_i-X_i) ≤1/t|X_i|.We suppose that X_1 is not separating in G_1, and we observe that c(U- X) ≤ c(G_2-X_2) +1.If c(U- X) ≤ c(G_2-X_2), then X_2 is separating in G_2, and thus, c(G_2-X_2) ≤1/t|X_2| and the desired inequality follows since |X_2| ≤ |X|.We assume that c(U- X) = c(G_2-X_2) +1. Clearly, if N(v_2) ⊆ X_2, then c(G_2-X_2) +1 = c(G^+_2-X_2); so X_2 is separating in G^+_2 and we have c(G^+_2-X_2) ≤1/t|X_2| and the inequality follows.In addition, we assume that there is a vertex of N(v_2) not belonging to X_2. We recall that this vertex has at least ⌈ t ⌉ neighbours in N(v_1). Since c(U- X) = c(G_2-X_2) +1, we note that all these neighbours belong to X_1. Thus, |X_1| ≥ t and we have c(G_1-X_1) ≤1/t|X_1|.Similarly, we get that if X_2 is not separating in G_2, then c(G_2-X_2) ≤1/t|X_2|. We conclude that (no matter whether X_i is separating or not) we have c(G_i-X_i) ≤1/t|X_i| for both i = 1,2; and the inequality follows.We remark that a similar statement appears in <cit.>; and it is used in <cit.>. (Compared to Lemma <ref>, the main difference is that the minimum degree of the considered bipartite graph is required to be at least 1.) The graphs depicted in Figure <ref> show that this statement is false.We view Lemma <ref> as a fixed version of this statement, and we remark that Lemma <ref> can be applied in the arguments of <cit.> and <cit.>. We note that Lemma <ref> can be viewed as a generalization of a similar statement (for 1-tough graphs) presented in <cit.>. § TOUGHNESS OF THE CONSTRUCTED GRAPHSIn this section, we clarify that the graphs F_i, n have the properties stated in Proposition <ref>. We recall that the order of the constructed graphs and the length of their longest cycles are given by Corollary <ref> and by Proposition <ref>. For every i = 1,2, we show the toughness of the graphs F_i, n using induction on n. The case n = 0 is verified by Propositions <ref> and <ref>.By induction hypothesis, F_i, n-1 has the required toughness, and by Proposition <ref>, so does a graph obtained from F_i, n-1 by removing a simplicial vertex; we shall apply Lemma <ref>, and we view these two graphs as playing the role of G^+_1 and G_1 andwe view graphs F^+_i, 0 and F_i, 0 as playing the role of G^+_2 and G_2. We consider the graph obtained from F_i, n-1 by replacing oneof its simplicial vertices by a copy of F_i, 0 and by adding edges as in the present construction (we recall the construction of graphs F_i, n for i = 1,2; see Section <ref>). By Lemma <ref>, the resulting graph has the required toughness. Thus, we can replace a simplicial vertex of the resulting graph and apply Lemma <ref> again; and repeating this argument, we obtain that F_i, n has the required toughness.Similarly, we show the toughness of F_3, n by induction on n. By Corollary <ref>, the toughness of F_3, 0 is greater than 1. We consider Lemma <ref> (see below) applied on the graph F_3, n-1 playing the role of G (and then applied repeatedly on the resulting graph), and we obtain that the toughness of F_3, n is greater than 1. Similarly to Lemma <ref>, the following lemma can be used to construct large graphs from smaller ones while preserving certain toughness. Let G be a graph of toughness greater than 1 which contains a K_4-region and let G' be a graph obtained from G by replacing this K_4-region by a T-region (in the natural way). Then the toughness of G' is greater than 1.We let X' be a separating set of vertices in G', and we shall show that c(G'-X') < |X'|. We consider the set X obtained from X' by removing all inner vertices of the new T-region. For the sake of simplicity, we let c = c(G' - X') - c(G - X) and x = |X'X|, and we note that it suffices to show that c(G-X)+c < |X|+x.Considering the choice of X, we observe that c ≤ x. We conclude the proof by showing that c(G-X) < |X|.If X is separating in G, then the inequality is given by the toughness of G. We can assume that c(G-X) = 1. Consequently, c(G'-X') > c(G-X), so at least two outer vertices of the new T-region belong to X'. Thus, they belong to X, that is, |X| ≥ 2 and the inequality follows.§ NOTE ON LONGEST PATHSWe remark that (using similar arguments as in Section <ref> we obtain that) a longest path of F_i,n has p_i(n) vertices where * p_1(n) = 2 + c_1(n) + 2∑_k=0^n-1 c_1(k),* p_2(n) = 1 + sgn(n) + c_2(n) + c_2(n-1) + 2∑_k=0^n-2 c_2(k),* p_3(n) = 7 · 2^n+2 + 2sgn(n) -15n -19. § ACKNOWLEDGEMENTS The author would like to thank Petr Vrána for discussing different strategies for the graph constructions, and to thank the anonymous referees for their helpful comments. The research was supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports and by the project 17-04611S of the Czech Science Foundation. 99surv D. Bauer, H. J. Broersma, E. Schmeichel: Toughness in graphs — A survey, Graphs and Combinatorics 22 (2006), 1–35.74 D. Bauer, H. J. Broersma, H. J. Veldman: Not every 2-tough graph is Hamiltonian, Discrete Applied Mathematics 99 (2000), 317–321.chpl T. Böhme, J. Harant, M. Tkáč: More than one tough chordal planar graphs are Hamiltonian, Journal of Graph Theory 32 (1999), 405–410.3-conn G. Chen, X. Yu: Long cycles in 3-connected graphs, Journal of Combinatorial Theory, Series B 86 (2002) 80–99.chva V. Chvátal: Tough graphs and hamiltonian circuits, Discrete Mathematics 5 (1973), 215–228.dill M. B. Dillencourt:An upper bound on the shortness exponent of 1-tough, maximal planar graphs,Discrete Mathematics 90 (1991), 93–97.sef B. Grünbaum, H. Walther: Shortness exponents of families of graphs, Journal of Combinatorial Theory 14 (1973), 364–385.hara J. Harant: Toughness and nonhamiltonicity of polyhedral graphs, Discrete Mathematics 113 (1993), 249–253.5/4 J. Harant, P. J. Owens: Non-hamiltonian5/4-tough maximal planar graphs, Discrete Mathematics 147 (1995), 301–305.momo J. W. Moon, L. Moser: Simple paths on polyhedra, Pacific Journal of Mathematics 13 (1963), 629–631.owensTatra P. J. Owens: Non-hamiltonian maximal planar graphs with high toughness, Tatra Mountains Mathematical Publications 18 (1999), 89–103.owens P. J. Owens: Shortness parameters for polyhedral graphs, Discrete Mathematics 206 (1999), 159–169.tkac M. Tkáč: On the shortness exponent of 1-tough, maximal planar graphs, Discrete Mathematics 154 (1996), 321–328.TutteW. T. Tutte:A theorem on planar graphs, Transactions of the American Mathematical Society 82 (1956), 99–116. | http://arxiv.org/abs/1705.09475v3 | {
"authors": [
"Adam Kabela"
],
"categories": [
"math.CO"
],
"primary_category": "math.CO",
"published": "20170526082810",
"title": "An update on non-Hamiltonian $\\frac{5}{4}$-tough maximal planar graphs"
} |
Instituto de Ciencia de Materiales de Aragón and Departamento de Física de la Materia Condensada, CSIC-Universidad de Zaragoza, E-50009 Zaragoza, Spain Instituto de Física Fundamental, IFF-CSIC, Calle Serrano113b, Madrid E-28006, Spain Instituto de Ciencia de Materiales de Aragón andDepartamento de Física de la Materia Condensada, CSIC-Universidad deZaragoza, E-50009 Zaragoza, Spain Instituto de Física Fundamental, IFF-CSIC, Calle Serrano113b, Madrid E-28006, Spain Instituto de Ciencia de Materiales de Aragón and Departamento de Física de la Materia Condensada, CSIC-Universidad de Zaragoza, E-50009 Zaragoza, Spain Fundación ARAID, Paseo María Agustín 36, E-50004 Zaragoza, Spain In this work we discuss the emergence of approximate causality in a general setup from waveguide QED —i.e. a one-dimensional propagating field interacting with a scatterer. We prove that this emergent causality translates into a structure for the N-photon scattering matrix. Our work builds on the derivation of a Lieb-Robinson-type bound for continuous models and for all coupling strengths, as well as on several intermediate results, of which we highlight (i) the asymptotic independence of space-like separated wave packets, (ii) the proper definition of input and output scattering states, and (iii)the characterization of the ground state and correlations in the model.We illustrate our formal results by analyzing the two-photon scattering from a quantum impurity in the ultrastrong coupling regime, verifying the cluster decomposition and ground-state nature. Besides, we generalize the cluster decomposition if inelastic or Raman scattering occurs, finding the structure of the in momentum space for linear dispersion relations. In this case, we compute the decay of the fluorescence (photon-photon correlations) caused by this S-matrix. 42.50.Ct, 42.50.-p, 03.65.-w, 11.55.BqEmergent Causality and the N-photon Scattering Matrix in Waveguide QED D. Zueco December 30, 2023 ======================================================================§ INTRODUCTION Causality is expected to hold in every circumstance. The causality principle states that two experiments which are space-like separated, such that no signal travelling at the speed of light can connect them, must provide uncorrelated results <cit.>. In Quantum Field Theory (QFT), strict causality imposes that two operators A(x,t) and B(y,t') acting on two space-like separated points (x,t) and (y,t'), must commute, [ A(x, t), B(y,t^')]= 0 if|x-y| -c |t-t^'| > 0,where c is the speed of light (we restrict ourselves to 1+1 dimensions).Another consequence of causality in QFT appears in the study of scattering events or collisions: scattering matrices describing causally disconnected events must “cluster”, or decompose into a product of independent scattering matrices <cit.>. In fact, all acceptable QFT interactions must result in S-matrices fulfilling such a decomposition <cit.>.Nonrelativistic quantum mechanics is an effective theory which allows signals to propagate arbitrarily fast, but which may give rise to different forms of emergent approximate causality. The typical examples are low-energy models in solid state, where quasiparticle excitations have a maximum group velocity. In this case, there exists an approximate light cone, outside of which the correlations between operators are exponentially suppressed. This emergent causality was rigorously demonstrated by Lieb and Robinson <cit.> for spin-models on lattices with bounded interactions that decay rapidly with the distance. Lieb-Robinson bounds not only imply causality in the information-theoretical sense <cit.>, but lead to important results in the static properties of many-body Hamiltonians, such as the clustering of correlations and the area law in gapped models <cit.>.In this work we demonstrate the existence and explore the consequences of emergent causality in the nonrelativistic framework of waveguide QED <cit.>. Theses systems consist of photons propagating in low-dimensional environments —waveguides, photonic crystals, etc—, interacting with local quantum systems. Such models do not satisfy Lorentz or translational invariance, they are typically dispersive, and the photon-matter interaction may become highly non-perturbative.Experimental implementations include dielectrics <cit.>, cavity arrays <cit.>, metals <cit.>, diamond structures <cit.>, and superconductors <cit.> interacting with atoms, molecules, quantum dots, color centers in diamond or superconducting qubits. The focus of waveguide QED is set on quantum processes involving few photons and scatterers. In this regard, it is not surprising that there exists an extensive theoretical literature for waveguide-QED systems <cit.>, which develops a variety of analytical and numerical methods for the study of the N-photon <cit.>.The main result in this work is the structure of the N-photon in waveguide QED, rigorously deduced from emergent causality constraints. Our result builds on a general model of light-matter interactions, without any approximations such as the rotating-wave (RWA), the Markovian limit, or weak light-matter coupling. To derive the decomposition we are assisted by several intermediate and important results, of which we remark (i) the freedom of wave packets far away from the scatterer, (ii) Lieb-Robinson-like independence relations and approximate light-cones for propagating wave packets, (iii) a characterization of the ground state correlation properties, and (iv) a proper definition and derivation of scattering input and output states.We illustrate our results with two representative examples. The first one is a numerical study of scattering in the ultrastrong coupling limit <cit.>, where we demonstrate the clustering decomposition and the nature of the ground state predicted by our intermediate results. The second is an analytical study of a non-dispersive medium interacting with a general scatterer, which admits exact calculations. Here, we find the shape of the from general principles, including the inelastic processes. We recover the nontrivial form computed by Xu and Fan for a particular case in <cit.> and find the natural generalization of the standard cluster decomposition. The paper has the following organization. Sect. <ref> presents the nonrelativistic Hamiltonian that models the interaction between propagating photons and quantum impurities, the concept of wave packet, a review of the scattering theory needed, and two conditions necessary for the validity of our results.Sect. <ref>summarizes our formal theory arriving to the general N-photon scattering compatible with causality.Sect. <ref>presents the examples applying the theory. We close this work with further comments and outlooks. Intermediate lemmas, theorems, andtechnical issues are discussed in the appendices.§ MODEL AND SCATTERING THEORY §.§ Waveguide QEDmodel The simplest model that describes a waveguide-QED setup consists of a one-dimensional bosonic medium and a scatterer. Using units such that ħ=1, it readsH = H_0+H_sc + ∫ (g_k G^† a_k + g_k^* G a_k^†)dk .The first term stands forthe free-Hamiltonian of the photonsH_0 = ∫ω_k a_k^† a_k dk,with frequency ω_k for momentum k,which is created (annihilated) by the corresponding Fock operator a_k (a_k^†), satisfying [a_k, a_k^'^†]= δ (k - k^'). Thelast two terms are the Hamiltonian H_sc of the finite-dimensional system, which is the scatterer, and the dipolar interaction term described by the bounded operators G and the coupling strengths g_k. We assume that the coupling strengths in position spaceg_x = 1/√(2 π)∫ dke^i k x g_khave a finite support centered around x_ sc=0. The model(<ref>) is not exactly solvable in general. For instance, if the scatterer is a two-level system, H_sc∝σ_z and G=σ_x the model is the celebrated spin-boson model<cit.>, which results in a nontrivial ground state with localized photonic excitations around the scatterer.The discussion below assumes a single photonic band ω_k ∈ [ω_min, ω_max] and typically a chiral medium k≥ 0, ∂_kω_k ≥ 0. This is a rather standard simplification which does not affect the generality and applicability of our results. More generic dispersion relations and non-chiral medium can be taken into account by introducing additional degrees of freedom in the photons (chirality, band index, etc.) and keeping track of those quantum numbers in a trivial extension of our results.§.§ Localized wave packetsIn order to talk about causality, we introduce a set of localized wave packets to which an approximate position can be ascribed. As we will see below, approximate localization becomes essential in the discussion, allowing us to discuss the order in which photons interact with the scatterer.Let us introduce the creation operator ψ_k̅x̅(t)^† for a wave packet asψ_k̅x̅(t-t_0)^† = ∫ e^ik x̅-iω_k(t-t_0)ϕ_k̅(k) a_k^† dk.The wavefunction ϕ_k̅(p)=ϕ(p-k̅)∈ℒ^2 is normalized and centered around the average momentum k̅. The exponential factor e^ikx̅ ensures the wave packet is centered around x̅ in position space at time t=t_0.As wave packets we will use both Gaussianϕ_k̅(k) = 1/√(2π)√(σ)exp[-(k-k̅)^2/4σ^2],and Lorentzian envelopesϕ_k̅(k) = √(σ/π)1/k-k̅ + iσ.These wave functions are only approximately localized in the sense that the probability of finding a photon decays exponentially far away from the center x̅. The width σin momentum space implies a localization length 1/σ in position space. Note that our definition of the wave packet lacks factors such as √(ω_k) or ω_k^1/2 which typically appear when transforming back to position space from a linear bosonic problem that was diagonalized in frequency space. This is a convenient definition that avoids divergences when computing things such as the number of photons. The choice of prefactors is ultimately irrelevant when we take the limit σ→0 in many of the argumentations below.Fig. <ref> illustrates the collision of two approximately localized wave packets against a quantum impurity in a chiral medium. The average momentum of the wave packets k̅_1 or k̅_2 determines the group velocity at which the photons move v_g(k) =∂_kω_k. The wave packets may be distorted due both to the dispersive nature of the medium and the interaction with the scatterer.§.§ Scattering operatorIn the typical scattering geometry, the interaction occurs in a finite region. Besides, it is assumed that asymptotically far away from that region the field is a linear combination of free-particle states (generated via creation operators on the non-interacting vacuum) even in the presence of the scatterer-waveguide interaction.A sufficient condition for this is that both the ground state and any non-propagating excited state accessible by scattering |Ω_μ⟩ are indistinguishable from the vacuum state |vac⟩ far away from the scatterer. Mathematically this occurs whenlim_x̅→±∞⟨Ω_μ|O(x̅,Δ)|Ω_μ|=⟩⟨vac| 𝑂(x̅,Δ) |vac|,⟩where O(x̅,Δ) is an operator with compact support in the finite interval x̅-Δ/2 < x <x̅+Δ/2 and the vacuum state | vac⟩ is such that a_k| vac⟩=0 ∀ k. Besides, the free particle states must satisfy the asymptotic condition <cit.>:‖ U(t) | Ψ⟩ - U^0(t) |Ψ_ in /out⟩‖t→∓∞⟶ 0,with U(t) the evolution operator of the full Hamiltonian (<ref>) and U^0(t) = e^-iH_0t the free-evolution operator.The scattering operator S relates the amplitude of the output and input fields through|Ψ_ out⟩ = S|Ψ_ in⟩ ,which, using (<ref>), has the formal expression:S = lim_t_±→±∞ U_I(t_+,t_-) .Here, U_I(t_+, t_-)= e^i H_0 t_-e^-i H (t_+- t_-)e^-i H_0 t_+ is the evolution operator in the interaction picture. Using again Eq. (<ref>) leads to |Ψ_ in /out⟩ = U_0^† (t_-/+) U (t_-/+)| Ψ⟩≡ |Ψ (t_-/+) ⟩_I, which shows that the input and output fields are represented in the interaction picture.Related quantities are the scattering amplitudes.For example, the single-photon amplitude is defined as:A ≡⟨Ω_μ | ψ^ out (t_+)Sψ^ in (t_-)^† | Ω_ν⟩with ψ^ in (t_-)^†= ψ_k̅x̅(t_-)^† and an analogous definition for ψ^ out (t_+) and the photon mean position x̅ being well separated from the scatterer.One of the goals of this work is to find the most general form for the amplitude A compatible with causality, thus providing a more clear understanding of the structure of the scattering matrix.§.§ Sufficient conditions for having a well-defined scattering theoryGiven a general Hamiltonian (<ref>), it is not generally known whether the condition (<ref>) is satisfied. Thus, the existence of scattering states must be assumed. In this work, we provide a further evidence of the validity of this assumptions by demonstrating a limited version of Eq. (<ref>) (see App. <ref>) for the unique ground state of Hamiltonian (<ref>), which reads ⟨Ω_0|ψ_k̅x̅^†ψ_k̅x̅|Ω_0|≤⟩𝒪(|x̅|^-n), | x̅|→∞provided that (i) for all k, |g_k/ω_k| <∞ and (ii) that the correlators C_kp =⟨Ω_0|a_k^† a_p|Ω_0|$⟩ aren-differentiable functions. Unfortunately, this result is insufficient for treating the most general case. It is well known that the Hamiltonian (<ref>) may support excited eigenstates which are localized around the scattering center <cit.>, which in the literature are usually referred as ground states. Two paradigmatic examples of scatterer with multiple ground states are the three-levelΛatom, with two electronic ground state, and a two-level system coupled to a cavity array in the ultrastrong coupling regime <cit.>.However, we have been unable to find a general proof that (<ref>) is satisfied (and thus that input and output states can be defined) for non propagating excited states that appear in these systems. In order to make any progress, and as usual in the literature, we have instead assumed a plausiblefirst condition:the Hamiltonian (<ref>) hasa finite set of ground states,{|Ω_μ⟩},which are localized in the sense of Eq. (<ref>).Notice that with this assumption (<ref>) has a well defined theory [See. App. <ref>]. This condition allows the expression of the elements ofSin the momentum basis:(S_𝐩𝐤)_μν = ⟨Ω_μ|∏_ia_p_i S∏_j a_k_j^†|Ω_ν|.⟩In this paper we will also assume asecond condition: the N-photon scattering process conserves the number of flying photons in the input and output states. We only provide results for the sector of the scattering matrix that conserves the number of excitations, excluding us from considering other scattering channels, such as downconversion processes. Notice, however, that a large number of systems fulfill this condition. For instance, the unbiased spin-boson model (whereH_sc ∝σ_zandG=σ_x) exactly conserves the number of excitations within the rotating-wave-approximation, which is valid when the coupling strength is much smaller than the photon energy. But even in the ultrastrong coupling regime, when counter-rotating terms are important, numerical simulations have shown that the scattering process conserves the number of flying excitations within numerical uncertainties [cf. Refs. <cit.> and Sect. <ref>]. § CAUSALITY AND THE N-PHOTON SCATTERING MATRIX§.§ Approximate causality We are describing waveguide QED using nonrelativistic models for which strict causality (<ref>) does not apply. However, as a foundational result we have been able to prove that the waveguide-QED model (<ref>) supports an approximate form of causality. This form states that there exists an approximate light cone, defined by the maximum group velocity,c=max(∂_kω_k). Two wave-packet operators which are outside their respective cones and far away from the scatterer approximately commute.To be precise, we define the distanced (x-y, t-t^')=|x̅ - y̅| - c|t-t'|and prove in App. <ref>that‖[ψ_k̅x̅(t),ψ_p̅y̅(t')^†]‖ = 𝒪(1/|D|^n)+𝒪(1/|D_0|^n-1) ,withD≡d (x-y, t-t^')andD_0 ≡min{d (x̅, t), d (x̅, t_0),d(y̅, t), d (y̅, t_0) }the distance between the packets andthe minimum distance between them and the scatterer respectively. The powernstands because we use that the dispersion relation isn-times differentiable. A sketch of the proof is as follows.First, we prove (<ref>) for free fields, i.e. for wave packets moving underH_0. In the Heisenberg picture, the phasesi k (x̅ -y̅) -i ω_k (t - t^')can be bounded by the distanced(x-y, t-t^'). Using the Riemann-Lebesgue lemma (∫e^ i k z f(k) dk →0, asz→∞) we find thepower law decay,|D|^-n. Causality is thereby linked to the cancellation or averaging of fast oscillations in the unitary dynamics. Applying a similar technique to the interaction term in (<ref>) allows us to prove that packets away the influence of the scatterer evolve freely, producing the second algebraic decay term|D_0|^1-n.This leads the second decay|D_0|^1-n. If their evolution can be approximated by the evolution underH_0, what we found for the commutator of free-evolving packets holds also in the interacting part.This result is analogous to Lieb-Robinson-type bounds that were initially developed for a lattice of locally interacting spins <cit.>, and which were later generalized to finite-dimensional models, anharmonic oscillators, master equations, and spin-boson lattices <cit.>. It is important to remark that the approximate causality in Eq. (<ref>) is not obtained for the free theory, but for the full waveguide-QED model. As a consequence, it can be used to derive important results on the photon-scatterer interaction.§.§ Causality and thescattering matrix Causality imposes restrictions on the<cit.>, among which is the cluster decomposition that we summarize here. For now, let us consider the case of a unique ground state and split theinto a free partS^0and an interacting partT, both in momentum space S_pk = S^0_pk+ iT_pk.The interacting partTaccounts for processes in which two or more photons coincide and interact simultaneously with the scatterer. Causality is then invoked to argue that they cannot influence each other if the input events are space-like separated. Thus,Tdoes not contribute to the scattering amplitude as wave packets fall apart|x̅_i-x̅_j|→∞.This, together with energy conservation, imposes the constraintiT_pk=iC_pkδ(E_p-E_k)<cit.>.In this limit the only term contributing to the scattering amplitude is the free part,S^0.In QFT (typically) occurs momentum conservation which implies thatS^0_pk = 1/N!∏_n=1^N S_p_nk_n+permutations[k_n↔ k_m,p_n↔ p_m],withS_p_nk_n∝δ(ω_p_n-ω_k_n)the one-photonS-matrix. This is nothing but the cluster decomposition. Fourier transformingS^0_pk, this structure also holdsS^0_yx = 1/N!∏_n=1^N S_y_nx_n+permutations[x_n↔ x_m,y_n↔ y_m].This shall be relevant in the following section, where we will work in position space. §.§ Generalized cluster decomposition Our goal is to explain how approximate causality (<ref>) implies a cluster decomposition for theS-matrix. We will also show that in waveguide QED the photonmomenta need not be conserved and thatS^0may not have the structure given by Eq. (<ref>). To understand how causality fixes the form ofS^0we refer to our Fig. <ref> where two well separated wave packets interact with a scatterer. The scattering amplitude is,A=⟨Ω_μ|∏_m=1^2 ψ_p̅_m y̅_m^ out (t_+) ∏_n=1^2ψ_k̅_n x̅_n^ in(t_-)^†|Ω_ν⟩.Note that for a sufficiently large separation of the wave packets, the output state of the first packet must be causally disconnected. This implies that the input operator for the first wave packet must commute with the output operator for the second packet [see Eq. (<ref>)].Notice that the second output and the first input will not commute in general.We can then approximate, at any degree of accuracy, the above amplitude as,A ≃⟨Ω_μ|ψ_p̅_2y̅_2^ out (t_+) ψ_k̅_2y̅_2^ in(t_-)^† ψ_p̅_1x̅_1^ out (t_+) ψ_k̅_1x̅_1^ in(t_-)^†|Ω_ν⟩.Let us know insert the identity between the operatorsψ_k̅_2x̅_2^in(t_-)^†andψ_p̅_1y̅_1^out (t_+). Recalling the conditionsdiscussed in Sect. <ref>, namely the localized nature for the ground states together with the fact that there is not particle creation, just{|Ω_λ⟩}_λ=0^M-1will contribute to the identity.The final result is:A_12 = ∑_λ=0^M-1 A_1,ν→λ A_2,λ→μ,withA_1, ν→λ=⟨Ω_λ| ψ_p̅_1 y̅_1^out (t_+) ψ_k̅_1 x̅_1^in(t_-)^†|Ω_ν⟩and similarly forA_2,λ→μ. We can generalize this expression toNphotons, with initial average positionsx̅_1> x̅_2>…> x̅_Nand asymptotic ground statesλ_0:= νandλ_N:=μA = ∑_λ_1,…,λ_N-1=0^M-1∏_n=1^N A_n,λ_N+1-n→λ_N-n,withA_n,λ_N+1-n→λ_N-n=⟨Ω_λ_N-n|ψ_p̅_n y̅_n^ out (t_+) ψ_k̅_n x̅_n^ in(t_-)^†|Ω_λ_N +1-n⟩.Thesketched constructive demonstration (a complete demonstration is given in App. <ref>) has confirmed that causality imposes that the amplitude can be built from single photon events whenever those are well separated.Inelastic processes yield the sum over intermediate states.If only one ground state is considered, the amplitude is the productA=Π_n A_n.In this case, thein momentum space recovers the typical structure in QFT (see Eq. (<ref>)). However, when inelastic-scattering events occur, the sum in (<ref>) leads to a particular structure for the free part of the scattering matrixS^0that we discuss now.We now find the structure forS^0in position space compatible with the amplitude (<ref>). For the sake of simplicity, we work with chiral waveguides and a monotonously growing group velocity,∂_kω_k≥0. Therefore, we can order the events using step functions, eliminating unphysical contributions (e.g. the wave packetψ_k̅_2 x̅_2arriving beforethanψ_k̅_1 x̅_1, see Fig. <ref>).Some algebra, fully described in Appendix <ref> yields thatS^0has the following structure( S^0_yx )_μν= ∑_λ_1…λ_N-1=0^M-1∏_n=1^N (S_y_nx_n)_λ_n-1λ_n∏_m=1^N-1θ(y_m+1-y_m) + permutations[x_n↔ x_m,y_n↔ y_m],The sum over intermediate states and the Heaviside functions are a direct consequence of causality, since they order the different wave packets and keep track of the state of the scatterer for each arrival. Nevertheless, if the ground state is unique (M=1), the step functions cancel out and we recover the structure described by (<ref>). However, strikingly, forM>1thiscannot be written as a product of one-photon scattering matrices, up to permutations, due to the Heaviside functions.In order to shed light on this, it is convenient to move to momentum space. Although(S_pk^0)_μνcannot be analytically calculated for a general dispersion relation, a mathematical expression can be found for a linear one. This calculation will be presented in Sect. <ref> The final result is that(S_pk^0)_μνcannot be written as a product of one-photonS-matrices. This has been recently pointed out in the particular example of aΛatom by Xu and Fan <cit.>.§ APPLICATIONS The set of previous theorems and conditions create a framework that describes many useful problems and experiments in waveguide QED. We are now going to illustrate two particular problems which are amenable to numerical and analytical treatment, and which highlight the main features of all the results.The first problem consists of a two-level system that is ultrastrongly coupled to a photonic crystal. The scattering dynamics has to be computed numerically. The simulations fully conform to our our framework, showing the fast decay of photon-qubit dressing with the distance, the independence of space-like separated wave packets, and the decomposition of the two-photon scattering amplitude as a product (for the chosen parameters, the one-photon scattering is elastic).The second problem consists of a general scatterer with several ground states that is coupled to a non-dispersive medium and it serves to illustrate the breakdown of thedecomposition in momentum space §.§ Ultrastrong scattering Let us consider a system described by the following HamiltonianH=Δσ^+σ^- + ϵ∑_x a_x^† a_x - J∑_x (a_x^† a_x+1 + a_x+1^† a_x) + g(σ^- + σ^+)(a_0 + a_0^†).The scatterer is a two-level system described by the ladder operatorsσ^±and the level splittingΔ. The lattice tight-binding Hamiltonian, describes an array of identical cavities with frequencyϵ, cavity-cavity couplingJ, and bosonic modes[a_x,a_y^†]=δ_xy.The lattice model is diagonalized in momentum space, giving raise to a cosine-shaped dispersion relation,ω_k=ϵ-2Jcosk.The scatterer-waveguide interaction, which is described by the last term, is point-like andgis the coupling constant.The light-matter interaction term can be expressed as a sum of the rotating-wave part,g(σ^+a_0 + σ^- a_0^†), and the so-called counter-rotating terms,g(σ^- a_0 + σ^+ a_0^†). The latter can be neglected ifgis small enough compared to the other energies of the full system. This is known as the rotating-wave approximation (RWA). It is well known that the RWA simplifies the problem because (i) the new effective model conserves the number of excitations and (ii) the ground state is the trivial vacuum|vac⟩withσ^- | vac ⟩= a_x |vac ⟩=0∀x.However, when the coupling strength is large enough –the so-called ultrastrong coupling regime–, the RWA fails to describe the dynamics and one has to use the full Rabi model (<ref>). This regime not only represents an interesting and challenging problem where we can test our theoretical framework, but it describes a family of current experiments <cit.> for which the following simulations are of interest. An important remark is that, despite the fact that the number of excitationsN̂ = ∑_x a_x^†a_x + σ^+σ^-is not a good quantum number, i.e.[H,N̂]≠0, numerical simulations indicate that the total number of flying photons is asymptotically conserved throughout the simulation <cit.>. Therefore, the second condition needed for proving our results is fulfilled (see Sect. <ref>).We have studied this model using the matrix-product-state variational ansatz, a celebrated method for describing the low-energy sector of one-dimensional many-body systems <cit.>, which has been recently adapted to the photonic world in <cit.>. Using this ansatz, we computed the nontrivial minimum-energy state <cit.>, which consists of a photonic cloud exponentially localized around the qubit, see Fig. <ref>. This result confirms our theoretical predictions from Eq. (<ref>) and implies that the minimum-energy state|Ω_0⟩can be approximated by the vacuum far away from the qubit. According to the previous result, we can generate free wave packets, such as input and output states of Eqs. (<ref>) and (<ref>)) by inserting photons far away from the scatterer. We have used the MPS ansatz to study the evolution of input states which consist of a pair of photons, see Eq. (<ref>), with|Ω_ν⟩=|Ω_0⟩. Both wave packets will be Gaussians, Eq. (<ref>), with mean momentumk̅and widthσ. The numerical simulations show that the scattering is elastic for the chosen parameters (ϵ=1,J=1/π,Δ=ϵ=1, andg=0.3) <cit.>.We have also demonstrated numerically that the correlation between output photons vanish as the separation between the input wave packet increases. Our study aimed at computing the two-photon wave function in momentum space,ϕ_p_1,p_2(t) = ⟨Ω_0|a_p_1a_p_2|Ψ(t)⟩. This was used to compute the fluorescenceFat timet_+, the number of output photons whose energy and momentum differ from the input wave packets. More precisely F = ∫ dp_1dp_2 |ϕ_p_1,p_2(t_+)|^2,withp_1andp_2such thatω_p_1+ω_p_2=2(ω_k̅±σ_ω)andω_p_1,ω_p_2∉(ω_k̅-σ_ω,ω_k̅+σ_ω), beingσ_ωthe width of the input wave packets in energy space.Fig. <ref>(g) showsFas function of the distance between the incident wave packets. When the wave packets are close enough the fluorescence maximizes and the output wave function shows a nontrivial structure, withϕ_p_1,p_2(t_+)≠0even though|p_1|≠k̅or|p_2|≠k̅(see panels (a) and (c)). The wave function has also a rich structure in position space, with antibunching in the reflection component and superbunching in the transmission one (see panels (b) and (d)). This structure was already found in the RWA <cit.>. For long distances, the fluorescenceFvanishes [see panels (e) and (f)]. In these cases, the output state is clearly uncorrelated: in position space it is formed by two well-defined wave packets andϕ_p_1,p_2(t_+)goes to zero if|p_1|≠k̅or|p_2|≠k̅. All this is aconsequence of the cluster decomposition, see Eq. (<ref>) and Th. <ref> in App. <ref>.§.§ Inelastic scattering and linear dispersion relation: the cluster decomposition revisited We setω_k=c|k|inH_0. The scatterer and interaction are described byH_ sc = ∑_ν=0^M-1 E_ν|Ω_ν⟩⟨Ω_ν| + ∑_J=0^M'-1Ẽ_J|J⟩⟨J|,H_ int = ∑_J=0^M'-1∑_ν=0^M-1 g_J,ν(|J⟩⟨Ω_ν| a_0 +H.c.),where{|Ω_ν⟩}and{|J⟩}are the ground and decaying states of the scatterer, respectively,{E_ν}and{Ẽ_J}are their energies,MandM'is the number of ground and excited states, respectively, andg_J,νis the coupling strength corresponding to the transition|Ω_ν⟩↔|J⟩(see Fig. <ref>).This is a prototypical situation in waveguide QED. E.g., if there are two ground states,M=2, and the decaying state is unique,M'=1, the scatterer is aΛatom. From now on, we work in units such thatc=1. We further assume chiral waveguides: the scatterer only couples tok > 0, which simplifies the final expressions, so we can start from Eq. (<ref>). Before writing down the two-photonS^0-matrix in momentum space, we need the one-photon scattering matrix. Imposing energy conservation, it has to be(S_pk)_μν=t_μν(k)δ(p+E_μ-k-E_ν),withkandpthe incident and outgoing momenta, respectively, and|Ω_ν⟩and|Ω_μ⟩the initial and final ground states. The factort_μν(k)is the so-called transmission amplitude. The Dirac delta guarantees energy conservation. Then, the two-photonS^0-matrix, Eq. (<ref>) in momentum space is(S_𝐩𝐤^0)_μν =1/(2π)^2∬ (S_𝐲𝐱^0)_μν e^-i𝐩^T𝐲+i𝐱^T𝐤 d^2𝐲d^2𝐱= i/2π∑_n,m=1^2 ∑_λ=0^M-1t_μλ(k_n) t_λν(k_n')/p_m+E_μ -k_n -E_λ + i0^+××δ(p_1+p_2+E_μ - k_1-k_2-E_ν).Here,n^'≠n, e.g.,n^'=2ifn=1. The computation is detailed in Appendix <ref>. This structure has recently been found by Xu and Fan for aΛatom (M=2,M'=1) within the RWA and Markovian approximations <cit.>. At first sight (<ref>) may look striking. The matrixS^0is not the product of two Dirac-delta functions conserving the single-photon energy, as discussed in Sect. <ref>.The mathematical origin of the structure can be traced back to its form in position space, Eq. (<ref>). The Heaviside functions set the order in which the different wave packets impinge on the scatterer. The product of Dirac-delta functions is recovered ifM=1[see App. <ref>]. Besides, Eq. (<ref>) is also remarkable because presents the natural generalization of the cluster decomposition for theS-matrix [Cf. Eqs. (<ref>) and (<ref>)] when inelastic processes occur in the scattering.A consequence of (<ref>) is thatS^0contributes to the fluorescenceF, Eq. (<ref>). This seems to contradict our previous arguments, sinceS^0is built from causally disconnected one-photon events (they do not overlap in the scatterer).To solve the apparent paradox we recall that (<ref>) is a matrix element in momentum space (delocalized photons). For wave packets (<ref>), the scattering amplitude is the integral of these wave packets with (<ref>). In doing so we find that the fluorescence decays to zero as the separation grows, thus solving the puzzle.In what follows the fluorescence decay is discussed within the fullS-matrix, i.e. we consider the contributions toFfromS^0andT(see Eq. (<ref>)). Energy conservation imposes that(T_p_1p_2k_1k_2)_μν = (C_p_1p_2k_1k_2)_μν δ(p_1+p_2+E_μ- k_1-k_2-E_ν). Since the contribution ofTvanishes as the photon-photon separation increases,Cmust be sufficiently smooth, at least smoother than a Dirac delta <cit.>. Then, we assume that(C_p_1p_2k_1k_2)_μνhas simple poles with imaginary parts{γ^C_n}. Similarly, we expect that divergences oft_μν (k)come from simple poles with imaginary parts{γ^t_n}. As far as we know, this structure has been found for allS-matrices in waveguide QED <cit.>.Let us write down the input state in momentum space|Ψ_ in⟩ = ∫ dk_1 dk_2 ϕ_1(k_1) ϕ_2(k_2)e^ik_2l a_k_1^† a_k_2^†|Ω_ν⟩.The functionsϕ_1(k)andϕ_2(k)are localized far away the scattering region in position space. The exponential factore^ik_2lensures the separation between both wave packets isl. The output state reads|Ψ_ out⟩=S|Ψ_ in⟩ =∑_μ∫ dp_1 dp_2 ϕ_μ^ out(p_1,p_2)a_p_1^† a_p_2^†|Ω_μ⟩with the two-photon wave functionϕ_μ^out(p_1,p_2)ϕ_μ^ out(p_1,p_2)∝∑_n=1^2 ∑_m=1^2 ∫ dk_n(i/2π∑_λt_μλ(k_n) t_λν(p_1+p_2+E_μ - k_n - E_ν)/p_m+E_μ-k_n-E_λ + i0^+. . + i(C̃_p_1p_2k_n)_μν) (ϕ_1(k_n)e^i(p_1+p_2+E_μ - k_n-E_ν)lϕ_2(p_1+p_2+E_μ-k_n-E_ν) + ϕ_1(p_1+p_2+E_μ - k_n-E_ν)e^ik_n lϕ_2(k_n) ) ,being(C̃_p_1p_2k_n)_μν = ∫dk_n̅ (C_p_1p_2k_nk_n̅)_μνδ(p_1+p_2+E_μ- k_n - k_n̅ - E_ν), withn̅≠n. Even though this expression is cumbersome, we can clearly identify the contribution ofS^0andT. We solve this integral by means of the residue theorem. Each poleγ_n^tandγ_n^C, together with the exponentialse^ik_n lande^i(p_1+p_2+E_μ-k_n-E_λ)l, gives an exponentially decaying term,e^-|γ_n^t|lore^-|γ_n^C|l. We choose Lorentzian envelopes for the wave packets. They have a pole atk̅ - iσ[see Eq. (<ref>)]. In consequence, the wave packets will give a term proportional toe^-σl. Lastly, the imaginary part of the pole of the first term vanishes,∼i0^+, so it gives a nondecaying term,e^-0^+ l=1. The real part of this denominator imposes the single-photon-energy conservation. Thus, it results in the amplitude for the single-photon events,∑_λA_1,ν→λA_2,λ→μ. Therefore,norS^0neitherTcontains fluorescent terms as the separation between the wave packets grows. The technical details are in App. <ref>. As a final application, one can find experimentally the poles of the one- and two-photon scattering matrices{γ_n^t}and{γ_n^C}by measuring the decay ofFwith the distance. § FINAL COMMENTS Our work represents a significant evolution over the field-theoretical methods <cit.> that have been so successfully adapted to the study of waveguide QED. Developing an extensive set of theorems shown in the appendices, we have completed a program that derives the properties of theN-photonfrom the emergent causal structure of a nonrelativistic photonic system. This, together with the fact that the ground states of the Hamiltonian are trivial far away from the scatterer and the asymptotic independence of input and output wave packets, allows us to build a consistent scattering theory. Among the consequences of this framework, we have explained how the existence of Raman (inelastic) processes modifies the usual form of the cluster decomposition to produce a structure that includes the particular example developed in <cit.>. Our formal results also provide insight in the outcome of simulations for problems where no analytical derivation is possible, such as a qubit ultrastrongly coupled to a waveguide <cit.>.As a second example, we have considered a non-dispersive mediaω_k = c |k|, where we found the general form for the scattering matrix in momentum space (independent of the scatterer and the coupling to the waveguide), which has been recently calculated for aΛatom <cit.> as a particular case.On top of that, we have clarified how fluorescence decays in a general scattering experiment. Throughout the previous discussion we have focused our attention to scattering processes which involve the same number of flying photons both at the input and the output [See Sect. <ref>], but this is just a convenient restriction that can be lifted. One may incorporate more scattering channels for the photons using extra indices to keep track of the photon-annihilation and creation processes, which results in a slightly more involved version of Theorem <ref>. In particular, we can incorporate photon-creation events (see e.g. <cit.>). Finally, our program can be extended to treat other systems, deriving a cluster decomposition for the scattering of spin waves in quantum-magnetism models or for fermionic excitations in many-body systems.We acknowledgesupport by the Spanish Ministerio de Economía y Competitividad within projects MAT2014-53432-C5-1-R and FIS2015-70856-P, the Gobierno de Aragón (FENOL group), and CAM Research Network QUITEMAD+ S2013/ICE-2801.§ THE GROUND STATE OF THE LIGHT-MATTER INTERACTION In this appendix we demonstrate that the ground state converges to the trivial vacuum far away from the scatterer, Eq. (<ref>). The nextlemma is neccessary to proof the main theorem.Given the waveguide-QED model (<ref>), we have the following bounds for the expectation values on its minimum-energy state |Ω_0⟩,| ⟨Ω_0|a_k^†a_p|Ω_0||⟩≤√(|g_k g_p/ω_kω_p| )⟨Ω_0|G G^† |Ω_0|.⟩Let us assume that |Ω_0⟩ is the minimum-energy state of H as given by Eq. (<ref>), and thus (H-E_0)|Ω_0⟩ = 0. The energy of the unnormalized state | χ⟩ = O |Ω_0⟩,created by any operator O must be larger or equal to that of the ground state, ⟨χ | (H - E_0) | χ|≥⟩0. Using (<ref>) ⟨χ|( H -E_0 )|χ⟩= ⟨Ω_0|O ^† H O-O^† O H|Ω_0|⟩we conclude with the useful relation⟨χ| H -E_0 |χ|=⟩⟨Ω_0| O ^† [H, O] |Ω_0|≥⟩0.Let us take O=a_k. The previous statement leads to⟨Ω_0|a_k^† (-ω_k a_k - g_k G)|Ω_0 |≥⟩0,or equivalently0 ≤⟨Ω_0|a_k^†a_k|Ω_0|≤⟩-g_k/ω_k⟨Ω_0| G a_k^† |Ω_0|.⟩Using Cauchy-Schwatz, this translates into the upper bound⟨Ω_0| a_k^† a_k |Ω_0|≤⟩|g_k|/ω_k√(⟨Ω_0| G G^†|Ω_0|⟨%s|%s⟩⟩Ω_0 |a_k^† a_k |Ω_0). Once the diagonal elements of the correlation matrix are bounded the nondiagonal can also be bounded.The correlation matrix is positive C ≥ 0 with C_kp = ⟨Ω_0|a_k^†a_p|Ω_0|$⟩.A property of positive matrices is <cit.> | C_kp | ≤√(|C_kk| |C_pp|)which implies (<ref>). With this lemma at hand we state: Let us define ψ_k̅x̅x^† as the operator (<ref>) removing the time-dependent part, where ϕ_k̅(k) is infinitely differentiable with a finite support K centered around k̅. Then, the expected value of ψ_k̅x̅^†ψ_k̅x̅ in the minimum-energy state fulfills⟨Ω_0|ψ_k̅x̅^†ψ_k̅x̅|Ω_0|→⟩0, |x̅|→∞,where we choose x_sc=0. Moreover, if we can assume that ⟨a_k^† a_p|$⟩ is ann-times differentiable function ofkandp, the bound will be improved⟨Ω_0|ψ_k̅x̅^†ψ_k̅x̅|Ω_0|≤⟩𝒪(|x̅|^-n), | x̅|→∞.Let us compute the expectation value of the number operator for a wave packet N := ⟨Ω_0|ψ_k̅x̅^†ψ_k̅x̅|Ω_0|$⟩,N= ∬⟨Ω_0|a_k^† a_p|Ω_0|e⟩^i(k-p) x̅ϕ_k̅(k)^*ϕ_k̅(p)dk dp.We can rewriteNas the Fourier transform of another functionN = ∫ e^i u x̅ F(u)du, whereF(u) := 1/2∫ϕ_k̅((u+v)/2)^*ϕ_k̅((u-v)/2)××⟨a_(u+v)/2^† a_(u-v)/2|d⟩v.We are now going to assume thatϕ_k̅(k)is a test function with compact supportKof size|K|centered aroundk̅, and infinitely differentiable. We will also assume that within its support|g_k/ω_k|^2 ⟨ GG^†⟩≤ C_ϕfor some constantC_ϕ. Then we can bound∫ |F(u)|du ≤ |K|^2 C_ϕ.Assuming that⟨Ω_0| a_k^† a_p|Ω_0⟩isn-times differentiable andusing the Riemann-Lebesgue theorem, we have then that|∫ e^i u x̅ F(u)du| ≤𝒪(|x̅|^-n)at long distances. § APPROXIMATE CAUSALITY§.§ Free-field causality We first prove causal relations in a free theory. In order to do so, we work with localized wave packetsψ_k̅x̅(t), Eq.(<ref>). Actual calculations are done with Gaussian wave packets, Eq. (<ref>). The following two lemmas are used in the demonstration of the theorem. Let the dispersion relation ω_k have an upper bounded group velocity v_k=∂_k ω_k:| v_k |≤ c.Then, the function f(k) = k x - ω_k t only has stationary points if the distance to the light cone is nonnegative. In other wordsd_c(x,t) = | x | - c | t | > 0 ⇔ | f^' (k) | > 0, ∀ k.Solving the equation f'(k)=x - ∂_k ω_k t = 0 leads to the condition x/t = v_k or |x/t|=|v_k|≤ c. Then, provided f'(k)=0, it follows |x|≤ c|t|⇒ d_c(x,t)≤ 0, which shows (<ref>).Assume that ω_k is n-times differentiable and that every derivative |ω_k^(r≤ n)| is upper bounded by an m-th order polynomial in |k|. Then the following integral bound applies|∫ e^i k x - 1/σ^2(k-k_0)^2 - i ω_k t p(k)dk| =max(σ^m+n + r,1) max(t^n,1) 𝒪(1/|x|^n).where p(k) is a polynomial of degree r. Result 5.1 from Ref. <cit.> states that the integral I(x) = ∫_a^b e^i k x q(k) dk may be integrated by parts n times, obtainingI(x) =∑_s=0^n-1(i/x)^s+1[e^iax q^(s)(a) - e^ibx q^(s)(b) ] +ϵ_n(x),where the error term satisfiesϵ_n(x) = (i/x)^n∫ e^ikx q^(n)(k)dk = o(x^-n)provided that q(k) is n-times differentiable and that q^(n)∈ L^1. Based on the conditions of the lemma, this is satisfied since q(k)=e^- 1/σ^2(k-k_0)^2 - i ω_k t p(k). The limits of the integral may be easily extended to ±∞, as explained in Result 5.2 from <cit.>. Since x^-sq^(s)(a) → 0 when a→±∞,∀ x, we obtainI(x) = ∫ e^i k x q(k) dk = (i/x)^n ∫ e^ikx q^(n)(k)dk,Moreover, q^(n), resulting from a product of derivatives of ω_k t,-k^2/σ^2 and the polynomial p(k) of degree r, is bounded by a polynomial of at most (m+n + r)-th order in |k|. Such a polynomial is integrable together with the Gaussian wave packet giving a constant prefactor. In estimating this factor, we can take the worst-case scenario for the terms in t, which appears at most n times together with (∂_kω_k)^n, and the monomials in|k|, which produce another prefactor σ^m+n+ r.Note that it would suffice to consider q(k) as a test function or even a Schwartz function since in this case all the differentiability requisities are fullfilled and x^-sq^(s)(a) → 0 → 0 when a→±∞,∀ x still holds, because these functions and their derivatives are rapidly decreasing. With these lemmas at hand we can prove Let the Hamiltonian be given just by the photonic part, H_0 = ∫ dk ω_k a_k^† a_k. Let ψ_k̅x̅(t) and ψ_p̅y̅(t') denote two localized wave packets of the form (<ref>). We will assume that (i) the absolute value for the group velocity of these wave packets is upper bounded by a constant c within the domain of the wave packets (| v_k| = |∂_k ω_k |≤ c) and (ii) the dispersion relation is n-times differentiable and that each derivative is upper bounded by a polynomial of at most order m:|∂^(r≤ n)_kω_k|≤ a_r + (|k|/b_r)^m, 0< a_r,b_r<+∞.The commutator between these wave packets is small whenever they are outside of their respective light cones, that is, whenever d = |y̅-x̅| - c|t'-t|≫ 0, ‖[ψ_k̅x̅(t),ψ_p̅y̅(t')^†]‖ = 𝒪(1/|d|^n), d→∞.Let us assume that the model evolves freely according to the free Hamiltonian H_0 = ∫ d kω_k a_k^† a_k. In this case, our wave packet operators have the simple formψ_k̅x̅(t) = ∫ e^ik x̅-iω_ktϕ_k̅(k)^* a_k(0)dk,and analogously for ψ_p̅y̅(t'). The commutator between operators readsI :=[ψ_k̅x̅(t),ψ_p̅y̅(t')^†] =∫ e^ik(x̅-y̅)-iω_k(t-t')ϕ_k̅(k)ϕ_p̅(k)^*dk.Let d= d_c(x̅-y̅,t-t') = |x̅-y̅| -c|t-t'|> 0, using Lemma <ref> we know that the exponent has no stationary point.Assuming w.l.o.g. x̅> y̅, t>t^' (other combinations are analogous) andwriting ω̃_k = ω_k -c k, we obtainI = ∫ e^ik(x̅- y̅)-iω_k(t-t')ϕ_k̅(k)^*ϕ_p̅(k)dk=∫ e^ikd_c(x̅-y̅, t-t')-iω̃_k(t-t')ϕ_k̅(k)^*ϕ_p̅(k)dk.The exponent ω̃_k = ω_k -c k is n-times differentiable and is upper bounded in modulus by a polynomial of degree m≥ 1. Lemma <ref> therefore allows us to bound the commutator by a term 𝒪(d^-n). Note that for a linear dispersion,ω_k=c |k|, we can rewrite this integral as a function of the distance between world lines from Eq. (<ref>),d=(x̅- y̅)-c(t-t'). Introducingk_±=(k̅±p̅)/2and using our Gaussian wave packets (<ref>), we obtain|I| = exp[-k_-^2/σ^2-d^2σ^2/4].This bound is better than the one we have found but it is compatible with Lemma <ref> and Theorem <ref>. §.§ Full model causality Causal relation (<ref>) can be extended to the full model (<ref>). Let H be the light-matter Hamiltonian given by Eq. (<ref>). We assume the conditions of Theorem <ref>: differentiable, polynomially bounded functions ω_k and g_k, with degrees n≥ 2. Then, all wave packets outside the light cone of the scatterer evolve approximately with the free Hamiltonian, H_0. More precisely, if (x̅,t_1) and (x̅,t_0) are two points outside the light coneψ_k̅x̅(t_1) = U_0(t_1,t_0)^†ψ_k̅x̅(t_0)U_0(t_1,t_0) +𝒪(1/|d_min|^n-1),where d_min = min{d(x̅,t_1),d(x̅,t_0)}≫ 0 andU_0(t,t_0) = exp (-i(t-t_0)H_0)is the free-evolution operator for the photons at time t_0.We start by building the Heisenberg equations for the operators∂_t a_k(t) = -iω_k a_k(t) - i g_k G(t).Making the change of variables a_k(t) = e^-iω_k tb_k(t), we have∂_t b_k(t) = -ig_k G(t) e^iω_k t,so that the wave packet operators evolved from some initial time t_s areψ_k̅x̅(t)= ∫ e^ikx̅ - i ω_k t[b_k(t_s) -i∫_t_s^t g_k G(τ)e^+iω_k τdτ]ϕ_k̅(k)dk=U_0(t,t_s)ψ_k̅x̅(t_s)U_0(t,t_s)^† -i∫_t_s^t [∫ e^ikx̅-ic(t-τ)g_kϕ_k̅(k)dk ] G(τ)dτ=U_0(t,t_s)ψ_k̅x̅(t_s)U_0(t,t_s)^†-i∫_0^t-t_s[∫ e^ikx̅-icτ'g_kϕ_k̅(k)dk ] G(τ)dτ'.The first part corresponds to free evolution, while the second part is an error term ε(t), which can be bounded. We will assume without loss of generality ‖G‖=1, with ||· || the Hilbert-Schmidt norm, and |t_1|>|t_0|. We have to choose the integration limits t and t_s so that sign(τ')=sign(x). If x>0 then t_1>t_0>0 and (t,t_s)=(t_1,t_0) is a good choice. If x<0 then 0>t_0>t_1 and again (t,t_s)=(t_1,t_0) is also a valid choice (τ'<0). This means we can introduce τ”=sign(x)τ'≥ 0 and bound|ε(t_1)|≤∫_0^|t_1-t_0||∫ e^i sign(x̅)kd_c(|x̅|,τ”)q(k)dk|dτ”≤∫_0^|t_1-t_0|𝒪(1/d_c(|x̅|,τ”)^n)dτ”≤𝒪( . 1/c(n-1)1/(|x̅|-cτ)^n-1|_τ=0^τ=|t_1-t_0|)≤𝒪(1/d_c(|x̅|,|t_1-t_0|)^n-1).Here we have taken into account that d_c(|x̅|,τ”)≥ d_c(|x̅|,|t_1-t_0|)> 0 in the domain of integration. We can now use the fact that d_c(|x̅|,|t_1-t_0|)≥ d_c(|x̅|,|t_1|)≥min{d_c(x̅,t_1),d_c(x̅,t_0)}, obtaining the expression in the theorem. §.§ Asymptotic Condition One important limitation of Theorem<ref> is that it is focused on the operators, not on the states themselves.This is a key point. For having a well defined scattering theory, the asymptotic condition must holds [See Sect <ref> and Eq. (<ref>)]. However, using Theorems <ref> and<ref> we have that, given a state|Ψ⟩≡ψ_k̅, x̅ (t_0)^† |Ω_ν⟩, thenU(t_±) |Ψ⟩=U(t_±) ψ_k̅, x̅ (t_0) U(t_±)^†|Ω_ν⟩ =U_0(t_±) ψ_k̅, x̅ (t_0)U_0(t_±)^†|Ω_ν⟩≡ U_0 (t_±) |Ψ_ in⟩Thefirst equality is up to a global phase. In the second line, we have used Theorem <ref>. In the last line, we can introduce input (output) states since the wave packets are well separated (t_±→±∞) from the scatterer and, by means of Theorem <ref> and the conditions presented in <ref> they are well defined free particle states. This last result warrants that, under rather general conditions, the light-matter Hamiltonian (<ref>) gives a physical scattering theory.§ SCATTERING AMPLITUDE DECOMPOSITIONLet us suppose the input state is|Ψ_ in⟩ = ψ_ in^†|Ω_ν⟩ = (∏_n=1^N ψ_k̅_n x̅_n^ in †)|Ω_ν⟩,with |x̅_n-x̅_m|→∞∀ n≠ m. Thus, the scattering amplitude of going to|Ψ_ out⟩ = ψ_ out^†|Ω_μ⟩ = (∏_n=1^N ψ_p̅_m y̅_m^ out †)|Ω_μ⟩,with |y̅_n-y̅_m|→∞∀ n≠ m, is reduced to a product of single-photon events:A= ∑_λ_1,…,λ_N-1=0^M-1∏_n=1^N ⟨Ω_λ_n-1|ψ_p̅_n y̅_n^ out (t_+) ψ_k̅_n x̅_n^ in(t_-)^†|Ω_λ_n⟩,being λ_0 = μ and λ_N=ν, with the wave packet operators given in the Heisenberg picture for t=t_±→±∞. The proof is based directly on causality. Therefore,we find convenient to discuss it here. The proof is done for the two-photon scattering. The generalization for N photons is straightforward.The scattering operator S is nothing but the evolution operator in the interaction picture, cf. Eq. (<ref>). This permits to write the scattering amplitudes as,A= ⟨Ψ_ out | S |Ψ_ in⟩ =⟨Ω_ν | ψ_ out U_I (t_+, t_-) ψ_ in^† | Ω_μ⟩=⟨Ω_ν | ψ_ out (t_+) ψ_ in (t_-)^† | Ω_μ⟩,In the second equality we have dropped an irrelevant global phase. Here, ψ_ in^† and ψ_ out^† are operators creating wave packets localized far away from the scatterer. Because of Theorem <ref>, they are well defined N-photon wave packets. Using Eqs. (<ref>) and (<ref>) the amplitude is given byA=⟨Ω_μ|∏_m=1^2 ψ_p̅_m y̅_m^ out (t_+) ∏_n=1^2ψ_k̅_n x̅_n^ in(t_-)^†|Ω_ν⟩.As |x̅_1 -x̅_2| can be arbitrarily large, we can always choose a time t_1 such thatψ_p̅_1 y̅_1^out(t)^†|Ω_μ⟩ is well separated from the scatterer for t>t_1, so ψ_p̅_1y̅_1^out(t) ≅ U_0(t,t_1)^†ψ^out_p̅_1 y̅_1(t_1) U_0(t,t_1).Besides, t_1 is such that the second wave packet is still far away from the scatterer.Therefore ψ_k̅_2x̅_2^in(t') ≅ U_0(t',t_1)^†ψ_k̅_2 x̅_2^in(t) U_0(t',t_1), for t'<t_1.Using Theorem <ref>, [ψ_p̅_1 y̅_1^out(t_+) , ψ_k̅_2y̅_2^in(t_-)^† ]→ 0 and Eq. (<ref>), the amplitude equals toA =⟨Ω_μ|ψ_p̅_2y̅_2^ out (t_+) ψ_k̅_2y̅_2^ in(t_-)^† ψ_p̅_1x̅_1^ out (t_+) ψ_k̅_1x̅_1^ in(t_-)^†|Ω_ν⟩.Finally, we insert the identity between the operators ψ_k̅_2x̅_2^ in(t_-)^† and ψ_p̅_1y̅_1^ out (t_+). Assuming there is not particle creation and just the ground states {|Ω_λ⟩}_λ=0^M-1 will contribute to the identity, ∑_λ=0^M-1|Ω_λ⟩⟨Ω_λ|, and we arrive to (<ref>).This comes because ψ_k̅_2x̅_2^ in(t_-)^† and ψ_p̅_1y̅_1^ out (t_+) asymptotically commute but not ψ_k̅_1x̅_1^ in(t_-)^† and ψ_p̅_2y̅_2^ out (t_+). This is a clear signature of causality, saying which one is arriving first. Lastly, notice that if the ground state is unique, |Ω_λ_n⟩=|Ω_0⟩, this ordering is not important as the amplitude is simply the product of single-photon scattering amplitudes. § SCATTERING AMPLITUDE FROM EQ. (<REF>) In this appendix, we prove that (<ref>) is consistent with the amplitude factorization from Theorem <ref>, Eq. (<ref>). We do it in the two-photon subspace.Before, we need the one-photon amplitude as an intermediate result. §.§ One photon We first need to compute the one photon amplitude. Let the one-photon input state be,|Ψ_in^1⟩=ψ_k̅_1,x̅_1^in † |Ω_ν⟩,with the creation operatorψ_k̅_1,x̅_1^in †given by Eq. (<ref>), removing the time dependence. For simplicity, we absorb the factore^ikx̅_1into the wave packet:ϕ_k̅_1,x̅_1(k) = e^ikx̅_1ϕ_k̅_1(k). In position space, the output state will read|Ψ_out^1⟩ = S|Ψ_in^1⟩ = ∑_μ=1^M ∫ dy dx (S_yx)_μνϕ_k̅_1,x̅_1(x)|y,Ω_μ⟩.Definingϕ_1,μν(y)= ∫ dx(S_yx)_μνϕ_k̅_1,x̅_1(x)and|ξ_out^1⟩_1,μν= ∫ dy ϕ_1,μν(y) |y;Ω_μ⟩,being|y;Ω_μ⟩= a_y^†|Ω_μ⟩the state with a photon atyand the scatterer in the ground state|Ω_μ⟩, the output state (<ref>) can be rewritten as|Ψ_out^1⟩ = ∑_μ=1^M |ξ_out^1⟩_1,μν.The probability amplitude will readA_1,ν→μ= ⟨Ω_μ| ψ_p̅_1,y̅_1^out S ψ_k̅_1,x̅_1^in † |Ω_ν⟩= ∫ dy ϕ_p̅_1,y̅_1(y)^* ϕ_1,μν(y).If the wave packets are monochromatic with momentak_1andp_1, respectively, this amplitude isA_1,ν→μ = (S_p_1k_1)_μν.§.§ Two photons The two-photon wave packet, as sketched in Fig. <ref>, is|Ψ_in^2⟩ = ψ_k̅_1,x̅_1^in †ψ_k̅_2,x̅_2^in †|Ω_ν⟩.By definition, the output state is|Ψ_out^2⟩ = S|Ψ_in^2⟩.Here, we are interested in he limit of well separated incident photons. Thus, only the linear partof the scattering matrixS^0is considered. We introduce the identity operator|Ψ_out^2⟩ = 𝕀S𝕀|Ψ_in^2⟩,with𝕀=1/2∑_μ=1^M ∫ dx_1dx_2 |x_1x_2;Ω_ν⟩⟨ x_1x_2;Ω_ν|,being|x_1x_2;Ω_ν⟩ = a_x_1^† a_x_2^†|Ω_μ⟩the symmetrized state with two photons atx_1atx_2and the scatterer at|Ω_ν⟩.Introducing (<ref>) in (<ref>) and considering (<ref>) and (<ref>) we get|Ψ_out^2⟩ =1/4∫ dy_1dy_2dx_1dx_2 ∑_μ,λ=1^M∑_n,m=1^2(S_y_nx_m)_μλ (S_y_n^'x_m^')_λνθ(y_n^'-y_n) (ϕ_k̅_1,x̅_1(x_1)ϕ_k̅_2,x̅_2(x_2)+ϕ_k̅_1,x̅_1(x_2)ϕ_k̅_2,x̅_2(x_1))|y_1y_2;Ω_μ⟩.with n^'≠ n and m^'≠ m. Now, we have to compute integrals asC=∫ dx_1dx_2 ∑_n,m(S_y_nx_m)_μλ (S_y_n^'x_m^')_λνϕ_k̅_i,x̅_i(x_1)ϕ_k̅_j,x̅_j(x_2)θ(y_n^'-y_n).Using Eq. (<ref>) C=∑_n=1^2( ϕ_i,μλ(y_n)ϕ_j,λν(y_n^') + ϕ_j,μλ(y_n)ϕ_i,λν(y_n^'))θ(y_n^'-y_n).Following the sketch drawn in Fig. <ref>, if x_m<x_m^', then ϕ_1(x_m)ϕ_2(x_m^') is zero, so ϕ_1,μν(y_n)ϕ_2,μν(y_n^') is zero if y_n < y_n^'. Therefore, choosing i=1 and j=2, the integral C readsC=∑_n=1^2ϕ_2,μλ(y_n)ϕ_1,λν(y_n^').One can easily show that the same expression holds if we take i=2 and j=1. The output state, Eq. (<ref>), then reads|Ψ_out^2⟩ =1/2∫ dy_1dy_2 ∑_μ,λ=1^M(ϕ_2,μλ(y_1)ϕ_1,λν(y_2)+ϕ_2,μλ(y_2)ϕ_1,λν(y_1))|y_1 y_2;Ω_μ⟩.Finally, the probability amplitude of going to the output stateψ_p̅_1,y̅_1^out †ψ_p̅_2,y̅_2^out †|Ω_μ⟩will be the overlap between this state and (<ref>). Using (<ref>) A_in→out= ⟨Ω_μ|ψ_p̅_1,y̅_1^outψ_p̅_2,y̅_2^outS ψ_k̅_1,x̅_1^in †ψ_k̅_2,x̅_2^out †|Ω_ν|⟩ =∑_λ=0^M-1 A_1,ν→λ A_2,λ→μ,as expected. In the calculations, we have set⟨Ω_μ| ψ_p̅_i,y̅_i^out S ψ_k̅_jx̅_j^in † |Ω_ν⟩=0fori≠ j, since we assume that both incident wave packets are far away.A final comment is in order. Without the step functions in (<ref>), the unphysical amplitudeA_2,ν→λ A_1,λ→μwould appear in the final probability amplitude.§ S^0 IN MOMENTUM SPACEHere, we show S^0 in momentum space follows Eq. (<ref>). After that, we prove the Dirac-delta structure is recovered if the ground state is unique.Let us write (S_p_1p_2k_1k_2^0)_μν as the Fourier transform of (S_y_1y_2x_1x_2^0)_μν (S_p_1p_2k_1k_2^0)_μν=1/(2π)^2∫ dy_1dy_2dx_1dx_2(S_y_1y_2x_1x_2^0)_μνe^-i(p_1y_1+p_2y_2) e^i(k_1x_1+k_2x_2).Due to the form of (S_y_1y_2x_1x_2^0)_μν, (<ref>), we have to compute integrals asI=∫ dxe^ikx(S_yx)_μν.Notice that (S_yx)_μν is the Fourier transform of (S_pk)_μν, Eq. (<ref>).Therefore,I=e^i(k+E_ν-E_μ)yt_μν(k).Considering this in (<ref>), we get(S_p_1p_2k_1k_2^0)_μν=1/(2π)^2 ∫ dy_1 dy_2e^-i(p_1y_1+p_2y_2) ∑_n,m=1^2 ∑_λ=0^M-1 e^i(k_ny_1 + k_n^' y_2)e^i[(E_λ-E_μ)y_m+(E_ν-E_λ)y_m^']t_μλ(k_n)t_λν(k_n^') θ(y_m^'-y_m),with n^'≠ n and m^'≠ m. The Fourier transform of the step function is1/√(2π)∫ dy e^-iqyθ(∓(y-y_0))=±i/√(2π)e^-iqy_0/q± i0^+.Therefore, integrating Eq. (<ref>) first in y_1 and later in y_2, we get(S_p_1p_2k_1k_2^0)_μν= i/(2π)^2∫ dy_2e^-i(p_1+p_2+E_μ-k_1-k_2-E_ν)y_2∑_n=1^2 ( t_μλ(k_n)t_λν(k_n^')/p_1+E_μ-k_n-E_λ+i0^+-t_μλ(k_n)t_λν(k_n^')/p_1+E_λ-k_n-E_ν-i0^+) = i/2πδ(p_1+p_2+E_μ-k_1-k_2-E_ν)∑_n=1^2 ∑_λ=0^M-1( t_μλ(k_n)t_λν(k_n^')/p_1+E_μ-k_n-E_λ+i0^+ -t_μλ(k_n)t_λν(k_n^')/p_1+E_λ-k_n-E_ν-i0^+)= i/2π∑_n,m=1^2 ∑_λ=0^M-1t_μλ(k_n) t_λν(k_n^')/p_m+E_μ -k_n -E_λ + i0^+δ(p_1+p_2+E_μ - k_1-k_2-E_ν),which is the expression given in the main text, Eq. (<ref>). This result has been recently reported for a Λ atom by Xu and Fan in <cit.>. Here, we show this is completely general due to our ansatz (Eq. (<ref>)).Lastly, we prove that Eq. (<ref>) is formed by two Dirac-delta functions if M=1. To do so, we use the following identity1/k+i0^+ = -iπδ(k) + 𝒫(1/k),with 𝒫 the principal value. Applying this identity to Eq. (<ref>) we get, (S_p_1p_2k_1k_2^0)_μν= i/2π∑_n,m=1^2 t(k_n) t(k_n^')(-iπδ(p_m -k_n) +𝒫(1/p_m-k_n)) δ(p_1+p_2 - k_1-k_2).Now, we sum over n and m (S_p_1p_2k_1k_2^0)_μν =1/2t(k_1)t(k_2)δ(p_1+p_2-k_1-k_2) (δ(p_1-k_1)+δ(p_1-k_2)+δ(p_2-k_1)+δ(p_2-k_2) + 𝒫(1/p_1-k_1)+𝒫(1/p_1-k_2)+𝒫(1/p_2-k_1)+𝒫(1/p_2-k_2)).Applying the constraint imposed by the global Dirac delta to p_2 to the second row, it is straightforward to see that they cancel each other, arriving to(S_p_1p_2k_1k_2^0)_μν =1/2t(k_1)t(k_2)δ(p_1+p_2-k_1-k_2)(δ(p_1-k_1)+δ(p_1-k_2)+δ(p_2-k_1)+δ(p_2-k_2)) =1/2t(k_1)t(k_2)(δ(p_2-k_2)δ(p_1-k_1) + δ(p_2-k_1)δ(p_1-k_2)+ δ(p_1-k_2)δ(p_2-k_1) + δ(p_1-k_1)δ(p_2-k_2))= t(k_1)t(k_2)(δ(p_1-k_1)δ(p_2-k_2) + δ(p_1-k_2)δ(p_2-k_1)),which is the usual expression in translational invariant (momentum conserving) QFT for the cluster decomposition, which also holds in waveguide QED if the ground state is unique. § FLUORESCENCE DECAY In this appendix, we calculate how the correlations and thus the fluorescence decay as the distancelbetween the packets grows (See Figs. <ref> and <ref>).The input state (<ref>) in momentum space is given by,|Ψ_ in⟩ = ∫ dk_1 dk_2 ϕ^ in(k_1,k_2)a_k_1^† a_k_2^†|Ω_ν⟩,withϕ^ in(k_1,k_2) = ϕ_k̅_1(k_1)e^ik_2 lϕ_k̅_2(k_2).In these expressions, the wave packetsϕ_k̅_n(k)are Lorentzian functions [see Eq. (<ref>)]. The out state is computed by means of Eq. (<ref>) |Ψ_ out⟩ = S|Ψ_ in⟩ = 𝕀S 𝕀|Ψ_ in⟩.With𝕀the identity operator in the two-photon sector:𝕀 = 1/2 ∫ dp_1 dp_2 ∑_μ a_p_1^† a_p_2^†|Ω_μ⟩⟨Ω_μ| a_p_1a_p_2. The scattering matrixSin momentum space is(S_p_1p_2k_1k_2)_μν = (S^0_p_1p_2k_1k_2)_μν + i (T_p_1p_2k_1k_2)_μν, with(S^0_p_1p_2k_1k_2)_μνgiven by Eq. (<ref>) and(T_p_1p_2k_1k_2)_μν = (C_p_1p_2k_1k_2)_μνδ(p_1+p_2+E_μ - k_1-k_2-E_ν)yielding|Ψ_ out⟩ = ∫ dp_1 dp_2 ∑_μϕ_μ^ out(p_1,p_2)a_p_1^† a_p_2^†|Ω_μ⟩,withϕ_μ^ out(p_1,p_2)∝∑_n=1^2 ∑_m=1^2 ∫ dk_n(i/2π∑_λt_μλ(k_n) t_λν(p_1+p_2+E_μ - k_n - E_ν)/p_m+E_μ-k_n-E_λ + i0^+. . + i(C̃_p_1p_2k_n)_μν) (ϕ_k̅_1(k_n)e^i(p_1+p_2+E_μ - k_n-E_ν)lϕ_k̅_2(p_1+p_2+E_μ-k_n-E_ν) + ϕ_k̅_1(p_1+p_2+E_μ - k_n-E_ν)e^ik_n lϕ_k̅_2(k_n) ) .Which is nothing butEq. (<ref>) that we have rewrittenhere for the discussion. As said in Sect. <ref>, we assume thatt_μν(k)and(C_p_1p_2k_nk_n̅)_μνhave simple poles with imaginary parts{γ_n^t}and{γ_n^C}respectively. Then, this integral is solved by taking complex contours and applying the residue theorem. In order to integrate the term proportional toe^i(p_1+p_2+E_μ - k_n-E_λ)l, we take the contour shown in Fig. <ref>(a) so that the exponential factor does not diverge.For the same reason,for that proportional toe^i k_n lwe take the contourof Fig. <ref>(b). AstandChave first-order poles, when integrating each pole, we just have to evaluate the rest of the function at the pole. Then,tandCgive terms proportional toe^-|γ_n^t|lande^-|γ_n^C|l, respectively. Now we consider the contribution to the integral of the wave packets,ϕ_k̅_n(k). We choose Lorentzian functions, with a simple pole atk=k̅_n - iσ(see Eq. (<ref>)). In consequence, we have a term proportional toe^-σ l. Lastly, the denominator in the first term has a pole with zero imaginary part.Therefore, itscontribution does not decay withl.Importantly enough, this pole enforces single-photon energy conservation givingsingle-photon amplitudes,∑_λ A_1,ν→λA_2,λ→μ.Finally, let us mention that we do not need to impose that thatt_μν(k)and(C_p_1p_2k_nk_n̅)_μνhave simple poles. Higher order poles, by virtue of the Cauchy Integral formula for the derivatives, also would yield exponential decay. apsrev4-1 | http://arxiv.org/abs/1705.09094v1 | {
"authors": [
"Eduardo Sánchez-Burillo",
"Andrea Cadarso",
"Luis Martín-Moreno",
"Juan José García-Ripoll",
"David Zueco"
],
"categories": [
"quant-ph",
"math-ph",
"math.MP"
],
"primary_category": "quant-ph",
"published": "20170525090245",
"title": "Emergent Causality and the N-photon Scattering Matrix in Waveguide QED"
} |
The Extragalactic Background Light Revisited University of Padova, Physics and Astronomy Department, Vicolo Osservatorio 3, I-35122 Padova, Italy [email protected] addition to its relevant astrophysical and cosmological significance, the Extragalactic Background Light (EBL) is a fundamental source of opacity for cosmic high energy photons, as well as a limitation for the propagation of high-energy particles in the Universe. We review our previously published determinations of the EBL photon density in the Universe and its evolution with cosmic time, in the light ofrecent surveys of IR sources at long wavelengths.We exploit deep survey observations by the Herschel Space Observatory and the Spitzer telescope, matched to optical and near-IR photometric and spectroscopic data, to re-estimate number counts and luminosity functions longwards of a few microns, and the contribution of resolved sources to the EBL.These new data indicate slightly lower photon densities in the mid- and far-infrared and sub-millimeter compared to previous determinations. This implies slightly lower cosmic opacity for photon-photon interactions. The new data do not modify previously published EBL modeling in the UV-optical and near-IR up to several microns, while reducing the photon density at longer wavelengths.This improved model of the EBL alleviates some tension that had emerged in the interpretation of the highest-energy TeV observations of local blazars, reducing the case for new physics beyond the standard model (like violations of the Lorenz Invariance, LIV, at the highest particle energies), or for exotic astrophysics, that had sometimes been called for to explain it. Applications of this improved EBL model on current data are considered, as well as perspectives for future instrumentation, the Cherenkov Telescope Array (CTA) in particular.The extragalactic background light revisited and the cosmic photon-photon opacity Alberto FranceschiniGiulia RodighieroReceived September 15, 2016; accepted March 16, 2017 ================================================================================= § INTRODUCTION The extragalactic background radiation at various electromagnetic frequencies, from radio to gamma-rays, is a fundamental constituent of the Universe, and was demonstrated to permeate it quite uniformly (e.g., Longair 2000). Such radiations have a key role during most of the history of universal expansion and the formation of cosmological structures. Given their ubiquity, radiations are a fundamental source of opacity for the propagation of high-energy cosmic-ray particles and photons throughout space-time (Nikishov 1962; Gould and Schreder 1966).The most intense and cosmologically relevant among the diffuse radiation components is the Cosmic Microwave Background. As discussed below (Sect. 2), these numerous photons make a wall to the propagation of ultra-energetic particles and photons in the PeV regime (the Greisen–Zatsepin–Kuzmin effect), that has been confirmed by many experiments (see e.g., the Pierre Auger Collaboration 2010).Another particularly important component of this radiation is the Extragalactic Background Light (EBL) in the wavelength interval between 0.1 and 1000 μm, produced by cosmic sources during the evolutionary history of the Universe. Interactions between these and very high-energy photons from astrophysical sources, and the consequent pair production, produce strong and observable opacity effects (e.g., Stecker et al. 1992, among others). The corresponding high-energy exponential cutoffs are customarily identified in the high- and very-high energy (HE [0.2 to 100 GeV] - and VHE [above 100 GeV]) spectra of several blazars observed with imaging atmospheric Cherenkov telescopes, now operating between a few tens of GeV to tens of TeV (e.g., HEGRA, HESS, MAGIC, see e.g., Dwek and Slavin 1994, Stanev and Franceschini 1998, Aharonian et al. 2006, Albert et al. 2006). Of course, not only photons from cosmic sources interact with EBL photons, but any particles of sufficiently high energy, from cosmic rays to neutrinos, in principle. Therefore, the issue of a high-precision determination of the time-evolving EBL is of central importance.Several efforts to model the EBL and its time evolution have been published, including those based on physical prescriptions about the formation and evolution of cosmic structures, in particular the semi-analytical models of galaxy formation. This is the so-called “forward evolution” approach, advocated by Gilmore et al. (2012), among others. The alternatives are the more empirical approaches essentially based on local properties of cosmic sources and educated extrapolations back in cosmic time (e.g., Stanev & Franceschini 1998,Pei, Fall, and Hauser 1999, Stecker et al. 2006, Stecker, Malkan & Scully 2006, Franceschini, Rodighiero, and Vaccari 2008, henceforth AF2008), the “backward evolution” modeling.Once the background photon densities are defined, the calculation of the cosmic photon-photon opacity and its effects on very high-energy spectra of blazars are standard physical practice (e.g., Stecker et al. 1992). Many papers have compared the predictions for such attenuated spectra with multi-wavelength blazar observations, and the results appear often rather consistent, for both local and high-redshift objects.An interesting example, illustrativeof what is currently feasible, is reported in Ackermann et al. (2012), who analyzed years of Fermi Large Area Telescope (LAT) observations of a large sample of blazars over a wide range of redshifts to investigate the effects of the pair-production opacity in various redshift bins, assuming smooth spectral extrapolations towards the highest energies and the EBL evolutionary models by AF2008.These LAT-normalized blazar spectra show indeed cutoff features increasing with redshift just as expected by the model, providing remarkable proof of the overall validity of our understanding of the local EBL photon density and its time evolution.Similarly, the multi-wavelength high-energy spectra of both BL Lac and Flat-spectrum radio quasars (the Active Galactic Nuclei - AGN - populations emitting at the highest energies, blazars henceforth) are usually fairly well reproduced in terms of standard blazar emission models (like the Synchrotron Self Compton or the External Compton processes), once corrected for photon-photon absorption based on the most accurate EBL models. There are however still some areas of concern related with observations of the highest energy blazar spectra.One is the reported relative independence (or only moderate dependence) of the observed spectral indices of blazars in the limited redshift interval currently probed at TeV energies, while a faster increase might be expected due to the strong dependence of the opacity on the source distance (De Angelis et al. 2009; Persic & de Angelis 2008). Another related consideration was concerning the detection of variable emission in the energy range 70 to 400 GeV from the flat spectrum radio quasar PKS 1222+216, where such high-energy photons would be expected to be absorbed in the broad line region (Tavecchio et al. 2012).Another source of possible concern are some apparent upturns of the spectra of blazars at the highest energies, once the observed spectra are corrected for EBL absorption. A characteristic instance would have been shown by the HEGRA observations of Mkn 501, whose EBL-corrected VHE spectrum appeared to show a statistically significant hardening above 10 TeV, as reported for example in Costamante (2012, 2013).All the above considerations may have important implications for fundamental physics, as they have raised an important question about photon propagation in space. It has been suggested that photons can oscillate into axion-like particles (ALPs), which are a generic prediction of extensions of the Standard Model of elementary particles, in particular the super-string theories. Photons can oscillate with axions in a similar way as massive neutrinos do, which would require an external field, for example a nano-Gauss magnetic field, to compensate for the photon and ALP spin difference. Then a fraction of VHE photons could escape absorption, because ALPs do not interact with ambient photons (De Angelis et al. 2007; Tavecchio et al. 2012).ALPs may even be an important constituent of the cosmos because, while not contributing to dark matter, they are a possible candidate for the quintessential dark energy. It has also been proposed that photons and particle interactions at very high energies might reveal deviations from predictions of the Standard Model of particle physics. A violation of the Lorentz transformation of special relativity might happen, for example, in the framework of quantum gravity theories (Jacobson et al. 2003), but would only manifest itself at energies comparable to the Planck energy. As suggested by many, this violation of the Lorentz invariance (LIV) might be testable for photons of energies above ∼ 10 TeV interacting with the EBL, by either increasing or decreasing the standard photon-photon opacity. The famous luminous outburst of the local blazar MKN 501 in 1997 has offered the opportunity to test LIV effects in the spectrum above 10 TeV, corresponding to interactions with EBL photons of wavelengths >10 ,if we consider the relation between interacting photon energies at the maximum of the cross section: λ_max≃ 1.24 (E_γ [TeV]) μ m .Based on the above considerations, several experiments are now being planned or under development, targeting the highest-energy photons, such as the ASTRI mini array (Di Pierro et al. 2013; Vercellone et al. 2013), HAWC (Abeysekara et al. 2013); HiSCORE (Tluczykont et al. 2014), LHAASO (Cui et al. 2014), all well sensitive in the photon energy interval 1-100 TeV, and above. Ultimately, the Cherenkov Telescope Array (CTA) will offer vast, factor 10 sensitivity improvement compared to current instrumentation over a very wide energy range (Acharya et al. 2013).All this calls for a revision of the EBL modeling, with a particular consideration of the longer wavelength infrared and submillimetric regime. Our previous effort in AF2008 has revealed remarkable success in reproducing absorption spectral features for sources over a wide redshift interval and up to photon energies of a few TeV. However that EBL model was only partly optimized for wavelengths above several μm,including only a preliminary account of the Spitzer MIPS mid-IR data.The present paper is dedicated to reconsidering the issue of the propagation in space-time of the highest-energy cosmic photons. In particular, we update AF2008 by including new very extensive data in the far-IR and submillimeter obtained with the Herschel Space Observatory, in a spectral region where the EBL and its evolution with time cannot be directly measured. The updated model also makes full use of deep survey data from the Spitzer telescope.Section 2 summarizes the new data needed for the improved EBL model. Section 3 reports on our data-fitting scheme and multi-wavelength spectral corrections. Section 4 describes the updated EBL model and the cosmic photon density. This accounts only for the contributions by known sources, like galaxies, active galactic nuclei and quasars, and does not consider various published attempts to measure the total-light EBL in the near-IR. Section 5 summarizes our new results on the cosmic photon-photon opacities and applications to a few known blazars. Section 6 contains some discussions on our results and prospects for future observations, including some detailed simulations for observations with the Cherenkov Telescope Array (CTA) concerning in particular the effects of a truly diffuse background from primeval sources.We use in the following a standard cosmological environment with H_0=70 km s^-1 Mpc^-1, Ω_m=0.3, Ω_Λ=0.7. We indicate with the symbol S_24 the flux density in Jy at 24 μm (and similarly for other wavelengths). L_λ/L_⊙ is the ν L_ν luminosity in solar units. § THE NEW DATA With reference to the analysis of AF2008, the new data made available during the recent years for the estimate of the contributions of extragalactic sources to the EBL concern in particular the source's infrared emission. Instead, the available data in the optical and near-IR portion of the galaxy spectrum have largely remained unchanged. Considering also that the cosmic opacity originating from UV-optical background photons in AF2008 have proved so far quite successful in reproducing the blazar spectra at about 1 TeV or less, we have chosen not to modify the model at wavelengths shorter than 5 (corresponding approximately to the spectral coverage of the Spitzer IRAC surveys).Many of the data at longer wavelengths that we will exploit in the present analysis were already discussed in detail in Franceschini et al. (2010, AF2010). We concentrate here on the new data only, that have mostly come from two infrared observatories in space, the NASA Spitzer telescope and the ESA Herschel observatory. §.§ The Spitzer MIPS number counts and luminosity functionsThe Spitzer MIPS imaging camera has been used to map several deep sky regions with the 24 μm channel and, to much shallower depths, at 70 μm. The former observations benefited from the excellent instrumental sensitivity and good spatial resolution, while the latter observations were limited in both senses. From the local EBL viewpoint, what matters particularly are the source number counts at both wavelengths that are reported in differential normalized Euclidean units in Figures <ref> and <ref>.MIPS observations at 70 (and those at 160 μm) are limited by sensitivity and source confusion due to the poor angular resolution. Galaxy number counts based on MIPS data by Frayer et al. (2006) and Bethermin et al. (2010) are reported in differential Euclidean-normalized units in Figures <ref>.Figure <ref> also includes data from the Herschel observatory that will be presented below. All these are compared with the model predictions discussed later in the paper.Important to note is that both these counts, as well as those discussed in following sections, show a flat, roughly Euclidean behavior at bright fluxes, a well-defined maximum at a given flux density value, and then a rather quick convergence towards faint fluxes. This has the implication that the source contribution to the total EBL intensity at the two wavelengths is mostly resolved at the limiting fluxes of the deepest surveys (see Madau & Pozzetti 2000 for a similar consideration about the optical counts), and the residual contribution by the fainter sources is negligible. We estimate the latter, in any case, from our model fit to the data in Sect. <ref>, easily extrapolated to the faintest fluxes. Since we are interested not only in the local EBL intensity, but also in its time evolution, we need information on the source spatial density and emissivity as a function of redshift. The classically used descriptors for this are the redshift-dependent luminosity functions, giving the source number density per comoving volume as a function of luminosity and cosmic time. Various flux-limited Spitzer samples have been used for deriving redshift surveys with good degrees of completeness, all suitable for the calculation of redshift-dependent luminosity functions. Rodighiero et al. (2010) have worked on a complete 24 μm selected sample and used it to calculate galaxy luminosity functions at the rest-frame wavelengths of 15 μm, shorter than the observation wavelengths to minimize the K-corrections for the typical source redshift (z∼ 1). We report these redshift-dependent luminosity functions in Figure <ref>, together with the predictions of our evolutionary model in Sect. <ref>. Functions at longer wavelengths are discussed in the following section.§.§ The Herschel multi-wavelength number counts and luminosity functions The two imaging instruments onboard Herschel, PACS (Poglitsch et al. 2010), and SPIRE (Griffin et al. 2010), offer capabilities of deep imaging in the far-IR and sub-millimeter at thewavelengths of 70, 100, 160 μm and 250, 350, 500 μm, respectively, with large improvement in sensitivity and angular resolution compared to previous space facilities. SPIRE has carried out cosmological surveys in the submillimeter over tens of square degrees below the confusion limits (the HerMES survey, Oliver et al. 2010, 2012), and over hundreds of square degrees in a contiguous fashion to sensitivities of S_250∼ 30-50 mJy (HerMES and H-ATLAS surveys, Eales et al. 2010). PACS observations (the PEP programme, Lutz et al. 2011, and H-GOODS, Elbaz et al. 2011), in particular, have exploited the diffraction-limited imaging capability of the instrument to reduce the confusion noise and ease source de-blending and identification. PACS imaging, however, requires long integrations and has been performed to such faint fluxes only in small areas, such as GOODS and COSMOS. We report in Figure <ref> number count data at the effective wavelengths of 100 and 160 μm from Berta et al. (2010) and Magnelli et al. (2013), which account for all various corrections of sampling incompleteness, effective area, and fraction of spurious identifications. The 160 μm number counts include also data collected on wider areas by the Spitzer telescope and reported in Bethermin et al. (2010), these latter quite consistent with the Herschel-PACS data.The SPIRE number count data at longer wavelengths are reported in Fig. <ref>. Although they are not directly related to our photon-photon opacity estimate, they are relevant to further constrain our model of the IR emissivity of cosmic sources and its evolution with redshift.Because we are interested in estimating absorption effects at energies up to a few tens of TeV, from eq. 1 and from consideration of the pair-production cross-section, this means that we require knowledge of the EBL from a wavelength of approximately 10 up to approximately 100 μm, that is, the interval where the EBL will never be measured because of the overwhelming brightness of the foreground interplanetary dust emission. For a precise determination of the EBL intensity, we then need to know both the local IR source emissivity and its evolution with redshift, which is constrained by the deep number counts in the far-IR and sub-mm. As we see, in all cases, such differential normalized counts display a fast increase when going from flux densities of a few hundred mJy to ∼10 mJy. The slope of the counts is monotonically increasing with the effective wavelength. The SPIRE counts are limited at the faint fluxes by the source confusion.A significant extension to fainter limits is possible based on the analysis of the fluctuations of the background integrated intensities [the probability distribution of deflections, P(D), see, for example, Franceschini 1982], as recently discussed in AF2010 and Glenn et al. (2010).An application to the deep Herschel images is reported by Glenn et al. (2010), who analyzed three fields included in the HerMES programme at all three SPIRE bands (250, 350, and 500 μm).The differential number counts determined in this way of thenumber counts are reported in Figure <ref> (green open squares). These estimates are consistent with those based on individually detected SPIRE sources, and reveal clear evidence for a break in the slope of the differential counts, with a maximum and fast convergence at low flux densities. Obviously, these data from the P(D) analysis are less reliable than those from individual source detections (in red). Nevertheless, they provide us with useful information about the convergence of the SPIRE counts at faint fluxes. We note that only the green data points in Figure <ref>, and the black asterisks and red arrows in Figure <ref> below, are based on this kind of stacking analysis; all others in the present paper are from individually detected sources at the respective wavelengths. With this extension, the total counts in the three SPIRE bands account for as much as 64, 60, and 43% of the far-infrared background intensity observed by COBE (red arrows in Fig. <ref>).Indeed, in the wavelength interval covered by the SPIRE observations, the COBE/FIRAS surveys have allowed direct determination of the EBL intensity, after subtraction of the Galactic and Inter-Planetary dust emissions (Puget et al. 1996, Hauser et al. 1998, Lagache et al. 1999; see also AF2008). All this information about the integrated number counts and the direct EBL estimates, when available, is summarized in Figure <ref>, including our updated modeling reported as a thick continuous line. The data points and limits appearing in this figure are also reported in the review by Dwek & Krennrich (2013), to which the reader is referred if interested in the tabulated values. The Herschel cosmological surveys have also allowed us to make fundamental progress towards the determination of the redshift-dependent luminosity functions at long wavelengths. Particularly relevant are the deep multi-wavelength PACS and SPIRE observations of the GOODS-N, GOODS-S and COSMOS fields, including also deep coverage with MIPS at 70 μm. The former two cover an area of approximately 300 square arcminutes each, and the latter an area of approximately 2 sq.deg., all including a vast amount of complementary data in all e.m. bands (Giavalisco et al. 2004; Scoville et al. 2007; Sanders et al. 2007), optical spectroscopic surveys (Lilly et al. 2007) and highly precise photometric redshifts (Ilbert et al. 2010, 2013).Berta et al. (2010, 2011) reported a detailed description of the Herschel source extractions and data catalogs. Further deep IR surveys have been performed in the Extended Groth Strip and Extended Chandra Deep Field South for a total of more than 1000 square arcmins (Magnelli et al. 2009).Gruppioni et al. (2013) and Magnelli et al. (2009) published independent multi-wavelength luminosity functions at the rest-frame 35 μm wavelength, over a wide redshift interval from local to z≃ 4, well consistent with one another.These results are reported in Figure <ref> as red datapoints from Gruppioni et al. (2013) and black datapoints from Magnelli et al. (2013), both comparedwith our model fit.Luminosity functions at longer wavelengths were also calculated by many authors based on the deep fields (Gruppioni et al. 2010, 2013; Magnelli et al. 2013; Eales et al. 2010). In addition, Herschel SPIRE data over a total of 39 deg^2 within five high-latitude fields have been used by Marchetti et al. (2016) to compute luminosity functions at 250/350/500 μm, and provide us with a census of the luminosity density in the low-redshift Universe at these wavelengths. All these data are relevant to constrain our models of the IR source emissivity discussed in the following Section. § THE DATA-FITTING SCHEME To interpret the previously discussed statistical data, we have adopted the multi-wavelength modeling scheme of IR galaxy evolution reported in AF2010 and used in AF2008, with few changes and some parameter optimization.Our aim consists of obtaining a phenomenological fitting scheme with the best possible adherence to this large variety of data in order to use it to accurately quantify the source IR emissivity. For this reason, and considering its good ability to reproduce the data, we did not attempt to substantially update our evolutionary model, by, for example removing our distinction between LIRG and ULIRG populations (see below) that might be seen as somewhat schematic. §.§ Population components Our model considers four population components characterized by different physical and evolutionary properties.The first class are normal spirals, dominating the local source populations and assumed to contribute only at z<1.The presence of this non-evolving population has alreadybeen identified in AF2010 in order to explain the observed number counts at bright fluxes and the low-redshift luminosity functions.A second class, already introduced by Franceschini et al. (2001) among others, is a population of star-forming galaxies of moderate luminosities (LIRG). In order to explain the fainter number counts at all IR wavelengths, this population needs to evolve fast in redshift up to z∼ 1. The LIRG's evolution rates in both luminosity and number density are slightly modified compared to AF2010 to account for the new Herschel data. The local fraction of the moderately-luminous star-forming population is assumed to be ∼ 10% of the total galaxy population, in line with the observational constraints.A third evolutionary population introduced by AF2010 to explain the Spitzer 24 μm and SCUBA data at high-redshifts are the ultra-luminous infrared galaxies (ULIRGs), dominating the cosmic IR emissivity above z≃ 1.5 and, as such, less relevant for our present analysis.This class of sources is essential to reproduce the statistics, particularly the number counts, in the submillimeter (Herschel) and millimeter (JCMT, IRAM, CSO, APEX). We note that, as explained in AF2010, our used terminology for LIRG and ULIRG indicates objects of approximately 10^11 L_⊙ at z∼ 1 and ∼ 10^12 L_⊙ at z > 1.5, respectively.Finally, we consider the contributions of AGNs, in particular type-I AGNs that are easily identified with simple combinations of optical-IR colors (Polletta et al. 2006, Hatziminaoglou et al. 2005).The type-II category, instead, is much more difficult to disentangle among the star forming galaxy population, and we do not treat them separately here.Our choice to distinguish two separate populations of star forming galaxies, LIRGs and ULIRGs, with different luminosity functions and evolutionary histories, was suggestedin order to fit the large observational dataset in the IR, that appear to require at least two components, one of moderate luminosity dominating at low z and a second high-luminosity class at higher redshifts. This choice allowed us the flexibility to fit statistical data in the mid-IR (e.g., the 24 μm counts) and in the sub-millimeter simultaneously. Simpler realizations, for example, including only one component of star forming galaxies, would not allow us to achieve acceptable fits (see also M. Rowan-Robinson 2009).§.§ The spectral modelIn consideration of the multi-wavelength nature of the data used, a key ingredient in our description of the IR-selected galaxy emission is the spectral model to be adopted for the various considered sources. This is needed to calculate both the K-corrections and the transformations and interpolations of the luminosity functions at various wavelengths, as well as their volume emissivities. Since AGNs of type-I mostly contribute in the mid-IR, and have little influence on the far-IR and sub-mm statistics, we have simply imposed a priori the spectral shape for them. We assume for type-I AGNs a spectral energy distribution (SED)corresponding to an emission model by a pure face-on dusty torus from Fritz et al. (2006), as reported with a green line in Figure <ref>. As for the normal spiral population, we have simplified the treatment by AF2010 of a luminosity dependent spectral shape, by assuming a single constant spectrum, shown as a dotted line in Figure <ref>, and dominated by cold dust.Concerning the LIRGs and ULIRG populations, modeling their IR spectra is critical to achieve good fits to the data. Our adopted SEDs for both classes, that we assumed to be independent of redshift and luminosity, are reported in Figure <ref> as the cyan and red dashed lines. They have been obtained by slightly modifying the spectrum of the starburst galaxy M82 (see e.g., Polletta et al. 2007): while in the range from 5 to 18 μm they are fixed to the mid-IR spectrum of M82 observed by ISOCAM (Franceschini et al. 2001), at longer wavelengths the two spectra have been modified so as to best-fit all the multi-wavelength data reported in Section <ref>. The two average SEDs for LIRGs and ULIRGs needed for reproducing the data are not very dissimilar to one another, but the ULIRG spectrum appears to be broader, with a significant enhancement at λ∼ 60μm and in the submillimeter. This is likely due to significantly different physical conditions in galaxies at different epochs.Now, it is important to confirm our adopted spectral shapes with further direct observational constraints.Fadda et al. (2010) published ultra-deep mid-IR spectra of a representative and complete sample of 48 infrared-luminous galaxies obtained with the Spitzer IRS spectrograph. Half of these are LIRGs at z<1 and the other half are ULIRGs at z>1.The average spectra for the two classes of sources, as obtained by Fadda et al., are reported in Figure <ref> as small open squares, covering the ranges of 4-10 and 5-12 μm.As already remarked by Fadda et al. and shown in the figure, Spitzer IRS observations are in good agreement with our adopted mid-IR spectral shapes for both LIRGs and ULIRGs.To complement the Spitzer's data with longer wavelength data, Lo Faro et al. (2013) analyzed deep Herschel SPIRE and PACS photometric data in the GOODS-South field for 10 LIRGs and 21 ULIRGs of the Fadda et al. sample.We show in Figure <ref> the measurement of the average flux for the two subsamples as the blue filled circles for the LIRGs and red filled circles at 100 μm for the ULIRGs. Overall, we find excellent agreement between these average fluxes and our synthetic spectra, which supports our adopted spectral model.Our model luminosity functions are calibrated to primarily fit those estimated with Herschel in the far-IR and sub-mm, as discussed in Section <ref>. The spectral model is particularly critical to reproduce well LFs in the mid-IR, like those from the IRAS 12 μm survey, and in the mid- and far-IR by Spitzer and Herschel, that are reported above. §.§ Model to data comparisonA partial comparison of our model fitting with the statistical dataset is reported as thick lines in Figure <ref> to <ref>. As we see, the fits are from `acceptable' to `excellent' in all cases. The same happens to data at longer wavelengths, that are less relevant to our analysis however.For completeness, we also summarize in Figure <ref> our best guess evolution model for the time-dependence of the comoving IR galaxy emissivity (black continuous line), compared with existing literature data, where we see a fast increase of it from z=0 to 1.Our estimated IR luminosity density shown as a thick black line uses all IR data reported in the review by Madau & Dickinson (2014), plus some other independent determinations. The black dashed line in the Figure is their best-fit to the star-formation rate density translated back to IR bolometric luminosity with the Kennicutt (1998) recipes (see Figure caption). Further to a fair overall match between our analysis and Madau & Dickinson's, there are some small differences that are apparent in the figure, in particular our estimate is slightly higher than theirs at 0.5<z<1.5 (and slightly lower above). This results from the fact that Madau & Dickinson's fit, trying tointercept both the IR and UV data points, suffers some tension between the two sets of data (the IR density shows a maximum at slightly lower redshifts than the UV, even corrected for dust extinction). Our analysis reproduces more closely the IR data of our interest here. The result reported in Figure <ref> is important to further constrain our modeled EBL intensity between 10 and 300 .While we have adopted a standard χ^2 test to evaluate the goodness-of-fit to individual datasets (like e.g., the number counts in a given waveband or the luminosity function at a rest-frame wavelength and redshift) and to verify the occurrence of possible serious misfits,we have not attempted to run global χ^2 minimization and automatic χ^2 searches of the model parameter space to look for degeneracies in the solutions and to provide quantitatively defined uncertainty intervals. This was because of the huge number of datapoints with largely inhomogeneous error estimates, and hidden and unaccounted-for systematic uncertainties in specific data. The latter is particularly the case for the redshift-dependent LFs, for which a reliable error estimate free of systematic uncertainties is virtually impossible. In some other cases, like for the FIRAS-CIRB intensity (see Fig. <ref>), errors are not even defined except for an uncertainty range, not suited for a χ^2 test. The consequence is that we will not offer in the following a real discussion of uncertainties and parameter degeneracies. Note that our analysis benefits from an extreme adherence to a vast amount of data, at the cost of an incapacity to provide a well-defined quantitative confidence range. However, considering only the constraints set by the source multi-wavelength number counts, we can estimate a global uncertainty of about ±10% on them, which directly translates onto a ±10% uncertainty on the photon optical depth up to z≃1 and increasing above. The linear relation between number counts, background intensity, and photon-photon optical depth can be seen in eqs. 5 to 13 in AF2008.§ EBL MODEL IMPROVEMENTS AND THE COSMIC PHOTON DENSITYOur current modification of the EBL model by AF2008 is based on new data at long wavelengths, λ > 10 obtained from the Herschel deep and wide-area surveys, improving previous determinations based on the Spitzer data. Indeed, the latter required some substantial extrapolations from the observed mid-IR to longer wavelengths, that are not confirmed by the new far-IR data. Elbaz et al. (2010) and Rodighiero et al. (2010) found, in particular, that the highest-luminosity galaxies over a wide redshift interval have a relatively lower IR luminosity with respect to previous estimates. Consequently, both the new estimated contribution of sources to the EBL intensity at IR wavelengths and the photon number densities at λ > 8μm are slightly lower than reported by AF2008.Figure <ref> shows our current assessment of the local EBL intensity (background intensity at redshift z=0) after the above mentioned modifications. We see that all the spectral range from the UV to approximately 8 has remained unchanged as a consequence of the unchanged modeling of the galaxy populations at these wavelengths. Instead, the mid-IR portion from ∼8 to 40 is now lower because of the slightly reducedIR emissivity of galaxies. More significant is the decrease of the whole far-IR and submillimetric peak of dust emission from galaxies; the peak EBL emission at 160 μm, while estimated to exceed 20 nW/m^2/sr by AF2008 based on Spitzer data, is now lowered to ν I(ν)∼ 14 nW/m^2/sr, in good agreement with several other independent determinations (e.g., Lagache et al. 1999; Berta et al. 2011). As already stressed, our work accounts only for the known source contributions, like those from galaxies, active galactic nuclei and quasars.We do not attempt to consider here various published measurements of the total-light EBL in the near-IR based on COBE (Gorjian et al. 2000, Wright & Reese 2000, Cambresy et al. 2001, Hauser & Dwek 2001, Dwek & Krennrich 2005, 2013), and IRTS (Matsumoto et al. 2005; Kashlinsky 2006) observations, and rocket experiments (Zemkov et al. 2014). These measurements suffer substantial uncertainties due to the poorly understood contribution of the Zodiacal Sun-reflected light. The occurrence of a truly diffuse additional component will only be discussed in Sect. 6 as signals potentially reachable by future observations with Cherenkov telescopes. Unfortunately, the whole interval from 24 to 70 was not covered by any one of the space observatory missions during the last 20 years (ISO, Spitzer, Herschel), and this is reflected in the lack of the corresponding data in Fig. <ref>. However, thanks to our spectral analysis in Sect. <ref>, the excellent available information at 24, 70, and 100 is easily interpolated inside that wavelength range, as we have done for the luminosity functions in Figure <ref>. Note that our local EBL model at λ=70 μm fits well the two published independent estimates in Figure <ref>. In conclusion, we are completely confident that we control in an accurate and robust way the background intensity produced by sources over the whole IR domain, both the local EBL and its time evolution.Figure <ref> illustrates our model prediction for the time-dependent number density of diffuse cosmic photons. Again the difference with AF2008 is confined to photon energies ϵ < 0.2 eV. At the energy of 0.01 eV, corresponding to the peak of the photon density produced by cosmic sources (see Fig. 4 in AF2008), this difference is approximately a factor 2.5, in the sense of lower density values from the present analysis. For completeness, in our Figure <ref> we have added the contribution of the Cosmic Microwave Background (CMB) photons, showing up at wavelengths λ >100 μm and ϵ < 0.01 eV as a very steep rise in photon density with wavelength and redshift. § THE COSMIC PHOTON-PHOTON OPACITYOur previously discussed modifications of the local EBL intensity and its evolution impact on the estimate of the cosmic opacity for photon-photon interaction and pair-production, and on the HE and VHE photon horizon.All detailed formalism and calculations of the optical depths as a function of photon energies and cosmic distances are reported in AF2008, to which we defer the reader.The detailed dependences of the photon-photon optical depth τ_γγ on the photon energy for sources over a range of distances are reported in Figure <ref>, and compared there to those calculated by AF2008.As we see, differences due to our modified EBL concern mostly very high-energy observations of cosmic sources, from approximately 1 TeV to few tens of TeV photon energies, in agreement with the usual rule-of-thumb ofEq. <ref>, where our currently reduced background photon densities at long wavelengths imply reduced opacities.At lower energies the differences with AF2008 become essentially null.The effects of the inclusion of CMB photons in the optical depth calculations are instead illustrated in Figure <ref>. These effects start to be prominent at E_γ≥ 100 TeV for local sources, and at progressively lower energies at increasing source distances.The cosmological horizons for different reference values of the photon-photon optical depth τ_γγ and of the photon energy are reported in Figure <ref>. At an energy of E_0≃ 1 TeV there is a maximum rate of the decrease of E_0 with redshift for a fixed τ because of the maximum in the EBL at λ∼ 1 μm. The flattening at the highest E_0 values is due to the CMB, while that at the highest redshifts is due to the decreased density of photons produced by galaxies at z>1.Based on the constraints from the number counts only, we estimate at roughly ±10% the uncertainties on τ(E_γ,z) for z<1, and larger for z≥ 1.Our improved EBL modeling at the long IR wavelengths is particularly relevant for the interpretation of the highest-energy observations of blazars with Cherenkov telescopes. Because of the strong dependence of τ_γγ on distance, such high-energy multi-TeV photons can be better observed from local or low-redshift sources. Data on the two best known such examples, Markarian 501 (Mkn 501, z=0.034) and Markarian 421 (Mkn 421, z=0.031), are reported in Figure <ref>. In the top panels, our best-guess extinction factors e^τ_γγcorresponding to the two source distances are reported as a function of the photon energy. In the bottom panels, we compare the observed (black) and extinction-corrected (red) spectral data, together with the corrected best-fit slopes. The data on the former source, particularly relevant, come from a re-analysis of HEGRA atmospheric Cherenkov observations of the famous 1997 flares of Mkn 501, based on algorithms providing improved energy resolution to best constrain the highest energy portion of the spectrum (Aharonian et al. 2001).As for the Mkn 421 data, these were taken during the strong TeV gamma-ray outbursts registered during the observational season of 2000-2001 (Aharonian et al. 2002). In neither case do we see evidence in the absorption-corrected spectra for excess flux or upturns at the highest energies that might be indicative of an improper spectral correction.Some high-energy upturns for the two sources that were indicated by the analysis in AF2008 (their Fig. 9), and more significantly in Costamante (2013), are not confirmed by the present analysis. While the differences in τ_γγ between this latter and those reported in AF2008 are small at such low redshifts (see Fig. <ref>), because of the exponential dependence of the absorption correction on τ, these moderately significant upturns are removed. Aharonian et al. (2002) have performed a systematic comparison of the VHE spectra of the two sources and, based on the fact that they are essentially at the same distance, concluded that the exponential convergence of the two spectra is not consistent with being due to the EBL photon-photon absorption only, and an intrinsic cutoff is required. Our present analysis suggests that the pair-production effect predicted by our re-evaluated EBL intensity is sufficient to explain the VHE spectra for both sources,assuming intrinsic power-law shapes (that in the case of Mkn 421 are one unit, in the spectral index, steeper than for Mkn 501).§ DISCUSSION AND CONCLUSIONS §.§ Possible indications for new physics One of the important topics still open after several published analyses and long debate concerns the detailed spectral forms of the few local blazars whose Cherenkov spectra have been observed up to and above E_0 ∼ 10 TeV. These analyses (see e.g., Horns & Meyer 2012; Costamante 2013) reported evidence for substantial upturns in the VHE spectra of Mkn 501 and Mkn 421, once corrected for pair-production absorption according to standard EBL models - like if such absorption corrections would be too large. Even our previous analysis in AF2008 did not exclude such an effect in the two examined local blazars. These inferred spectral upturns at several TeV to tens of TeV energies are not consistent with standard blazar emission models (except assuming rather ad-hoc effects of photon absorption and pile-up). Indications for anomalies in the photon propagation and pair-production have been claimed by comparing the energy distributions for VHE photons received from distant cosmic sources (Horns & Meyer 2012).As discussed in several papers (e.g., De Angelis et al. 2007), this is an important issue for fundamental physics because, considering the very high energies of photons and particles involved, these effects might reveal deviations from the Standard Model and might require new physics. The most obvious possibility for increasing the transparency of the universe would be to decrease the photon-photon collision cross-section as a consequence of a violation of the Lorentz invariance, LIV. This may be a natural possibility in the framework of Quantum-Gravity theories, which predict a breakdown of classical physics at energies on the scale of the Planck energy (Jacob and Piran, 2008). For example, a modification of the photon dispersion relation might lead to a shift of the energy threshold for pair production at the highest energies, and a decrease of the optical depth at increasing photon energy.An alternative possibility considered for increasing the photon mean free path is to assume that photons oscillate into and from hypothetical axion-like particles (ALPs) when traveling in the presence of a magnetic field (e.g., De Angelis et al. 2007). During the ALP phase in their path to the Earth, photons would not suffer pair-production effects and would travel undisturbed, then increasing the universal transparency. There may, however, no longer be a convincing need for new physics emerging from VHE observations of distant sources. This request appears now to be alleviated by the present analysis because, compared with the results in AF2008, our re-analysis of the IR source emissivities at long IR wavelengths, with the corresponding slightly lower inferred EBL intensity values, imply somewhat reduced photon-photon opacities at the highest photon energies at > 1 TeV, relaxing the claimed potential inconsistencies. The pair-production extinction corrections to the observed spectra of Mkn 421 and Mkn 501 reported in Fig. <ref> based on our best-guess EBL are consistent with simple spectral extrapolations from the lower photon energies. The high-energy upturns in the corrected spectrum indicated in Costamante (2012) are not apparent here. Our current results are in line with those reported as fiducial models in Primack et al. (2011) and in Stecker et al. (2016).In our case, the modeling of the extragalactic source contribution to the EBL is purely empirical and based on deep photometric imaging data and the source luminosity functions at all wavelengths from UV to the sub-millimeter. Overall, our improved EBL modeling reduces, or even removes, the tension between VHE observations of blazars and standard physical interpretations of their spectra. §.§ Revising the optical and near-IR EBL (COB) intensity Interactions between the highest-energy photons produced by cosmic sources and the background light will likely remain a hot topic in coming years. While the pair production effect offers interesting constraints on the EBL intensity in the UV to the near-IR range, the exact level of this background is still subject to significant uncertainties. Zemcov et al. (2014), for example, report an observed excess signal of near-IR background fluctuations at 1.1 and 1.6 μm on scales of approximately 10 arc-minutes from a sounding rocket experiment, a signal that may correspond to diffuse light produced by stars in the intergalactic space outside galaxies, potentially implying an optical-NIR EBL around two times higher than our estimated galaxy contribution in Fig. 4. Abramowski et al. (2013) present a joint analysis of the spectra of a sample of blazars totaling approximately 10^5 VHE gamma-ray events. Further to the evidence of a clear, highly significant signature of the EBL absorption, the team reports an overall best-guess estimate for the EBL intensity over almost two decades of wavelengths from UV to near-IR with a peak value of ν I(ν)≃ 15 nW/m^2/sr at 1.4 μ, with essentially the same spectral shape of AF2008. This figure is slightly (∼ 40%) higher than our predicted value contributed by resolved sources as in Figure <ref>. The same analysis reveals that this excess flux corresponds to a pair-production optical depth larger than reported in AF2008 by a factor ∼1.34.One of the interesting sources where one can look for EBL absorption effects is the high-redshift blazar PKS1441+25 (z=0.94) that was detected with high significance by MAGIC during an outburst event in 2015 (Ahnen et al. 2015). We show in Figure <ref> results of our analysis of the VHE spectrum of this source. The cyan data-points are the observed fluxes from Fermi and MAGIC, red datapoints are corrected for EBL absorption based on our best-guess optical depth. As can be seen in the Figure, the latter are well fit by a theoretical blazar spectrum (red curve, in E^2 dN/dE units) with a peak energy at approximately 1 GeV, a spectrumthat is typical for the Intermediate or Low Synchrotron Peaked (ISP, LSP) objects. High-luminosity blazars at large redshifts tend indeed to be of this category.In the same figure, we also show (in blue) the MAGIC flux data corrected assuming an EBL intensity in the whole optical/near-IR a factor 1.5 larger than our best-guess estimate, following the suggestion by Abramowski et al., among others. These corrected data would be largely consistent with a power-law extrapolation (black line) of the lower-energy Fermi and MAGIC points, however implying a serious misfit of the highest-energy datum. How seriously this bad fit should be taken, and how physically consistent a power-law continuation of this kind, like the black line, could be (note that this is a high luminosity object, like ISP or LSP kind), cannot be decided with the present data. Furthermore, it is difficult to be conclusive about the exact level of the EBL within the above mentioned boundaries. §.§ Future prospects Although atmospheric Cherenkov observatories, joining efforts with the Fermi all-sky surveys, are accumulating VHE photon detections at good rates from sources over a substantial range of redshifts, up to z∼ 1 as we have seen, it seems unlikely that this subject of photon propagation across the universe and EBL characterization will experience significant progress in coming years, further to already published results. This is due to the limited sensitivity, spectral resolution, and energy coverage of current instrumentation, on one side, and the complex interplay between physics and astrophysics on the other, as discussed in the previous subsection.New-generation instrumentation is clearly needed to make the next step. One such occasion will be offered by the Cherenkov Telescope Array (CTA) in a few years from now (Actis et al. 2011). The full array will offer over factor 10 improvement in sensitivity and spatial and spectral resolution over current telescopes. Among the countless contributions to physics and astrophysics that can be envisaged from such an instrument, we certainly expect substantially improved constraints on the total EBL intensity (sources + diffuse emissions) over a wide interval of wavelengths from UV to the far-IR, as well as relevant constraints on its evolution with cosmic time.As a particularly significant example of such an opportunity, we have considered using the pair-production absorption effect with an instrument like CTA to constrain the integrated emission of very high-redshift sources responsible for the cosmological re-ionization at redshifts of z∼8-10 (that we name Population III, or Pop III sources henceforth). While partly dedicated to this purpose, the new generation space telescope JWST itself will not directly detect them if they are single stars or stellar aggregates.Instead CTA might see their signatures in the spectra of high-redshift blazars. This also takes advantage of the fact that, while the EBL contribution by known sources, such as galaxies, starts fading away at z>1 (see Figs. <ref> and <ref>), the Pop III background proper photon number density increases very quickly and proportionally to (1+z)^3 up to z∼7-10.We have then performed simulations of future observations of a few tens of hours with CTA using the MAGIC and Fermi data of the z=0.94 blazar PKS1441+25as a reference. We then assumed, on top of our modeled galaxy background, the existence of a truly diffuse signal, like that due to Pop III sources active at very high redshifts. For the latter, we adopted a spectrum as measured by Matsumoto et al. (2005), with a peak at 1.4 μm, but scaled down in flux, following a similar procedure as in AF2008 (see also Raue, Kneiske & Mazin 2009).We have simulated CTA observations of PKS1441+25, first of all assuming its measured redshift z=0.94, a luminosity as observed during the 2015 outburst, and a spectral shape as our best-fit red line in Fig. <ref>.The simulation results are reported in Fig. <ref>: the upper continuous red line is the intrinsic source spectrum, while the upper sequence of red datapoints are the simulated data calculated assuming our best-guess EBL model for the photon-photon absorption. The corresponding 1σ error-bars are based on the predicted sensitivity and spectral resolution of the full CTA array for a 50-hour integration [CTA performances have been obtained from:https://portal.cta-observatory.org/Pages/Home.aspx]. The blue data-points correspond to the simulated spectral data corrected for absorption assuming, in addition to the best-guess known-source contribution to the EBL, that of a Pop III component with an intensity equal to only 5% of the IRTS background measured by Matsumoto et al. At the peak intensity wavelength of 1.4 μm, this corresponds to an intensity of 3.5 nW/m^2/sr, added to the known source contribution of 11.4 nW/m^2/sr (see Fig. <ref>). This Pop III contribution would amount to 1.5 nW/m^2/sr in bolometric units when integrated between 1 and 4 μm, corresponding to a comoving star-formation rate density of approximately 1 M_⊙/yr/Mpc^3 at z∼ 10, according to the analysis of Raue et al. (2009). As we see in Fig. <ref>, this small truly diffuse component implies a moderate hardening of the (blue) corrected spectrum, however with an overall spectral shape not overly dissimilar to a simple power-law extrapolated from the lower-energy data shown as a black line.The relative incidence of such a Pop III additional component would become much more evident for a 50-hour observation with CTA of the same source, assumed to be observed at z=1.8 with the same luminosity. To simulate this occurrence, we have scaled the spectrum of PKS1441 to that redshift and performed the same calculation as before.The lower continuous red line in Figure <ref> is the predicted intrinsic spectrum for z=1.8, while the lower red datapoints simulate the CTA observation including the EBL absorption for the known-source contribution only. As we see, the source becomes fainter with z and, after application of the EBL absorption, it is detectable by CTA only up to 150 GeV, while the lower-redshift counterpart could be observed up to almost 400 GeV. We have then corrected back these simulated spectral data assuming the small additional contribution of Pop III sources as above (5% of the IRTS background signal). The resulting absorption-corrected spectrum is shown as green datapoints in Figure <ref>. We see that, in this case, the inclusion of a small Pop III signal dramatically increases the τ_γγ for such a high-z source, as shown also by the comparison of the green and blue lines in the upper panel. At the photon energy of 200 GeV, the inclusion of the Pop III excess increases the τ_γγ only by a factor 2 for a source at z=0.9 (see blue line in Fig. <ref> versus the black one in Fig. <ref>). Instead, for the same source observed at z=1.8, the inclusion of the small Pop III excess dramatically increases the optical depth, as shown by the green line compared to the blue one in the upper panel of Figure <ref>. As mentioned, this is a consequence of the very fast (1+z)^3 increase of the Pop III photon density above z=1. This exercise illustrates how the low-z known source contributions to the EBL and residual high-z components could be disentangledby comparing CTA observations of blazars at different redshifts.Our simulation exercise shows that new-generation Cherenkov arrays, with the vast expansion of the VHE cosmological horizon that they will offer, will provide us with new unique opportunities for sampling the history of radiant energy production in the universe.We are glad to acknowledge useful discussions and exchanges with A. De Angelis, Mose' Mariotti, Michele Doro, Sara Buson, Abelardo Moralejo, and Daniel Mazin among others. 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"authors": [
"Alberto Franceschini",
"Giulia Rodighiero"
],
"categories": [
"astro-ph.HE",
"astro-ph.GA"
],
"primary_category": "astro-ph.HE",
"published": "20170526072018",
"title": "The extragalactic background light revisited and the cosmic photon-photon opacity"
} |
[][email protected] ^1INO–CNR BEC Center and Dipartimento di Fisica, Università di Trento, 38123 Povo, Italy ^2Politecnico di Torino, Dipartimento di Elettronica e Telecomunicazioni, Corso duca degli Abruzzi 24, 10129 Torino, ItalyBoundaries strongly affect the behavior of quantized vortices in Bose-Einstein condensates,a phenomenon particularly evident in elongated cigar-shaped traps where vortices tend to orientalong a short direction to minimize energy. Remarkably, contributions to the angular momentumof these vortices are tightly confined to the region surrounding the core, in stark contrast tountrapped condensates where all atoms contribute ħ. We develop a theoretical model anduse this, in combination with numerical simulations, to show that such localized vortices precessin an analogous manner to that of a classical spinning top. We experimentally verify thisspinning-top behavior with our real-time imaging technique that allows for the tracking of positionand orientation of vortices as they dynamically evolve. Finally, we perform an in-depth numericalinvestigation of our real-time expansion and imaging method, with the aim of guiding futureexperimental implementation, as well as outlining directions for its improvement.Observation of a Spinning Top in a Bose-Einstein Condensate F. Dalfovo^1 December 30, 2023 ===========================================================§ INTRODUCTIONBose-Einstein condensates (BEC) are ideally suited for the study of quantum vortices, owning to their purity and high-degree of tunability <cit.>, and this inherent flexibility has inspired experimental and theoretical works in a wide variety of settings. Vortex lattices provide the fundamental means for bulk superfluid flow in rotating BECs <cit.> while, on the other hand, vortices also lie at the heart of quantum turbulence in nonequilibrium systems <cit.>. Boundaries play a central role, and when a vortex line pierces a condensate's surface it does so at an angle perpendicular to it. When a vortex is positioned off-center, it tends to orbit around the condensate center at an increasing frequency as it spirals outward due to dissipation <cit.>. Under the influence of a pancake-shaped trapping potential, vortices tend to minimize their energy by aligning along the short direction and, at finite temperature, the vortex-unbinding Berezinskii-Kosterlitz-Thouless phase transition was studied <cit.>. In three dimensions, in addition to vortex lines <cit.>, vortices can fold to create rings <cit.> and even more exotic structures likehopfions <cit.> and Chladni solitons <cit.>. Spiralling undulations of the cores, known as Kelvin waves, are responsible for the so-called Kelvin-wave cascade <cit.>.In three-dimensional (3D) cigar-shaped traps the most stable defect is the so-called solitonic vortex, a short vortex line that pierces the condensate through its side <cit.>. While they are indeed vortices, solitonic vortices possess some solitonic characteristics such as being more localized and, on the coarse-grain scale, they cause a π phase jump between each end of the cigar which results in a planar density depletion after expansion <cit.>. Solitonic vortices, which were recently realized in experiments with bosons <cit.> and fermions <cit.>, tend to be long-lived and orbit about the condensate center on an elliptical path, along which the core remains surrounded by a roughly constant density. We experimentally produce solitonic vortices in cigar-shaped traps that are inherited from the condensate formation process, thanks to the Kibble-Zurek mechanism <cit.> (also see <cit.>).A BEC is formed by a cooling quench across the transition temperature, where a symmetry-breaking phase transition occurs. If the quench is fast enough, distant regions of the system do not have sufficient time to communicate and hence they randomly develop order parameters disparate from one other. Defects, such as dark solitons or quantized vortices, then become trapped at the boundaries between such regions. With an appropriate imaging technique we track the axial position and the orientation of the vortex lines which remain in our BECs, as remnants of the Kibble-Zurek mechanism and the subsequent post-quench dynamics. Some of these vortices exhibit a peculiar rotation of their core around the long axis of the trap as depicted in Fig. <ref> (a). In this work we show that such a rotation is caused by a tilt of the vortex line out of the radial plane and towards the symmetry axis, as shown in Fig. <ref> (b); the tilt implies an increase of the vortex line length, with a consequent energy cost and an induced torque. The torque produces the precession of the vortex around the axial direction, in an analogous manner to a classical spinning top.The analogy works well because the solitonic vortex is a localized object, in contrast to regular 3D vortices. We verify this spinning top behavior by performing numerical simulations, using the Gross-Pitaevskii equation (GPE). We find that a solution of the GPE exists corresponding to a tilted vortex, which is stationary in a reference frame rotating around the long axis of the trap. We then use such a stationary state as an input of real time GP simulations in the nonrotating BEC and we observe that the vortex line keeps rotating at a constant angular velocity. We use the GPE also to simulate the extraction and expansion of atoms as performed in the experiments, in order to reproduce our minimally-destructive imaging scheme that is able to track the orientation and position of the spinning vortices in real time.The paper is structured as follows: In Sec. <ref> we present our spinning top theory for the solitonic vortex. In Sec. <ref>, we outline our numerical approach for simulating real-time dynamics of 3D cigar-shaped condensates. We also explain how to obtain solitonic vortex initial states for our precession simulations.The focus of Sec. <ref> is the comparison of our spinning top model with our numerical results; in particular, we calculate the precession frequency versus the axial tilt angle θ (see Fig. <ref> (b) for the definition of θ). In Sec. <ref> we outline our experimental extraction procedure and provide details for how this is numerically simulated. Section <ref> presents our experimental observations of solitonic vortex spinning tops, alongside their numerical counterpart for comparison. We also discuss our extraction and imaging scheme, and suggest directions for its improvement. We conclude with section <ref>. § FORMALISM §.§ Theory of the BEC spinning topIn contrast to a vortex in an untrapped system the solitonic vortex is a highly localized object. This local character is what allows for a close analogy with the classical spinning top, and serves as the basis for our analytic approach.For an isolated straight vortex line in an untrapped condensate, each atom contributes ħ to the angular momentum regardless of the distance from the core. Consequently, the angular momentum per unit length rapidly diverges with increasing system size. Realistic trapped systems, however, offer qualitative differences: close boundaries in anisotropic traps act to restrict the superfluid flow, which limits the fraction of atoms that contribute to the angular momentum to be only those in the vicinity of the core. A solitonic vortex in a cigar-shaped condensate is an excellent example, and Figs. <ref> (c) and (d) show that the dominant contributions to the angular momentum are tightly localized about the core. Furthermore, as can be seen in Figs. <ref> (a) and (b), the kinetic energy density is also localized about the core.One way for a vortex to decrease its energy is for its core length to shorten. It follows, then, that a solitonic vortex that is tilted by an angle θ into the long axis of the trap, as shown in Fig. <ref> (b), experiences a restoring torque that acts to reduce θ. The torque τ modifies the angular momentum 𝐋 according toτ = d𝐋/dt = Ω×𝐋,causing a precession Ω about the x axis, as illustrated in Fig. <ref>. Note that we assume that the angular momentum associated with the precession is much smaller than the angular momentum of the vortex itself. The angular precession frequency, in radians per second, is then given byΩ≡ |Ω| = τ/Lcosθ,where τ = |τ| and L=|L|. The torque τ can be calculated as follows. The energy of an untilted solitonic vortex in a highly-elongated condensate (ω_⊥≫ω_x, for radial and axial harmonic trapping frequencies, respectively) has been calculated <cit.>, to logarithmic accuracy, to beE^0 =4/3π n_0ħ^2 R_⊥/mln(R_⊥/ξ),with ξ=ħ/√(2mμ) being the healing length, R_⊥ = √(2μ/mω_⊥^2) is the Thomas-Fermi radius in the tight-confinement direction, n_0 is the peak density and m is the mass.Tilting the solitonic vortex so that the core develops a nonzero axial (x) component, as shown in Fig. <ref> (b), increases its energy. To lowest order in θ this is described by E = E^0/cosθ, which produces a torque of strengthτ = dE/dθ = 4/3 A_E π n_0ħ^2 R_⊥/mln(R_⊥/ξ)sinθ/cos^2θ .The constant A_E, to be determined numerically by solving the GPE, is a correction factor to Eq. (<ref>) and is expected to be approximately unity. Its purpose is, both, a way to quantify the accuracy of Eq. (<ref>) for realistic chemical potentials and 3D trapping, and to improve the value of τ for the prediction of the precession frequency given by Eq. (<ref>). For the regime considered in our simulations (i.e., μ/ħω_⊥=9.72 and ω_⊥/ω_x=10) we find A_E = 0.944, which is indeed close to unity. As shown in Fig. <ref> (c) and (d), the atoms contributing to the angular momentum are predominantly confined to the vicinity of the core, within a radius R_⊥. If each of these atoms contributes approximately ħ then this gives a total angular momentum,L = A_L πR_⊥^3 n_0 ħ,where A_L is some constant, of order unity. We also calculate this numerically and find that for the regime of our simulations A_L = 0.995.Finally, by substituting Eqs. (<ref>) and (<ref>) into Eq. (<ref>), we obtain a prediction for the precession frequency,Ω_ A/ω_⊥ = A_E/A_L[ 2ln(2μ̃)/3μ̃] sinθ/cos^3θ≈A_E/A_L[ 2ln(2μ̃)/3μ̃] θ ,which is a function of the tilt angle θ and the dimensionless chemical potential μ̃= μ/ħω_⊥.§.§ Numerics Our analytic predictions are supported by full numerical simulations of the time-dependent GPE <cit.>,iħ∂ψ(𝐱)/∂ t = [-ħ^2/2m∇^2 + V(𝐱) + g|ψ(𝐱)|^2]ψ(𝐱) ,where interactions are characterized by g=4πħ^2a_s/m, with a_s being the s-wave scattering length. We consider a 3D harmonic trapping potential,V(𝐱)=1/2mω_x^2x^2 + 1/2mω_y^2y^2 + 1/2mω_z^2z^2 ,that is cylindrically symmetric and elongated along the x direction, i.e. ω_x ≪ω_y,z =ω_⊥.Initial states, for subsequent precession dynamics, are created by making use of the rotating-trap GPE <cit.>,μψ(𝐱) = [-ħ^2/2m∇^2 + V(𝐱) + g|ψ(𝐱)|^2 - Ω_trL̂_x]ψ(𝐱) ,where Ω_tr is the trap-rotation frequency and L̂_x = ħ/i(y∂_z-z∂_y), so that the axis of rotation is coincident with the long (x) axis. The procedure begins by imprinting an untilted solitonic vortex onto the ground state of the GPE [Eq. (<ref>)]. A tilted solitonic-vortex stationary state is then obtained by evolving this state according to Eq. (<ref>) with imaginary time evolution; the adjustment of Ω_tr acts as a control knob for the tilt angle θ. For the purpose of investigating the in-trap dynamics of precessing solitonic vortices we consider N = 8× 10^5 ^23Na atoms in a cigar-shaped trap, having ω_⊥ = 10ω_x. The scattering length is 54.54(20) a_0, for Bohr radius a_0 <cit.>, and the radial trapping frequency is given byω_⊥/2π=92 Hz which then corresponds to μ/ħω_⊥=9.72, a system well within the Thomas-Fermi regime.Time propagation is performed with a 4th order Runge-Kutta integration method and a time step size of 1.7 μs. The 3D numerical grid for the simulation of in-trap dynamics has size {L_x,L_y,L_z}={229,34.9,34.9}μm, and there are {N_x,N_y,N_z}={600,60,60} points in the respective directions. The grid has linear spacing and we employ fast Fourier transforms to evaluate the kinetic energy terms at each time step.§ RESULTS §.§ In-Trap Behavior: Numerics and AnalyticsRecall from Fig. <ref> (b) that a solitonic vortex that is tilted into the long (x) axis of the trap will feel a torque and precess about this axis without changing its shape; in this part we numerically investigate how the precession frequency Ω depends on the tilt angle θ. In Sec. <ref> we outlined how to construct tilted solitonic vortex states by solving the rotating-trap GPE [Eq. (<ref>)] and using its stationary states. It is a relatively straightforward extension to turn off the trap rotation and then to evolve these initial states in real time, i.e. by solving Eq. (<ref>). In fact, the resulting real-time precession frequencies Ω were compared with the corresponding values of Ω_ tr, used for initial state preparation, as a means to check the convergence.The numerical data are presented in Fig. <ref> as plus symbols. As expected from our analytic model [Eq. (<ref>)], for small tilt angles the precession frequency exhibits a linear dependence although, remarkably, this nearly-linear relationship extends up to around θ = 0.6 radians. The agreement with the analytic prediction given by Eq. (<ref>) (solid line) is also quantitatively reasonable, with the analytic prediction being around 27% smaller. It should be noted that this discrepancy is not entirely surprising given that the system is not a rigid body and, as the vortex tilts, the superfluid flow has to contend with the anisotropic boundary. Next, we investigate how the shape of a tilted solitonic vortex depends on a range of precession frequencies, from Ω/2π =0.5 Hz to 15.5 Hz, in Fig. <ref> (a). The precession phases are chosen such that the cores lie exclusively in the y=0 plane; incidentally, these were also used as initial states for the dynamical simulations displayed in Fig. <ref>. Remarkably, when the x-axis is multiplied by the central slope of the corresponding vortex core, i.e., x→ x/tanθ, all profiles neatly collapse onto a single curve as shown in Fig. <ref> (b).The tendency for a vortex line to exit its condensate at an angle perpendicular to the surface <cit.> suggests that the most natural deformation, for a tilted (θ>0) solitonic vortex, is a sine functionx = D sin (kz),with k=π/(2R_⊥) and amplitude D. It is worth noting that this deformation is analogous to the lowest Kelvin mode for a vortex of length 2R_⊥, but with the distinction that a Kelvin mode in a uniform superfluid is a helix whereas the states considered here lie in a plane.If one fits Eq. (<ref>) to the profiles in Fig. <ref>, one finds a value of k slightly smaller than expected, consistent with the fact that the density is not quite zero in the region |z|>R_⊥, where it vanishes smoothly. In detail, for the condensate in Fig. <ref>, the chemical potential is μ = 9.72 ħω_⊥ which corresponds to a transverse Thomas-Fermi radius R_⊥=4.41 a_⊥, where a_⊥=√(ħ/mω_⊥). The analytic Thomas-Fermi prediction is then π/(2R_⊥)=0.36/a_⊥, while a best fit to the GPE data gives k=0.30/a_⊥. Single bent vortices were studied in <cit.> for BECs in rotating traps. On the one hand, the so-called S-shape vortices in cigar-shaped condensates with a fast trap rotation about the long axis <cit.> can be seen as the high Ω_ tr counterpart of our tilted solitonic vortices. In fact, for large Ω_ tr,transverse vortices become so stretched that they develop long straight portions, aligned parallel to the trap's long axis, hence losing their solitonic-vortex character. Eventually, as the trap rotation speed increases further, such a vortex continuously evolves to become a perfectly straight line, coaxial with the trap's long axis. Furthermore, the cigar-shaped trap of <cit.> was not symmetric about the trap's long axis. This means that a vortex stationary state in the rotating frame would not simply precess if evolved in real time with the trap's rotation switched off. On the other hand, in the extremely dilute limit where kinetic energy dominates over interaction energy, Ref. <cit.> found that instead of a sine-shaped core the tilted ( θ >0) solitonic vortex remains as an almost rectilinear line. In contrast, our BECs are not dilute, i.e., they are well within the Thomas-Fermi limit, and we consider an axially symmetric nonrotating trap with slowly precessing solitonic-vortices. §.§ Extraction ProcedureIn our experiments we implement forced evaporation to produce sodium BECs of around 2× 10^7 atoms in the same cigar-shaped traps as used for our numerics, i.e., with ω_⊥ = 10ω_x and ω_⊥/2π = 92 Hz. A detailed description of our experimental procedure can be found in Refs. <cit.> while here we highlight the relevant points. The speed of the temperature quench is controlled such that a given condensate typically inherits one, or a few, solitonic vortices via the Kibble-Zurek mechanism <cit.>. For the purpose of investigating vortex dynamics in real time we utilize an imaging scheme, also presented in Ref. <cit.>, that periodically probes the condensate in a minimally invasive manner <cit.>. A small fraction ≈ 1% is extracted every 12 ms and after expansion this is imaged, leaving the trapped condensate otherwise intact. The extraction is performed by transferring atoms from the trapped |F = 1, m_F=-1⟩ state to the untrapped |1, 0⟩ state using a radio frequency field. Since only one component feels the magnetic trap, the energy difference (and hence the resonance condition) between the two states is position dependent and enables us to selectively address different spatial regions of the condensate. The gravitational sag, of 30 μm in the z direction, is larger than the condensate radius and this allows us to linearly sweep the radio frequency field to produce a single resonance front that travels from top to bottom. This sweep of the extraction front has the effect of compressing the extracted portion in the vertical direction and enhancing self-interference effects, which aids with the gathering of in situ information about the position and orientation of the solitonic vortices. As the extracted fraction expands it also falls under gravity while interacting significantly with the trapped condensate for about 3 ms, after which they become spatially separated. Finally, following a 13ms time of flight (TOF), the extracted portion is imaged.To better understand how the expansion images from the above procedure relate to the in situ positions and orientations of the vortices, we perform full numerical simulations using the time-dependent GPE [Eq. (<ref>)]. The interactions between trapped atoms, and between untrapped and trapped atoms, are of the same strength and are characterized by the scattering length 54.54(20) a_0; the subdominant interactions between extracted atoms have a scattering length of 52.66(40) a_0 <cit.>. On the one hand, the expansion dynamics is relatively fast and this allows us to treat the in-trap vortex positions as fixed during the extraction sweep. On the other hand, the global phase of the trapped condensate continues to evolve and it turns out to be crucial to account for this during the extraction.As was the case in Sec. <ref>, the in-trap part of this simulation treats N=8×10^5 sodium atoms in a harmonic trap with the same confinement parameters as for the experiment. The discrepancy of atom number between theory and experiment corresponds to a chemical potential difference of a factor of three and, hence, Thomas-Fermi radii that differ by a factor of √(3). It is not feasible for us to simulate the full experimental atom number since, as a reference, producing the results in this paper already consumed around four weeks of computer time on 100 cores. However, due to the findings in Ref. <cit.> and the comparisons between theory and experiment in this paper, we expect this discrepancy not to be of qualitative importance. To account for the different Thomas-Fermi radii between theory and experiment, when optimizing the vertical compression of the extracted portion, the numerical extraction sweep is 12 kHz/ms, while it is 10 kHz/ms for the experiment. The numerical results that follow assume a 10 ms TOF before imaging the extracted fraction (cf. 13 ms for the experiment); we have numerically checked that this results in only minor differences. While the creation of our initial states is described in Sec. <ref>, during the course of the expansion we interpolate and enlarge the grid such that the one used for the final part of the expansion has size {L_x,L_y,L_z} = {229,139,109} μm and {N_x,N_y,N_z} = {600,240,500} points, respectively. We have checked that all results presented here are numerically converged to ∼1% or better. §.§ Extraction Results: Experiments and NumericsNumerical simulations relating the in situ vortex orientations to the corresponding expanded extractions are shown in Fig. <ref> for a vertical (left) and a horizontal (right) vortex. The compression in the vertical (z) direction, evident in Figs. <ref> (c)(d), is remarkable given that this direction would normally see the greatest expansion if the extraction had instead been uniform (not swept). By considering a top-down view, i.e., the x-y plane in Figs. <ref> (e)(f), the extracted fraction is seen to be framed by a high density elliptical border that in turn surrounds an ellipse of low density. This effect is due to the interactions with the trapped condensate, and exists even in the absence of any vortex.Importantly, though, by contrasting the 2D extracted fractions for the two different vortex orientations, additional regions of constructive and destructive interference are apparent. The physical processes involved in the creation of these vortex-induced asymmetries are complicated but two important contributions are as follows. (i) The vortical superfluid velocity is greatest when forced to flow near a boundary and, after expansion, such a region of enhanced velocity tends to leave behind a hole and an adjacent bump <cit.>. This mechanism is important for, e.g., a vertical vortex as can be seen in Fig. <ref> (e) where two high-density bumps are positioned diagonally about the core. (ii) During an extraction sweep, the atoms that are released early experience a drop in potential since they no longer feel the trap. The result is that the wavefunction phase evolves faster for those atoms that remain trapped, and the eventual interference between the atoms released early and those released late causes destructive or constructive interference depending on the in situ phase pattern about the core. This process was important for the expanded horizontal vortex, seen in Figs. <ref> (d)(f), where constructive interference can be seen on the -x side, but not the +x side.Since each experiment periodically images a given condensate, it is useful to process the extraction images to determine parameters that keep track of the in situ vortices. The first step is to integrate either column-density over the remaining radial direction to obtain the 1D density, n_ 1D(x) = ∫∫ |ψ(𝐱)|^2dydz, as plotted in Figs. <ref> (g)(h). A 1D residual density, n_ res(x) = n_ 1D(x)-f_ poly(x), is then obtained by subtracting a fourth-order polynomial fit f_ poly(x). The residual is subsequently fitted by the function,f_ fit(x) = Acos[B(x-x_ν) + δ]/cosh^2[(x-x_ν)/C] ,for fit parameters A<0, B, δ, C, x_ν. It follows, then, that x_ν provides a measure of the vortex position along the long axis while the phase δ furnishes a means of tracking its orientation. For the cases considered in Fig. <ref> we find that for the vertical vortex, {δ,x_ν} = {0,0}, while for the horizontal vortex, {δ,x_ν} = {-1.35,-3.5μ m}. Note that for the latter case, the value of x_ν slightly misrepresents the position of the vortex, which lies in the x=0 plane. Since a horizontal vortex that is oriented in the -y direction has a phase δ = -1.35 then, by symmetry, a horizontal vortex of opposite sense must have δ = +1.35. An important point is that although this fitting procedure can ascertain that a vortex is vertical, it cannot determine its sense. However, Fig. <ref> (e) demonstrates that the sense of a vertical vortex can easily be attained from top-down images in the x-y plane if one notes the locations of the diagonal high-density bumps about the core. Furthermore, our simulations (not shown here) illustrate that adding a real-time imaging capability along the vertical direction, which was not feasible in the present experiments, would clearly reveal the y position of off-center vertical vortex cores (see Fig. <ref> (e) for comparison). A numerical calculation of the 1D residual n_ res(x), as a function of time, is presented for a precessing vortex in Fig. <ref> (a1). The central (red) color represents a negative value while the outer (green) color is positive. The initial state for this simulation, highlighted as the thick red line in Fig. <ref> (a), is evolved according to the time-dependent GPE [Eq. <ref>]. As the vortex precesses, the wavefunction is periodically saved and each of these is then used to initiate an extraction simulation to produce a single time slice of Fig. <ref> (a). This particular vortex has a tilt of θ = 0.22 radians and a precession frequency Ω/2π = 5.5 Hz, as can be seen from Fig. <ref>.The fitted phase δ [see Eq. (<ref>)], shown in Fig. <ref> (a2), is a smoothly varying function of time that maps to the in situ orientation of the precessing vortex.We present experimental evidence for a precessing solitonic vortex in Fig. <ref> (b). This has a precession frequency of ≈ 5 Hz, which is very similar to that of our numerical simulation in Fig. <ref> (a). Since the experiment has a chemical potential ≈ 3 times larger than in our simulations, the scaling of our analytic model, Eq. (<ref>), suggests that the tilt angle be ≈ 2 times larger for this experiment, i.e. θ∼ 0.4 radians. As was the case for the simulation, in Fig. <ref> (b2) the residual's phase displays an oscillatory behavior. However, a difference here is that δ now has an asymmetry, with some saturation near - π/2. A possible explanation for this bias is a slight tilt (∼ 1 degree) of the imaging camera, which looks down the y axis, effectively rotating the x-z plane relative to the direction of gravity.To help visualize this, consider the simulation in Fig. <ref> (d), where the density minimum of the vortex core exhibits a tilted, relatively narrow canyon in the vertical direction. The sensitivity to a camera tilt is expected to be even more pronounced in the regime of the experiment, for which the healing length is smaller, the Thomas-Fermi radii are larger and the TOF is longer. Two further experimental examples of precessing vortices are presented in Figs. <ref> (c) and (d). In addition to the precession evidenced by the changing order of colors (and the corresponding oscillations of δ), these vortices orbit about the BEC's origin, manifested here as oscillations of their x coordinate, as they follow contours of constant Thomas-Fermi density <cit.>. We note that it is not possible to directly obtain a vortex's axial tilt from our TOF images, due to complications from the interactions between the extracted portion and the trapped condensate, as well as interference effects within the extracted portion, thus prohibiting a quantitative comparison with our theoretical prediction (<ref>) for the precession frequency as a function of the tilt angle. An intriguing question is how the relationship between the residual and vortex orientation is modified for off-center vortices, i.e., those that do not pass through the x axis. The residual phase versus the in situ orientation angle is plotted for various off-center vortices in Fig. <ref> (a). These untilted vortices, which lie in the x = 0 plane, are represented as solid lines that vary in color from black to light blue (gray). To aid with their visualization, the positions of these vortex cores are plotted for three angles (the three vertical dashed lines in the main plot) in Figs. <ref> (b)(c)(d), with a matching color code. In the main plot, the stationary-state vortex (the straight line in the lower panels) has a δ that is symmetric about zero, as expected. For comparisson, we collapse the precessing vortex data from Fig. <ref> (a2) onto the main plot of Fig. <ref>, and mark these with plus symbols. The behavior is fairly similar to that of the stationary-state vortex, indicating that the axial tilt (θ>0) itself does not have a significant effect on the residual. The behavior changes radically, however, if a vortex is off-center. In particular, the values of δ become asymmetric and tend to become, either more negative or positive depending on the sense of the vortex relative to its closest boundary. For the most-off-center vortices, take the one indicated by the black line for example, changing the orientation angle has little effect on δ. Consequently, as a vortex becomes more off-center, δ is no longer a useful indictor of a vortex's orientation, but instead conveys the sense of the vortex relative to its closest boundary. As a final note, we can deduce that this off-center-vortex effect is not responsible for the δ<0 asymmetry in Fig. <ref> (b2). This is because such an off-center vortex would also orbit about the condensate center <cit.>, which would be evident as large oscillations of the vortex position along the x axis, contrary to observations in Fig. <ref> (b1).§ CONCLUSIONS Solitonic vortices are highly-localized objects, both in terms of their energy and angular momentum densities, in stark contrast to vortices of untrapped systems. With this as motivation, we developed a theoretical model that treats solitonic vortices on a similar footing to classical spinning tops. Using our minimally destructive imaging scheme we experimentally observed this spinning-top behavior by periodically imaging a given condensate in real time. We performed 3D Gross-Pitaevskii simulations to investigate how the precession frequency varies as a function of the axial tilt angle, and comparisons of these with our analytic prediction further supported our spinning-top model, while also quantifying its limitations. Finally, we carried out Gross-Pitaevskii simulations of our experimental extraction and imaging scheme. From these we suggested improvements, such as the addition of real-time imaging capability along the vertical (z) direction to keep track of the sense and radial position of vortices when they are vertical. Our simulations also demonstrated how to interpret expansion images when vortices are off-center, i.e. when they do not pass through the x axis.An interesting question arises regarding the role of a trap asymmetry in the radial plane or, in other words, the squashing of the cigar-shaped trap along one of its short directions. For small radial asymmetries the vortex is still expected to precess about the trap's long axis, albeit with a small periodic change of the tilt angle to preserve its length and hence conserve its energy. 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"authors": [
"R. N. Bisset",
"S. Serafini",
"E. Iseni",
"M. Barbiero",
"T. Bienaimé",
"G. Lamporesi",
"G. Ferrari",
"F. Dalfovo"
],
"categories": [
"cond-mat.quant-gas"
],
"primary_category": "cond-mat.quant-gas",
"published": "20170525092339",
"title": "Observation of a Spinning Top in a Bose-Einstein Condensate"
} |
On Star Coloring of Splitting Graphs]On Star Coloring of Splitting GraphsInstitute of Informatics University of Gdańsk Wita Stwosza 57 80-952 Gdańsk Poland [email protected] Part-Time Research Scholar (Category-B) Research & Development Centre Bharathiar University Coimbatore 641 046 TamilnaduIndia [email protected] Department of MathematicsUniversity College of Engineering Nagercoil(Anna University Constituent College)Konam Nagercoil - 629 004 Tamilnadu India [email protected] this paper, we consider the problem of a star coloring. In general case the problems in NP-complete.We establish the star chromatic number for splitting graph of complete and complete bipartite graphs, as well of paths and cycles. Our proofs are constructive, so they lead to appropriate star colorings of graphs under consideration. 05C15, 05C75[ Vernold Vivin.J===================§ INTRODUCTIONWe consider only finite, undirected, loopless graphs without multiple edges. The notion of star chromatic number was introduced by Branko Grünbaum in 1973. A star coloring <cit.> of a graph G is a proper vertex coloring in whichevery path on four vertices uses at least three distinct colors. Equivalently, in a star coloring, the induced subgraphs formed by the vertices of any two color classes has connected componentsthat are star graphs. Thestar graph is a tree with at most one vertex with degree larger than 1. Star coloring is a strengthening of acyclic coloring <cit.>, i.e. proper coloring in which every two color classes induce a forest. The star chromatic number χ_s(G) of G is the least number of colors needed to star coloring of G.Guillaume Fertin et al.<cit.> gave the exact value of the star chromatic number of different families of graphs such as trees, cycles, complete bipartite graphs, outerplanar graphs,and 2-dimensional grids. They also investigated and gave bounds for the star chromatic number of other families of graphs, such as planar graphs, hypercubes, d-dimensional grids(d≥3), d-dimensional tori (d≥2), graphs with bounded treewidth, and cubic graphs.In this paper we consider star coloring of some splitting graphs. For a givengraph G the splitting graph<cit.> S(G) of graph G is obtained by adding a new vertex v' corresponding to each vertex v of G such that N(v)=N(v'), where N(x) is the neighborhood of vertex x.For example, the splitting graph of K_2,3 is given in Fig. <ref>.Albertson et al.<cit.> showed that it is NP-complete to determine whether χ_s(G)≤ 3, even when G is a graph that is both planar and bipartite.Coleman and Moré <cit.> proved that finding an optimal starcoloring is NP-hard and remain so even for bipartite graphs. One can ask whether there is a subclass of planar graphs such that admits optimal star coloring in polynomial time.Sampathkumar and Walikar <cit.> posed an open problem: full characterization of graphs whose splitting graphs are planar. In this situation considering star coloring of splitting graphsseems to be desirable. Moreover, star coloring problem has application in combinatorial scientific computing. In particular, it has been employed since 1980's to efficientlycompute sparse Jacobian and Hessian matrices using either finitedifferences or automatic differentiation <cit.>.Additional graph theory terminology used in this paper can be found in <cit.>.For the completness of the reasoning given in this paper, we recall some known results. If C_n is a cycle on n ≥ 3 vertices, then χ_s(C_n)=4 when n=5 3otherwise.In this paper we prove results concerning the star chromatic number of splitting graph of complete graphs,paths, complete bipartite graphs andcycles. § MAIN RESULTS §.§ Star Coloring of Splitting graph of complete graphs Let K_n be a complete graph on n≥2 vertices. Thenχ_s(S(K_n))=n+1.Let V(S(K_n))={v_i:1≤ i ≤ n }∪{v_i^':1 ≤ i ≤ n }.We define star (n+1)-coloring σ of S(K_n) in the following way:σ(v_i)=c_i:1≤ i≤ n; σ(v^'_i)=c_n+1:1≤ i≤ n.Clearly the verticesof complete graph K_n needs n colors for a proper coloring and hence for star coloring. Thus,χ_s(S(K_n))≥ n. In the further part we will show that n colors are insufficient for star coloring of S(K_n).By definition of splitting graph, for 1≤ j≤ n, the vertex v_j is adjacent to all v_i except fori=j. Hence σ (v_j)≠σ(v_i^') for i ≠ j. Ifσ(v_i^')=σ(v_i),then there exist bicolored paths v_i^' v_i+1v_iv_i+1^' for 1≤ i ≤ n.A contradiction to proper star coloring. Thus, χ_s(S(K_n))=n+1. §.§ Star coloring of splitting graph of paths Let P_n be a path on n≥ 4 vertices. Thenχ_s(S(P_n))=4. Let V(S(P_n))={v_1,v_2,…,v_n, v_1', …, v_n'}. It is clear that vertices of P_n need three colors for proper star coloring. Thus χ_s(S(P_n))≥ 3.Now, we will show that three colors are insufficient for star coloring of S(P_n), n≥ 4. We start fromany star 3-coloring of P_n. we will denote it by the coloring c. Let v_i, v_i+1, v_i+2, v_i+3, 1 ≤ i ≤ n-3, be any four consecutivevertices of P_n. It is clear that there exists at least one pair of vertices of length two among these four ones with different colors assigned.Without lost of generality we may assume that c(v_i) ≠ c(v_i+2). Then color that may be used to color vertex v_i+1^' is determined toc(v_i+1). Let us try to assign appropriate color to v_i+2^'. We have two possibilities - we can use color c(v_i) or c(v_i+2). But anyof these two choices leads to 2-chromatic P_4. Thus χ_s(S(P_n))≥ 4.The star 4-coloring σ of S(P_n) is defined as follows: σ(v_i)= c_1i≡ 13 c_2i≡ 23 c_3i≡ 03and σ(v^'_i)=c_4for 1≤ i≤ n. It is easy to verify that any two color classes in coloring σ induce a forest whose components are K_1,2 and K_2,hence by definition σ is a proper star 4-coloring. Hence, χ_s(S(P_n))=4. It is easy to verify that χ_s(S(P_2))=χ_s(S(P_3))=3. §.§ Star coloring of splitting graph of complete bipartite graphsLet K_m,n, m ≤ n be complete bipartite graph. Thenχ_s(S(K_m,n))=2m+1. Let V(K_m,n)=X ∪ Y={v_1,v_2,…,v_m}∪{u_1,u_2,…,u_n}. Then we haveV(S(K_m,n))={X,X',Y,Y'}= {v_1,v_2,…, v_m ;v_1^',v_2^',…,v_m^'; u_1,u_2,…,. . u_n; u_1^',u_2^',…, u_n^'}. Note that in any star k-coloring of complete graph K_m,n, k < m+n, there is only one bipartite partition set, let say A, including at least two vertices from the same color class. Vertices in the second bipartitepartition set,let say B, must be colored with different |B| colors and these colors cannot be assigned to vertices in set A. We will named such multicoloredbipartite partition set B as rainbow.This implies that at least two out of four partition sets X,X',Y,Y' of S(K_m,n) must be rainbow:X,Y, X,X', Y,Y', or X',Y'.Case 1 X and Y are rainbow.Note that such coloring of X and Y uses different (m+n) colors and it may be extended to proper star (m+n+1)-coloring of S(K_m,n). Case 2 X and X' are rainbow.Note that such coloring of X and X' may use m colors, but then it cannot be extended to any star k-coloring of S(K_m,n) with k < m+n+1.We may extend it only to star k-coloring with k ≥ m+n+1. Indeed, Y must be also raibow in this case, otherwise 2-chromatic P_4 arises.If coloring of X and X' uses 2m colors, then it may be extended to star (2m+1)-coloring. Notice that 2m+1 ≤ m+n+1 for m ≤ n. Case 3 Y and Y' are rainbow.This case may be considered analogously to Case 2. We have two possibilities: star (n+m+1)- or (2n+1)-coloring of S(K_m,n). Case 4 X' and Y' are rainbow.In this cases at least one out of X and Y must be also rainbow partition set and we may look for optimal star coloring of S(K_m,n) due toCase 2 or Case 3. Summarizing, star coloring of S(K_m,n) with the smallest number of colors is mentioned in Case 2. In details, we define star (2m+1)-coloringσ of S(K_m,n) in the following way. σ(v_i)=c_i, 1≤ i≤ m σ(u_j)=c_m+1, 1≤ j≤ n σ(u^'_j)=c_m+1, 1≤ j≤ nσ(v_i^')=c_m+1+i, 1≤ i≤ m. §.§ Star coloring of splitting graph of cyclesLet C_n be a cycle graph on n≥ 3 vertices. Thenχ_s(S(C_n))= 4n ≢13andn ≠ 5 ≤ 5 Let V(C_n)={v_1,v_2,…, v_n} and V(S(C_n))={v_1,v_2,… v_n,v_1^',v_2^',. . …, v_n^'}. By Theorem <ref>, χ_s(C_n)=3and thus χ_s(S(C_n))≥ 3, for n≥ 6. We claim that 3 colors are insufficient for star coloringof S(C_n). First, we assign colors {c_1,c_2,c_3} to vertices of C_n to obtain proper star 3-coloring c of C_n. If wewant to extend this c coloring into whole S(C_n) without adding a new color, we may have at most three possibilities.Let v_i, v_i+1, v_i+2, v_i+3 be any four consecutive vertices of C_n, 1≤ i ≤ n-3. Since the cycle is star 3-colored, then exactly two out of four vertices v_i, v_i+1, v_i+2, v_i+3 are assigned the same color: * c(v_i)=c(v_i+3)Then the coloring of v_i+1^' and v_i+2^' is determined. Vertices v_i+1^' and v_i+2^' must obtain the samecolors as v_i+1 and v_i+2, respectively. But then 2-colored P_4 arises: v_i+2^', v_i+1, v_i+2, v_i+1^'. We need fourth color.* c(v_i)=c(v_i+2)It is clear that c(v_i+1)≠ c(v_i+3) and the coloring of v_i+2^' is determined while vertex v_i+1^' can be assigned two colors: c(v_i+1) or c(v_i+3). If we assign color c(v_i+1) to vertex v_i+1^', then we get 2-chromaticP_4: v_i,v_i+1^', v_i+2, v_i+1. In the second choice, also 2-chroma-tic P_4 arises: v_i, v_i+1^', v_i+2, v_i+3. Also in this case, we get additional color.* c(v_i+1)=c(v_i+3)This case is analogous to case with c(v_i)=c(v_i+2).Summarizing, χ_s(S(C_n)≥ 4. We will consider three cases depending on the value of n3, n≥ 3. Case 1 n ≡ 03.We define 4-coloring σ of S(C_n) in the following way. σ(v_i)= c_1i≡ 13 c_2i≡ 23 c_3i≡ 03and σ(v^'_i)=c_4, 1≤ i≤ n.Consider the color classes c_i and c_j, (1≤ i < j ≤ 4). The components of induced subgraph of these color classes are K_2 and K_1,2. Hence there exists no bicolored path on four vertices and thus σ is a proper star 4-coloring. Hence χ_s(S(C_n))=4. Case 2 n ≡ 13For n≥ 4, we define 5-coloring σ of S(C_n) in the following way. σ(v_i)= c_1i≡ 13 c_2i≡ 23 c_3i≡ 03for 1 ≤ i ≤ n-1, and σ(v_n)=c_2, σ(v^'_i)= c_4i≡ 03 c_5for 1≤ i≤ n.It is clear that every two out of three color classes corresponding to colors c_1, c_2, and c_3 avoid 2-chromatic P_4,similarly as color classes corresponding to colors c_4 and c_5. We have to only check pairs of color classes c_k;1 ≤ k ≤ 3 and c_4 or c_5. It is easy to check that every such two color classes induces forest whose components are isolated vertices, P_2 or K_1,2. Thus, σ is proper star 5-coloring. Hence, 4 ≤χ_s(S(C_n)) ≤ 5. Case 3 n=5Star 5-coloring of S(C_5) is given in Fig. <ref>b). Case 4 n ≡ 23 and n ≥ 8 Case 4.1 n=8Star 4-coloring of S(C_8) is given in Fig. <ref>c). Case 4.2 n ≥ 11Let n=8+3t, t ≥ 1. First, we color v_1, … v_8 in the way given in Fig. <ref>.Next, the remaining vertices of a cycle of S(C_n) are colored in the following way. σ(v_i)= c_1i≡ 03 c_2i≡ 13 c_3i≡ 23for 9≤ i≤ n, and σ(v_i^')= c_3c(v_i)=3 c_4 Similarly as it was in Case 1 we can check that σ is proper star 4-coloring. Hence, χ_s(S(C_n))=4. § CONCLUSIONIn the paper the problem of a star coloring for some splitting graphs has been considered. As an open question we put the problemof determining exact values of star chromatic number of splitting graphs of cycles C_n where n=5 or n ≡ 13. Moreover, considering star coloring of degree splitting graphs, defined by Ponraj and Somasundaram <cit.>, seems to be worth paying attention. 88 alberton Albertson, Michael O, Chappell, Glenn G, Kierstead, Hal A, Kündgen, André and Ramamurthi, Radhika , Coloring with no 2-Colored P_4's, The Electronic Journal of Combinatorics 11 (2004), Paper # R26,13. bm J.A. Bondy, U.S.R. Murty, Graph theory with Applications, London, MacMillan 1976. bg B. Grünbaum, Acyclic colorings of planar graphs, Israel J.Math 14(1973), 390–408. col T.F. Coleman, J. Moré, Estimation of sparse Hessian matrices and graph coloring problems, Mathematical Programming 28(3)(1984), 243–270. fertin G. Fertin,A. Raspaud, andB. Reed, On Star coloring of graphs, J. Graph Theory 47(3)(2004), 163–182. f F. Harary, Graph Theory, Narosa Publishing home, New Delhi 1969. ponraj R. Ponraj, S. Somasundaram, On the degree splitting graph of a graph, Nat. Acad. Sci. Letters 27(7-8)(2004), 275–278. sam E. Sampathkumar, H.B. Walikar, On splitting graph of a graph, J. Karnatak Univ. Sci., 25 and 26 (Combined) (1980-81), 13–16. | http://arxiv.org/abs/1705.09357v1 | {
"authors": [
"Hanna Furmańczyk",
"Kowsalya V",
"Vernold Vivin J"
],
"categories": [
"math.CO"
],
"primary_category": "math.CO",
"published": "20170525204931",
"title": "On Star Coloring of Splitting Graphs"
} |
*GREAT is a development by the MPI für Radioastronomie and the KOSMA/Universität zu Köln, in cooperation with the MPI für Sonnensystemforschung and the DLR Institut für Planetenforschung. 1Department of Physics & Astronomy, Johns Hopkins University3400 North Charles Street, Baltimore, MD 21218, USA 2Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA 2Department of Physics and Astronomy, San Jose State University, One Washington Square, San Jose, CA 95192-0106, USA 4Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53121 Bonn, Germany 5Univ. Grenoble Alpes, IPAG, and CNRS, F-38000 Grenoble, FranceWe report the discovery of water maser emission at frequencies above 1 THz.Using the GREAT instrument on SOFIA, we have detected emission in the 1.296411 THz 8_27-7_34 transition of water toward three oxygen-rich evolved stars: W Hya, U Her, and VY CMa.An upper limit on the 1.296 THz line flux was obtained toward R Aql. Near-simultaneous observations of the 22.23508 GHz 6_16-5_23 water maser transition were carried out towards all four sources using the Effelsberg 100m telescope.The measured line fluxes imply 22 GHz / 1.296 THz photon luminosity ratios of 0.012, 0.12, and 0.83 respectively for W Hya, U Her, and VY CMa, values that confirm the 22 GHz maser transition to be unsaturated in W Hya and U Her.We also detected the 1.884888 THz 8_45-7_52 transition toward W Hya and VY CMa, and the 1.278266 THz 7_43-6_52 transition toward VY CMa.Like the 22 GHz maser transition, all three of the THz emission lines detected here originate from the ortho-H_2O spin isomer. Based upon a model for the circumstellar envelope of W Hya, we estimate that stimulated emission is responsible for ∼ 85 % of the observed 1.296 THz line emission, and thus that this transition may be properly described as a terahertz-frequency maser.In the case of the 1.885 THz transition, by contrast, our W Hya model indicates that the observed emission is dominated by spontaneous radiative decay, even though a population inversion exists. § INTRODUCTION Maser action is widely-observed in Nature. It amplifies the emission from specific transitions of abundant astrophysical molecules in a variety of environments, including the interstellar medium, circumstellar envelopes, and the accretion disks within active galactic nuclei. Among the half-dozen astrophysical molecules that have been observed to exhibit the maser phenomemon, water vapor possesses the brightest known maser transitions, with brightness temperatures for its 22 GHz 6_16-5_23 transition often exceeding 10^10 K and in exceptional cases reaching 10^14 K.Thanks to these extraordinarily high brightness temperatures, this transition can be observed by means of Very Long Baseline Interferometry (VLBI), providing submilliarcsecond angular resolution that enables a variety of astronomical studies that have greatly expanded our understanding of the kinematics of protostellar outflows; helped elucidate the size, shape and kinematics of the Milky Way (e.g. Reid et al. 2009);and have provided among the best evidence yet obtained for the existence of supermassive black holes (e.g. Miyoshi et al. 1995). ccccc0ptObserved maser transitions within the ground vibrational state of waterTransition Frequency E_U/k ^a Discovery Environment(s)^b (GHz) (K) reference6_16-5_2322.235644 Cheung et al. (1969) IS, CSE, AGN 3_13-2_20 183.310205 Waters et al. (1980) IS, CSE, AGN 4_14-3_21 380.197323 Phillips et al. (1980) IS10_29-9_36321.2261861 Menten et al. (1990a) IS, CSE, AGN5_15-4_22 325.153575 Menten et al. (1990b) IS, CSE7_53-6_60 437.3471525 Melnick et al. (1993) CSE 6_43-5_50 439.1511089 Melnick et al. (1993) IS, CSE6_42-5_51 470.8891090 Melnick et al. (1993) IS, CSE5_33-4_40 474.689725 Menten et al. (2008) CSE, IS 5_32-4_41 620.701732 Harwit et al. (2010) IS, CSE 5_24-4_31 970.315599 Justtanont et al. (2012) CSE 7_43-6_52 1278.266 1340 Present work CSE8_27-7_34 1296.411 1274 Present work CSE 8_45-7_52 1884.888 1615 Present work CSE 5l^a Energy of upper state (temperature units) relative to the ground state of para-H_2O 5l^b IS = interstellar; CSE = circumstellar envelopes; AGN = active galactic nuclei In the decades following the first detection of interstellar water vapor through its masing 22 GHz transition (Cheung et al. 1969), additional water maser transitions have been discovered at higher frequencies.In Table 1, we provide a list of all observed maser transitions within the ground vibrational state of water that have been detected at high spectral resolution with the use of heterodyne receivers.NASA's airborne astronomy program played an early role in these discoveries, with the first detections of the 183 GHz (Waters et al. 1980) and 380 GHz (Phillips et al. 1980) transitions having been obtained using the Kuiper Airborne Observatory.In the 1990's and 2000's, thanks to the development of heterodyne receivers on ground-based submillimeter telescopes at dry, high-altitude sites, several additional maser transitions were detected in the 300 - 500 GHz range.More recently, two additional water maser lines – at frequencies in the 500 - 1000 GHz range – were detected using the HIFI instrument on Herschel.These transitions are indicated in the energy level diagram shown in Figure 1.Multitransition maser observations provide key constraints on the pumping mechanisms that lead to a population inversion and on the physical conditions in the masing region (Neufeld & Melnick 1991; hereafter NM91).All the masing transitions shown in Figure 1 have Δ J = -1 and Δ K_A = +1, a behavior predicted by models that invoke a combination of collisional excitation and spontaneous radiative decay as the origin of the population inversion.Although the quantitative predictions of such models do depend upon the details of the collisional rate coefficients, the question of which transitions become inverted is largely determined by the arrangement of the energy levels (e.g. NM91).Specific transitions become inverted because the upper state has a longer lifetime than the lower state for radiative decayvia non-masing far-infrared transitions.Any population inversion inevitably involves a departure from local thermodynamic equilibrium, and the magnitude of the required departure is an increasing function of the transition frequency, as are the spontaneous radiative decay rates for the masing transition.These considerations raise the following questions: what is the maximum frequency of water transitions in which a population inversion can occur in astrophysical environments?And, even if a population inversion is present, at what point does spontaneous radiative decay dominate stimulated emission?To address these questions, to test the collisional pumping scheme proposed as the origin of water maser emission, and to probe the physical and chemical conditions within circumstellar envelopes, we used the GREAT instrument on SOFIA to search for Terahertz maser transitions toward four oxygen-rich evolved stars: W Hya, U Her, R Aql, and VY CMa.The target sources have all been known to exhibit water maser emission in multiple lower-frequency transitions (e.g. Menten & Melnick 1991). The primary transition of interest was the 8_27-7_34 line at 1.296 THz, a transition predicted to be strongly masing (NM91; Gray et al. 2016); in addition, we targeted the 8_45-7_52 line at 1.885 THz toward all four sources at the same time as the 1.296 THz line, and the 7_43-6_52 line at 1.278 THz toward VY CMa.The observing strategy and data reduction methods are described in Section 2 below, and the results presented in Section 3.In Section 4 we discuss the results in the context of models for water maser emission from the circumstellar envelopes of evolved stars.§ OBSERVATIONS AND DATA REDUCTION Table 2 lists the evolved stars toward which the water transitions were observed, along with the positions targeted, the estimated distance to each source, the period of the stellar variability, the spectral type, the source systemic velocity relative to the Local Standard of Rest (LSR), and the estimated mass-loss rate.Also shown are the date of each SOFIA observation, the corresponding stellar phase, the velocity of Earth's atmosphere relative to the LSR, the observatory altitude at the time of the observation, and the total integration time.The 1.278 and 1.296 THz water lines were observed in the upper sideband of the L1 receiver of GREAT using the FFT4G backend; the latter provides ∼ 16,000 spectral channelswith a spacing of 283 kHz. The 1.885 THz water line was observed simultaneously in the lower sideband of the LFA receiver.At frequencies of 1.296 and 1.885 THz, the telescope beam has a diameter of ∼ 20^'' and ∼ 14^'' (HPBW) respectively.The observations were performed in dual beam switch mode, with a chopper frequency of 1 Hz and the reference positions located 60^'' on either side of the source along an east-west axis.The raw SOFIA data were calibrated to the T_A^* (“forward beam brightness temperature”) scale, using an independent fit to the dry and the wet content of the atmospheric emission.Here, the assumed forward efficiency was 0.97 and the assumed main beam efficiency for the L1 band was 0.67.The uncertainty in the flux calibration is estimated to be ∼ 20% (Heyminck et al. 2012).Additional data reduction was performed using CLASS[Continuum and Line Analysis Single-dish Software.]. This entailed the removal of a first-order baseline and the rebinning of the data to a channel width of 0.51 km s^-1 (1296 GHz line), 0.93 km s^-1 (1885 GHz line), or 1.03 km s^-1 (1278 GHz line).We used the Effelsberg 100-m telescope to carry out observations of the 22.23508 GHz 6_16 - 5_23 transition toward all four sources. In all cases, these observations were performed within 16 days of the SOFIA observations, and for U Her and VY CMa they were carried out on the same day. The 22 GHz observations were performed with the new secondary focus receiver, with a beam size of 41^'' (HPBW), which was connected to a FPGA-based FFT spectrometer providing a frequency resolution of 1.5 kHz. The data were calibrated by correcting for atmospheric opacity and the dependence of the telescope gain on elevation. For the conversion of the observed spectra into Jy, suitable calibration sources like 3C 286, NGC 7027, etc.were observed to determine the telescope's sensitivity (which was about 1 K/Jy for all observations). § RESULTS In Figure 2, we present the reduced spectra obtained from both SOFIA and Effelsberg for all four sources.The 22 GHz line was observed at very high signal-to-noise ratio in all sources, and the 1.296 THz line was unequivocally detected toward all sources expect R Aql.The 1.885 THz line was clearly detected toward W Hya and VY CMa.A search for the 1.278 THz line was only conducted toward one source, VY CMa, and led to a clear detection.The 1.278 and 1.296 THz H_2O lines had previously been detected toward VY CMa in spectrally-unresolved observations reported by Matsuura et al. (2014) using the SPIRE instrument on Herschel.In Table 2, we present the line-integrated fluxes measured for each source, as well as the velocity centroids and velocity dispersions for cases in which a line was detected.§ DISCUSSION§.§ Maser line ratios For the three stars toward which the 1.296 GHz line was detected, the 22 GHz/1.296 THz photon luminosity ratios listed in Table 2 lie in the range 0.012 to 0.137.We have compared these values with the predictions of a simple collisional-excitation model in which the steady-state level populations are computed as a function of H_2 density in the range 10^2 to 10^12cm^-3, H_2O column density in the range 10^10.5 to 10^18.5cm^-2 per km s^-1, and temperature in the range 10 to 5000 K.In this model, the radiative trapping of photons in non-inverted transitions is computed using an escape probability method.For inverted transitions, the effects of stimulated emission are initially neglected to obtain a solution applicable to the limit in which the maser gain is small and the population inversion is not reduced by maser emission. This yields the (negative) maser optical depths, τ( unsat), in the unsaturated limit, and also allows us to compute the maximum rate of maser photon emission that can be achieved while maintaining a population inversion.Full details of the method have been described by NM91.The one modification to the treatment discussed in NM91 is the substitution of the best currently-available rate coefficients for the collisional-excitation of H_2O by H_2; for transitions amongst the lowest 45 rotational states of ortho- and para-H_2, we adopt the excitation rate coefficients computed by Daniel et al. (2011), while for transitions involving higher-lying states we use an extrapolation method involving an artificial neural network (Neufeld 2010).A total of 120 states, all in the ground vibrational state of ortho-H_2O, are included.Radiative pumping in rotational and rovibrational lines is not included in this model.Although the infrared radiation field in circumstellar envelopes is typically much higher than in the interstellar medium, calculations by Deguchi & Rieu (1990) have determined that the excitation of water is still dominated by collisional excitation: in particular, they found that the water line fluxes expected from oxygen-rich evolved stars are typically altered by less than 20% as a result of vibrational radiative excitation.[Our purely collisional pumping model can explain many of the observed relative H_2O maser luminosity ratios.Nevertheless, clearly additional pumping mechanisms must be at work for some of the detected maser lines, most notably for the 437 GHz 7_53-6_60 transition. In this line, which has remained undetected in star-forming regions, Melnick et al. (1993) detected strong maser emission toward the Mira variable U Her with a flux density rivaling that of the 439 GHz line and much higher than that of the 325 and 471 lines (see Table 1). For the Mira star R Leo, Menten et al. (2008) found the 437 GHz to be by far the strongest of the 6 submillimeter H_2O maser lines they observed, with a 1200 times higher line luminosity than the 22 GHz line. In their comprehensive study on H_2O masers, Gray et al. (2016) included both radiative and collisional pumping to the ν_2=1 and 2 states of the bending mode. These calculations failed to reproduce the strong maser emission sometimes observed in the 437 GHz line. Apparently, the consideration of other vibrational states or other pumping processes is necessary for reaching a more comprehensive picture of H_2O maser pumping in evolved stars.] Moreover, in the extensive calculations of water maser emission reported recently by Gray et al. (2016), all three of the THz transitions discussed here were found to be in the class of collisionally-pumped masers.If both transitions are saturated, the 22 GHz/1.296 THz luminosity ratio is a decreasing function of gas temperature, as expected given the relative energies of the upper states (E_U/k = 644 K and 1274 K respectively for the 22 GHz and 1.296 THz transitions).However, for any combination of gas density, H_2O column density, and temperature, the minimum 22 GHz/1.296 THz photon luminosity ratio predicted for saturated maser emission is found to be 0.7, substantially larger than the observed values (0.012 to 0.137) in W Hya and U Her.This discrepancy confirms the suggestion, originally made by Menten & Melnick (1991) on the basis of 22 / 321 GHz maser line ratios, that the 22 GHz maser transition is often unsaturated in evolved stars.This suggestion may also explain the somewhat narrower line widths observed for the 22 GHz emission from W Hya and U Her (see Table 2). In the case of VY CMa, although the 22 GHz line shows a velocity dispersion similar to that of the THz lines, the 22 GHz spectrum exhibits more substructure than the THz line spectra.While a detailed comparison of the measured spectra is limited by the signal-to-noise ratio of the THz observations, it is clear, for example, that the narrow spike at v_ LSR∼ -1kms^-1 in the 22 GHz spectra is absent or relatively much weaker in the THz line spectra. §.§ Simple model for W Hya While the detailed modeling of the observed water maser emissions is beyond the scope of this paper, we present a simplified model for one of the sources, W Hya.Here, we adopted the gas temperature and velocity profiles given by Khouri et al. (2014), and used the excitation model described above to compute the following quantities for each transition as a function of distance from the star: themaser optical depth along a tangential ray in the unsaturated limit, τ( unsat); the photon emission rate per unit volume in the limit of saturation, Φ_p( sat); and the rate of spontaneous radiative decay per unit volume Φ_p( spont).These quantities were computed with the use of the Large Velocity Gradient (LVG) approximation. In upper left panel of Figure 3, the gas temperature and radial outflow velocity are shown by black and blue curves as a function of distance, R, from the center of the star.The other three panels, labeled by transition frequency, show 4 π R^3 Φ_p( sat) (black) and 4 π R^3 Φ_p( spont) (red) for each transition.Presented in this format, with the volumetric emission rates multiplied by 4 π R^3, these curves show the photon luminosity per logarithmic radial interval.In regions where a population inversion is present (τ( unsat) < 0), the total photon emission Φ_p is bracketed by Φ_p( sat) and Φ_p( spont).For the 22 GHz transition, Φ_p( sat) typically exceeds Φ_p( spont) by a factor ∼ 10^6, and a very large maser gain would be needed to achieve saturation.For the two THz maser transitions, Φ_p( sat) exceeds Φ_p( spont) by at most a factor 10, and only a relatively small gain is needed to achieve saturation.In our simple model, we estimate the actual photon emission rate, Φ_p, by considering maser amplification along tangential rays, assuming that maser amplifies seed radiation consisting of (1) the cosmic microwave background radiation and (2) radiation emitted by the spontaneous radiative decay.With these assumptions, we may obtain line flux predictions for all three transitions.The best agreement with the observed line fluxes, with all three line flux predictions within 15% of the observed values, was obtained for an assumed mass-loss rate of Ṁ=7 × 10^-8M_⊙ yr^-1 and an ortho-water abundance of x( o-H_2O)=3.1 × 10^-4 relative to H_2. The resultant gas density profile is shown by the red curve in the upper left panel of Figure 3, and the actual rate of photon emission is shown by the blue curves in the other three panels.Clearly, for the THz frequency transitions, the predicted line emission is very close to the maximum achievable for saturated masers.For the 22 GHz transition, however, the emission is unsaturated.In this case, the predicted line luminosity depends very sensitively upon Ṁ, x( o-H_2O) and the details of the velocity field. The best-fit values we obtained for both Ṁ and x( H_2O) are roughly a factor two below the best-fit estimates obtained by Khouri et al. (2014) from a fit to non-masing water transitions.This discrepancy is perhaps unsurprising given the simple nature of the model.A more detailed treatment would require the use of a detailed radiative transfer model for the maser radiation and the stellar and dust continuum radiation, with full inclusion of radiative pumping in rotational and rovibrational lines.Moreover, because the radial velocity gradient is zero within the region interior to the acceleration zone, an accurate calculation would require a treatment beyond the standard LVG approximation. §.§ Relative importance of stimulated emission In the context of the simple model described above for W Hya, we may compare the red (spontaneous emission) and blue (estimated total emission) curves in Figure 3 to determine the contribution of maser action to the emergent line fluxes for each of the three observed transitions.In the case of the 22 GHz transition, spontaneous radiative decay contributes 0.009% of the total emission, the remaining 99.99% being contributed by stimulated emission.For the 1.296 THz transition, the contribution of spontaneous radiative decay is 14%, and that of stimulated emission is 86%.The 1.885 THz transition, by contrast, is dominated by spontaneous radiative decay, which provides 69% of the emergent photons.These results confirm the declining importance of maser amplification as the transition energy increases.While the 1.296 THz transition may properly be described as a terahertz maser, the 1.885 THz transition, although inverted, is one for which maser amplification fails to play a dominant role according to our model for W Hya.This behavior likely reflects the tendency that as the frequency increases, stimulated emission becomes less important relative to spontaneous radiative decay, because the latter typically increases as the cube of the transition frequency. Based on observations made with the NASA/DLR Stratospheric Observatory for Infrared Astronomy, and the 100-m radio telescope of the MPIfR in Effelsberg. SOFIA Science Mission Operations are conducted jointly by the Universities Space Research Association, Inc., under NASA contract NAS2-97001, and the Deutsches SOFIA Institut under DLR contract 50 OK 0901. This research was supported by USRA through a grant for SOFIA Program 04-0023.We gratefully acknowledge the outstanding support provided by the SOFIA Operations Team and the GREAT Instrument Team. [Cheung et al.(1969)]1969Natur.221..626C Cheung, A. C., Rank,D. M., Townes, C. H., Thornton, D. D.,& Welch, W. J. 1969, , 221, 626 [Daniel etal.(2011)]2011A A...536A..76D Daniel, F., Dubernet, M.-L., & Grosjean, A. 2011, , 536, A76 [Gehrz & Woolf(1971)]1971ApJ...165..285G Gehrz, R. D., & Woolf, N. J. 1971, , 165, 285 [Gray et al.(2016)]2016MNRAS.456..374G Gray, M. D., Baudry, A., Richards, A. M. 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G. 1990a, , 350, L41 [Menten et al.(1990)]1990ApJ...363L..27M Menten, K. M., Melnick,G. J., Phillips, T. G., & Neufeld, D. A. 1990b, , 363, L27 [Menten & Melnick(1991)]1991ApJ...377..647M Menten, K. M., & Melnick, G. J. 1991, , 377, 647 [Menten etal.(2008)]2008A A...477..185M Menten, K. M., Lundgren, A., Belloche, A., Thorwirth, S., & Reid, M. J. 2008, , 477, 185 [Miyoshi et al.(1995)]1995Natur.373..127M Miyoshi, M., Moran, J.,Herrnstein, J., et al. 1995, , 373, 127[Neufeld& Melnick(1991)]1991ApJ...368..215N Neufeld, D. A., & Melnick, G. J. 1991, , 368, 215 (NM91) [Neufeld(2010)]2010ApJ...708..635N Neufeld, D. A. 2010, ,708, 635[Phillips et al.(1980)]1980IAUS...87...21P Phillips, T. G., Kwan,J., & Huggins, P. J. 1980, IAUS, 87, 21[Reid et al.(2009)]2009ApJ...705.1548R Reid, M. J., Menten, K. M., Zheng, X. W., Brunthaler, A., & Xu, Y. 2009, , 705, 1548 [Royer et al.(2010)]2010A A...518L.145R Royer, P., Decin, L., Wesson, R., et al. 2010, , 518, L145 [Vlemmings & van Langevelde(2007)]2007A A...472..547V Vlemmings, W. H. T., & van Langevelde, H. J. 2007, , 472, 547 [Waters et al.(1980)]1980ApJ...235...57W Waters, J. W., Kakar,R. K., Kuiper, T. B. H., et al. 1980, , 235, 57 [Zhang et al.(2012)]2012ApJ...744...23Z Zhang, B., Reid, M. J., Menten, K. M., & Zheng, X. W. 2012, , 744, 23 lcccc 0ptSource parameters, observational parameters, and observed line parametersSource: 000 000 W Hya 000000U Her 000 000 R Aql 000 000 VY CMa 000 3lSource parameters 000000RA (J2000) 13h 49m 01.99s 16h 25m 47.47s 19h 06m 22.25s 07h 22m 58.33s000000Dec. (J2000)) –28^o 22^' 03.5^'' +18^o 53^' 32.9^'' +08^o 13^' 46.9^''–25^o 46^' 03.2^'' 000000Distance (pc) 78± 6^a (K03^b) 266 ± 30 (V07) 214 ± 39 (K10) 1200 ± 115 (Z12) 000000Variability period^c (days) 361 406 270 956000000Systemic velocity^d (km s^-1) 40.0 ± 0.5 -13.4 ± 0.848.7 ± 0.8 17.6 ± 1.5000000Spectral type^c M7.5e-M9ep M6.5e-M9.5e M5e-M9IIIe M5eIbp(C6,3) 000000Estimated mass-loss rate (10^-6 M_⊙yr^-1) 0.13 (K14) 2.6 (G71) 0.8 (G71) 150 (R10)3lObservational parameters 000000Date of SOFIA observation 2016 May 27 2016 May 26 2016 Nov 10 2017 Feb 01000000Stellar phase^e 0.71 0.99 0.64 ... 000000Date of Effelsberg observation 2016 May 11 2016 May 26 2016 Nov 14 2017 Feb 01000000Source velocity relative to SOFIA (km s^-1)52 -3153 41000000Observatory altitude (kft) 43 40 – 41 39 – 40.5 42000000SOFIA integration time (min)^f 11 14 14 23 3lIntegrated line fluxes (Jy km s^-1) 00000022 GHz 86.5 167 46 93000000001.278 THz... ... ... 10500± 7900000001.296 THz 6970 ± 3301220 ± 170 < 1120^g 11220 ± 520 0000001.885 THz 3690 ± 530550 ± 300 < 1170^g11080 ± 660 3lPhoton luminosities (10^41 s^-1) 00000022 GHz 3.17 71 12.6 80700 0000001278 GHz... ... ... 91100 ± 69000000001296 GHz 255 ± 12519 ± 73 < 307^g 97400± 45000000001885 GHz 135 ± 19235 ± 128 < 322^g 96100 ± 5700 3lPhoton luminosity ratios00000022 GHz/ 1296 GHz 0.0124 ± 0.0006 0.137 ± 0.019> 0.041 0.829 ± 0.0038 3lLine centroids (km s^-1 relative to the LSR) 00000022 GHz 38.7 –14.646.5 15.1 0000001.278 THz... ... ...19.7 0000001.296 THz 38.7 –15.4 ... 19.60000001.885 THz 39.7–16.3 ... 22.63lLine velocity dispersions (in km s^-1) 00000022 GHz 1.85 1.171.08 7.93 0000001.278 THz... ... ... 10.950000001.296 THz 3.26 1.73 ... 8.770000001.885 THz 3.661.68 ... 8.125l^aWhere given, errors are 1 σ statistical errors 5l^bReferences: K03 = Knapp et al. (2003); V07 = Vlemmings & van Langevelde (2007); K10 = Kamohara et al. (2010); 5l000Z12 = Zhang et al. (2012); K14 = Khouri et al. (2014); G71 = Gehrz & Woolf (1971); R10 = Royer et al. (2010)5l^cGeneral Catalogue of Variable Stars, v5.1 (Samus et al. 2017) 5l^dRelative to the LSR, as determined from observations of thermal SiO emission (Dickinson et al. 1978) 5l^eDetermined from recent visual light curves generated by AAVSO (www.aavso.org) 5l^fOn source integration time 5l^g3σ upper limit obtained for the 43 - 52 km s^-1 v_ LSR range | http://arxiv.org/abs/1705.09672v1 | {
"authors": [
"David A. Neufeld",
"Gary J. Melnick",
"Michael J. Kaufman",
"Helmut Wiesemeyer",
"Rolf Güsten",
"Alex Kraus",
"Karl M. Menten",
"Oliver Ricken",
"Alexandre Faure"
],
"categories": [
"astro-ph.SR",
"astro-ph.GA"
],
"primary_category": "astro-ph.SR",
"published": "20170526180624",
"title": "SOFIA/GREAT Discovery of Terahertz Water Masers"
} |
Analytic estimates for source confusion and parameter estimation errors]Global analysis for the LISA gravitational wave observatoryeXtreme Gravity Institute, Department of Physics, Montana State University, Bozeman, MT [email protected] [email protected] Laser Interferometer Space Antenna (LISA) will explore the source-rich milli-Hertz band of the gravitational wave spectrum. In contrast to ground based detectors, where typical signals are short-lived and discrete,LISA signals are typically long-lived and over-lapping, thus requiring a global data analysis solution that is very different to the source-by-source analysis that has been developed for ground based gravitational wave astronomy. Across the LISA band, gravitational waves are both signals and noise. The dominant contribution to this so-called confusion noise (better termed unresolved signal noise) is expected to come from short period galactic white dwarf binaries, but all sources, including massive black hole binaries and extreme mass ratio captures will also contribute. Previous estimates for the galactic confusion noise have assumed perfect signal subtraction. Here we provide analytic estimates for the signal subtraction residuals and the impact they have on parameter estimation while for the first time incorporating the effects of noise modeling. The analytic estimates are found using a maximum likelihood approximation to the full global Bayesian analysis. We find that while the confusion noise is lowered in the global analysis, the waveform errors for individual sources are increased relative to estimates for isolated signals. We provide estimates for how parameter estimation errors are inflated from various parts of a global analysis. § INTRODUCTIONThe recent discovery of gravitational waves <cit.> and the outstanding success of the LISA pathfinder mission <cit.> have given new life to the LISA mission. Building on decades of study, an updated LISA mission concept <cit.> was recently submitted to address the European Space Agency's “Gravitational Universe” science theme with a launch scheduled for the early 2030's. The plan is to fly three identical spacecraft connected by six laser links forming a triangular detector with 2.5 million km long arms.It has long been recognized that the LISA mission will suffer from “an embarrassment of riches”, delivering data sets so packed with signals that extracting information about individual sources will require the development of unique data analysis techniques. Significant attention was given to this problem through the 2000's, culminating in a series of Mock LISA Data Challenges (MLDCs) <cit.> that produced some promising proof-of-principle solutions. The demise of the original LISA project in 2010 halted this effort, but work is now resuming following the re-birth of the mission. In addition to finding an implementable solution to the global analysis problem, there is also interest in producing reliable estimates for the science that can be achieved, including the number of sources of each type that can be resolved, and how well they can be characterized. A key input to these studies are estimates for the confusion noise from unresolved sources, as this adds to the instrument noise, and reduces the signal-to-noise ratio (SNR) of the resolved systems. These confusion noise estimates use variants of the idealized iterative source subtraction scheme introduced by Timpano et al <cit.>. We recently applied this technique to produce confusion noise estimates <cit.> that were used in the design study for the new LISA mission concept <cit.>. There are, however, several deficiencies with the simple confusion noise estimates: it assume that the confusion noise is stationary when in fact it oscillates with a 6 month period; it neglects the parameter estimation errors for the subtracted signals and the waveform residuals; and it also neglects the impact the removal has on other resolvable signals such as massive black holes. Here we generalize the Timpano et al <cit.> method to account for the waveform residuals and the impact they have on the detection of other signals. Some of our results were derived previously by Cutler and Harms <cit.> in studies of foreground subtraction for the Big Bang Observer, but many results are new, including analytic estimates for power spectrum of the waveform residuals, incorporating the process of noise modeling, and the impact on parameter estimation for other sources. We find that the parameter estimation errors caused by other resolved signals are typically small compared to those due to instrument noise and unresolved signals. The exception to this rule is when two signals have very high overlap, such as sometimes occurs for galactic binaries with near identical orbital periods and sky locations <cit.>.Electromagnetic observations have identified ∼50 galactic binaries with orbital periods that put their predicted gravitational signals in the LISA band <cit.>. Those that rise above the noise are refer to as “verification binaries”. Population synthesis models predict that there are far more detectable systems waiting to be discovered, though the estimates have been lowered in the past decade as on-going surveys have been used to re-calibrate the models <cit.>. It is estimated that there are hundreds of millions of galactic binaries GBs emitting gravitational waves in our galaxy. In the mid-band of the LISA sensitivity, between ∼ 0.5-3 mHz, gravitational waves from these systems are expected to dominate over instrument noise, with the unresolved component producing what is termed “confusion noise”. There have been several previous attempts at estimating the galactic confusion noise <cit.>. To characterize the confusion noise one must first determine how many galactic binaries are resolvable, but in order to figure out which binaries are resolvable, one must already have an estimate for the noise. The ideal solution is to perform a global fit, e.g. a full Bayesian analysis that fits both resolvable sources and noise at the same time as done by Littenberg <cit.>. Unfortunately, this procedure is extremely computationally intensive, and more efficient techniques are needed if we want to consider a range of population models and detector configurations for design studies. To this end, Timpano et al <cit.> developed an iterative subtraction scheme which starts with a simulated data set comprised of an instrument noise realization and the superposition of all gravitational waves produced by synthetic population of galactic binaries. The signal-to-noise ratio (SNR) of the GBs is calculated, and those above a specified threshold SNR are subtracted perfectly i.e. the true waveform is removed from the data stream using the simulated signal parameters. The noise estimate is updated after the bright signals are removed, and the SNRs of the remaining sources are re-computed. Those above the detection threshold are removed, and the whole process is repeated. It typically takes 5-6 iterations for the solution to converge. It is this assumption of perfect signal recovery we wish to address in this paper. In reality the instrument plus confusion noise realization will randomly perturb the estimated parameters for the resolvable systems, resulting in an inaccurate signal recovery. Here we use the Maximum-Likelihood approximation and Fisher information matrix to estimate the parameter errors and waveform residuals. The outline of this paper is as follows: In Section <ref> we briefly review the galactic population model used to produce realizations of the LISA data used in the analysis.Next, in Section <ref> we provide a review of relevant Maximum-Likelihood methods and how they can be used to estimation the noise-induced errors in signal recovery and parameter estimation of the resolved GBs. In Section <ref> we extend the usual ML analysis to include noise spectral estimation, and in Section <ref> we illustrate the relevance of the ML to a full Bayesian analysis using a simple model of a sinusoid in stationary Gaussian noise. In Section <ref> we apply the Maximum-Likelihood approach to the global fitting of multiple signals and drive expressions for how the interaction between the signals impacts waveform and parameter estimation errors. We conclude in Section <ref> by computing an improved estimate for the galactic confusion noise that takes into account parameter estimation errors in the bright source removal.§ INSTRUMENT AND GALACTIC POPULATION MODELS Our galaxy simulations use realizations of the the Toonen et al <cit.> population model provided by Valeriya Korol and Gijs Nelemans . The space density of interacting white dwarf binaries is reduced by a factor of ten relative to earlier models in response to the findings of recent observational studies<cit.>. The population has∼ 26 million systems with gravitational wave frequencies above 0.1 mHz. The signals from these systems are simulated using an improved version of the fast waveform generation algorithm of Cornish and Littenberg <cit.>.The improved algorithm removes the need to sum over terms in the Fourier convolution by referencing the carrier frequency f_0 to the nearest integer multiple of the sample frequency, such that f_0 = m/T_ obs +δ f, and absorbing the factor of e^2 π i δ f t into the slowly varying part of the signal. This removes the need for the sum in equation (A24) of Ref. <cit.>, and significantly speeds up the waveform generation. In our analysis we use the full set of first-generation time-delay interferometry (TDI) <cit.> variables X̃(f), Ỹ(f), Z̃(f) given in Ref. <cit.>, but when displaying results we show the more familiar Michelson-equivalent signals. To obtain the Michelson equivalent sensitivity we make use of the relation S_X = 4 sin^2(f/f_*)S_M where S_X is the noise as seen in the TDI X data channel and S_M is the equivalent Michelson noise and f_*=c/(2π L) is the transfer frequency. Our instrument noise model assumes white position noise S_p(f) and colored acceleration noise S_a(f) with spectral densitiesS_p(f)= 8.9 × 10^-23 m^2 Hz^-1S_a(f)= 9× 10^-30[1+(10^-4 Hz/f)^2+16(2×10^-5 Hz/f)^10] m^2 s^-4 Hz^-1.Under the assumption that the noise levels are the same in each link, we can form the noise-orthogonal {A,E,T}channels <cit.>A= 1/3(2 X - Y - Z) E= 1/√(3)(Z-Y) T= 1/3(X+Y+Z) .Below the transfer frequency f_*, where most signals are found, the T channel is far less sensitive to gravitational waves, and does not contribute to our analysis.Figure <ref> shows a realization of instrument noise (assumed to be Gaussian and stationary), combined with the signal from all of the relevant GBs in the population synthesis realization plotted as the Michelson-equivalent sensitivity. We see that the galactic foreground rises above the instrument noise across the frequency range 0.3-20 mHz. § PARAMETER ESTIMATION AND WAVEFORM ERRORS In the high SNR regime, the likelihood is strongly peaked about the true model parameters, which allows for a Maximum-Likelihood analysis. Many useful results can be derived from a Taylor expansion of the likelihood about the true parameters. Here we provide a brief review of the Maximum-Likelihood (ML) approximation, deriving results for the parameter estimation errors and waveform residuals.We follow with a discussion of how the ML analysis relates to a global Bayesian analysis. §.§ Maximum-Likelihood reviewConsider the simple case of data s comprised of a gravitational wave signal h_T = h(λ⃗_T) and stationary, Gaussian noise n. The likelihood of observing s given the presence of a gravitational wave signal h_T is thenp(s|h) = e^- G/2 e^-(s-h(λ⃗)|s-h(λ⃗))/2 = e^- G/2 e^-χ^2 /2 ,where(g|k)2∑_I={A,E}∫_0^∞g̃_I(f)k̃_I^*(f) + g̃_I^*(f)k̃_I(f)/S_n,I(f)df,defines the noise-weighted inner product taken across all independent data channels, and the one-sided noise spectral density in channel I is given by the expectation valueE [ñ_I(f) ñ_J^*(f^')] = 1/2δ(f-f^')S_n,I(f)δ_IJ .The noise S_n,I(f) will include instrument noise and unresolved gravitational wave signals. The normalization factor G is given byG = ∑_I={A,E}∫_0^∞ T log[π T S_n,I(f)] df .The traditional derivation of the maximum likelihood solution assumes that the noise model S_n,I(f) is known, and that the signal model, h(λ⃗) is close to the true signal h_T = h(λ⃗_T). The signal model is then Taylor expanded about the true parameters:h(λ⃗) = h_T+∂_ih_TΔλ^i + 𝒪(Δλ^2),where Δλ⃗ =λ⃗ -λ⃗_T. The chi-squared in the likelihood can then be expanded asχ^2 = (s-h|s-h) = (n|n) - 2(n|∂_ih_T)Δλ^i+(∂_ih_T| ∂_jh_T) Δλ^i Δλ^j +O(Δλ^3) .The maximum likelihood solution is found by setting ∂_i χ^2 = 0, which yieldsΔλ^j = (n|∂_ih_T)(Γ^-1)^ij + …where Γ_ij = (∂_ih_T| ∂_jh_T)is the Fisher information matrix. Using the identity E [ (n|g)(n|k)] = (g|k) we find that the error covariance matrix is given to leading order in the signal-to-noise ratio by the inverse of the Fisher information matrix:C^ij = E [Δλ^iΔλ^j] = (Γ^-1)^ij + 𝒪( SNR)^-1 ,where the SNR ρ is given by ρ^2(h) = (h|h). See Vallisneri <cit.> for a more in depth presentation that discusses some of the potential pitfalls in using the Fisher Information matrix approximationparameter error estimation. It is important to note that there are higher order corrections to the signal parameters and covariance matrix which appear in Cutler and Flanagan <cit.> as equations (A31) and (A35).§.§ Signal residuals We can use the maximum likelihood approximation to study noise induced errors in the parameter recovery and signal subtraction for galactic binaries. A closely related analysis was performed by Cutler and Harms <cit.> in the context of subtracting the signals from neutron stars to allow for the detection of a primordial stochastic background for the Big Bang Observer mission concept. We extend their analysis to include noise modeling, and derive new expressions for the impact the foreground removal has on parameter estimation for other sources such as massive black hole mergers and extreme mass ratio insprials (EMRIs).The noise-induced parameter estimation errors (<ref>) result in waveform errorsΔh = h_T - h≃- ∂_i h_T Δλ^i +…An example of the observed signal from a galactic binary and the noise-induced subtraction residual is shown in Figure <ref>.Note that the residual is below the reference noise level as the waveform error is down-weighted by the level of overlap between noise and parameter derivatives of the signal.The waveform error has zero mean, E [ Δh ] = 0, and varianceE [ρ_Δ h^2] E [(Δh|Δh)] = (∂_ih_T| ∂_jh_T) E [Δλ^iΔλ^j] ≈Γ_ij(Γ^-1)^ij= D .In the final step we have assumed that the error covariance matrix is approximated by the inverse of the Fisher matrix. The SNR of the residual depends only on the parameter dimension in the signal model and not upon the strength of the signal. Each term in the sum for Δ h is random walk induced by the noise realization, i.e. Δ h ∼ n √(D) as there are D terms. This means that |Δh̃|^2∼ (n^*n)D and the inner product is weighted by the RMS noise resulting in a dependence only on model dimension. It can be shown that the variance of ρ_Δ h^2 is 2D and the skew is 1/D^2. The expectation value of the chi-squared can be written asE[χ^2] =E[(n+Δh|n+Δh)] = N + E [ρ_Δ h^2] + 2E[(Δh|n)] = N - D,where the last step follows from E [(Δh|n)] = -E [(h_,i|n)(Γ^-1)^ij(n|h_,j)]= -D,and N is the number of data samples. We see that the signal residuals are anti-correlated with the noise, which results in a reduction in the chi-squared. Part of the noise gets absorbed by the signal model, which will ultimately result in a lowering of the confusion noise estimate relative to that found assuming perfect signal subtraction. Note that power spectrum of the residual s-h has expectation valueE[(s(f)-h(f))(s(f')-h(f'))^* ]= 1/2S_n,0(f)δ(f-f')-∂_ih̃_T(f)∂_jh̃^*_T(f') (Γ_0^-1)^ijδ(f-f')=1/2(S_n,0(f) - S_Δh(f)) δ(f-f'),where1/2δ(f-f^')S_Δh(f) =∂_ih̃_T(f)∂_jh̃^*_T(f') (Γ_0^-1)^ijδ(f-f') = E[Δh̃(f)Δh̃^*(f')]is the power spectral density of Δh. Note that we made use of E[ñ(f) (n|∂_ih_T)] = ∂_ih̃_T(f). Our waveform model for a galactic binary has D=9 parameters: the sky location (θ,ϕ); inclination and polarization angles (ι,ψ); amplitude A; reference phase ϕ_0 and reference gravitational wave frequency f_0; and first and second frequency derivatives ḟ_0 and f̈_0. For most galactic binaries the frequency derivatives are poorly constrained and the effective model dimension is closer to D=7. The relevant quantity for estimating which systems have detectable frequency evolution are the number of frequency bins of evolution, α = ḟ_0 T_ obs^2 and β = f̈_0 T_ obs^3, and the SNR. Roughly speaking, frequency evolution through ∼ (7/ SNR) bins is detectable <cit.>, and to leading post-Newtonian order we have α= 1.5 ( f_0/4 mHz)^11/3(M/ 0.25 M_⊙)^5/3(T_ obs/4 yrs)^2 β= 1.8 ( f_0/25 mHz)^19/3(M/ 0.25 M_⊙)^10/3(T_ obs/4 yrs)^3.The chirp mass M has been scaled to the mode of the population distribution <cit.>. From these expressions we see that only the loudest, most massive and highest-frequency systems will have a measurable second frequency derivative, and that most systems below 3 mHz will show no measurable frequency evolution at all. Including poorly constrained parameters in the model can lead to ill-conditioned Fisher matrices with badly behaved inverses. One solution is to reduce the model dimension by eliminating parameter λ_i whenever the inner product (∂_ih_T| ∂_ih_T) drops below some threshold. An alternative solution is to replace the Fisher matrix Γ with a matrix formed from the augmented Fisher matrix, K, which includes derivatives of the priors (see Section <ref> for details). We adopt the latter approach and include Gaussian priors on α,β centered on zero with width σ = 10. The priors condition the matrix and when the parameters are poorly constrained by the data, have the effect of reducing the model dimension. Using the more stable approximation E [Δλ^iΔλ^j] ≈(K^-1)^ij, yields E [ρ_Δ h^2] ≈ D_ eff, where D_ eff is the effective dimension of the model, defined by the number of parameters that have posterior distributions that differ measurably from their priors (a notion that can be made precise using the Kullback-Leibler divergence).Figure <ref> shows a histogram of square SNRs of the waveform residuals for the galactic binaries that are deemed detectable by the iterative subtraction scheme discussed in Section <ref>. The average value for there residual SNR^2 of 7.6 is less than the full model dimension D=9, and consistent with our estimate for the effective dimension.§ MAXIMUM-LIKELIHOOD APPROXIMATION WITH NOISE ESTIMATION The standard treatment of the maximum likelihood expansion assumes that the noise spectrum is known. If the detectable gravitational wave signals are infrequent and short-lived, as is currently the case for compact binary mergers in LIGO, then “off-source” data from times where no detectable signals are present can be used to estimate the noise spectrum. These estimates will include instrument noise and unresolved gravitational wave signals. The option of making off-source estimates will not be available for LISA, and the noise spectrum will have to be inferred along with the signal model. Our derivation we will assume that we have a parameterized model for the noise, such as the cubic spline model used by the BayesLine algorithm <cit.>.To get an understanding for how noise modeling impacts the maximum likelihood calculation, consider a simple example with zero mean, additive, white Gaussian noise and N data samples with likelihoodp(s|h) = ∏_k=1^N 1/√(2 πσ^2) e^-(s_k - h_k)^2/(2 σ^2).The un-perturbed (h = 0) noise level is given by sample varianceσ_0^2 = 1/N∑_k=1^N n_k^2 .We could expand σ^2 about the theoretical variance, σ_*^2, but it is simpler to expand σ^2 about the sample variance: σ^2 = σ_0^2 + Δκ so that Δκ = 0 when h = 0. The typical difference between the sample variance and the theoretical variance will be by an amount that scales as the standard deviation of the sample variance, Δσ_0^2 = √(2)σ_*^2/√(N). The log likelihood can be expanded:log p(s|h)= +(1 - Δκ/σ_0^2)((n|∂_ih_T)_0 Δλ^i-1/2(∂_ih_T| ∂_jh_T)_0 Δλ^i Δλ^j)-N/2( 1+Δκ^2/2 σ_0^4) + …,where the notation (a| b)_0 denotes that the noise weighted inner product is taken with respect to the un-perturbed noise level σ_0^2. Setting ∂_Δκlog p(s|h) = 0 and∂_Δλ^klog p(s|h) = 0, yields the ML solutionΔλ^j=(n|∂_ih_T)_0(Γ_0^-1)^ij Δκ =- σ_0^2/N (n|∂_ih_T)_0 (n|∂_jh_T)_0 (Γ_0^-1)^ij.We see that the leading order ML solution for the signal parameters is unchanged from the fixed noise case. The updated noise estimate σ^2_ ML = σ^2_0+Δκ is lowered relative to its true value, as can be seen by taking the expectation valueE [ σ^2] =σ^2_0( 1 - D/N) .While the ML waveform removes some of the noise, this is now accounted for in the ML estimate for the noise, such that the expected value of the chi-squared is again just the dimension of the data: E [χ^2 ] = N. From expanding the likelihood around the signal and noise parameters η⃗ ={λ⃗,κ} and maximizing the likelihood obtains the form p(s|Δη⃗) = 1/√(2πΓ^-1)e^-1/2Γ_μνΔη^μΔη^νwhere Γ_μν =-∂_μ∂_νlog p(s|h)|_. Note that we are using Greek indicies to denote the entire collection of parameters. One can read off the Fisher matrix from the log-likelihood: Γ = ( [ Γ_0,ij (n|∂_jh_T)_0/σ_0^2; (n|∂_ih_T)_0/σ_0^22 σ_0^4/N ]) ,where Γ_0,ij is the Fisher matrix obtained from the signal-only ML analysis discussed in the previous section. The full Fisher matrix can be inverted by recognizing that the off-diagonal terms (n|∂_jh_T)_0/σ_0^2 are smallcompared to the block diagonal terms.We find that the variances for the signal parameters are inflated: (Γ^-1)^ii≈(Γ^-1_0)^ii + 2/N(Γ^-1_0)^il(Γ^-1_0)^ki(n|∂_lh_T)_0(n|∂_kh_T)_0 .On average (Γ^-1)^ij→(Γ^-1_0)^ij(1+2/N) where in the limit of large N we obtain the original Fisher matrix. The signal model parameter variances are inflated by covariances with the noise model parameters as they both attempt to capture pieces of the signal. Note that covariance of the parameter shiftsΔλ^j and Δκ from their true values, as computed in (<ref>), does not equal the inverse of the Fisher matrix,E[Δη^νΔη^μ] ≠ (Γ^-1)^μν. This is because the noise modeling changes the shape of the likelihood, and not just the location of the peak. However, we see from (<ref>) that Γ^-1 does indeed describe the parameter covariances. The ML expansion for a colored noise model is considerably more involved, and we relegate the details to Appendix A. To keep the notation simple we suppress the sum over data channels. Introducing the parameterized noise modelS_n(f;θ⃗) = S_n,0(f) + Δθ^a∂_a S_n(f)+ 1/2Δθ^aΔθ^b∂_a ∂_b S_n(f) +…,(noise model derivatives are evaluated at the ML values after differentiation) where S_n,0(f) is some smooth estimate of the instrument noise and unresolved signals (assuming perfect subtraction of the resolvable signals), we find theleading-order solution for the signal parameters has the same form as in (<ref>), while the noise model parameters are given byΔθ^j≈ [ ∫S_n,iS_n,j/S_n,0^2(T S_n,0- 4 ñ^*ñ/S_n,0)df - ∫S_n,ij/S_n,0(T S_n,0- 2 ñ^*ñ/S_n,0)df ]^-1×[∫S_n,i/S_n,0(T S_n,0- 2 ñ^*ñ/S_n,0)df + 2(n|∂_ah_T)_i(n|∂_bh_T)_0(Γ_0^-1)^ab. . - (∂_ah_T|∂_bh_T)_i(n|∂_ch_T)_0(n|∂_dh_T)_0(Γ_0^-1)^ac(Γ_0^-1)^bd] .The notation (x|y)_a defines the inner product(x|y)_a=4 ∫ ( x̃ỹ^* + x̃^* ỹ) / S_n,0∂_a S_n/ S_n,0 df .The integrals with the factor (T S_n,0- 2 ñ^*ñ)/S_n,0 accounts for the difference between the theoretical noise model and fluctuation seen in a particular noise realization,i.e. the difference between σ_*^2 and σ_0^2 in the white noise toy model, as evidenced by its expectation value vanishing. Neglecting this difference and considering the expectation value of Δθ^j we obtain the simplification E[Δθ^j] ≈ - [ ∫ TS_n,iS_n,j/S_n,0^2df ]^-1 (∂_ah_T|∂_bh_T)_i(Γ_0^-1)^ab .The white-noise case (<ref>) can be recovered by setting S_n,0= 2/Tσ_0^2 and ∂_a S_n(f) = 2/T, so that [ ∫ TS_n,iS_n,j/S_n,0^2df ]^-1 = σ_0^4/N, and (x|y)_a= (x|y)_0/σ_0^2. We can nowcompute the expectation value of the noise perturbation Δ S_n(f) =Δθ^a∂_a S_n(f) +…E[ Δ S_n(f) ] =-[ ∫ TS_n,aS_n,b/S_n,0^2df ]^-1(Γ_0^-1)^ij(∂_ih_T |∂_jh_T)_a ∂_b S_n(f) .Note that the perturbation to the noise model is negative, as it must be given that the signal model absorbs some of the noise.One would expect that Δ S_n(f)should be a smoothed representation of -S_Δ h(f) for an effective noise model, mopping up errors introduced by the signal ML.Similar to the white noise model above we may obtain the signal model variances for a general noise model by making the same appeals to neglecting differences between the theoretical and sample variance and averaging over many noise realizations (Γ^-1)^ii≈(Γ_0^-1)^ii + 2 (Γ_0^-1)^im(Γ_0^-1)^ni (∂_mh_T|∂_nh_T)_ab×(∫_0^∞TS_n,aS_n,b/S_n,0^2df )^-1.An effective noise model would minimize the factors (∂_mh_T|∂_nh_T)_ab such that (Γ^-1)^ii→(Γ_0^-1)^ii. We can turn this into a more useful expression by taking advantage of the compact (in the frequency domain) nature of the GB signal and assume the noise PSD is roughly constant (Γ^-1)^ii≈(Γ_0^-1)^ii(1+2/TΔ f) ,where Δ f is the bandwidth of the signal such that TΔ f is the number of frequency bins the GB spans. For sources that span many frequency bins such that the noise PSD cannot be assumed to be constant this serves as an upper limit for the increase in the parameter errors. Note that other terms exist when considering covariances of the signal model. Again, we see that when the source occupies a large bandwidth we recover the variances for when the noise is known. For a 3 mHz source that experiences a Δ f ≈ 0.6 × 10^-6 Hz Doppler shift spreading due to LISA's orbital motion, the parameter variances will be inflated by 10% for a one year observation span, dropping to 3% after four years.§ RELATING BAYESIAN INFERENCE AND FREQUENTIST MAXIMUM LIKELIHOOD ESTIMATION The LISA data will include overlapping signals from an unknown number of sources of different types. Bayesian inference provides a powerful and flexible framework for inferring the number and properties of the resolvable sources. In addition to the ontological differences between the Bayesian and Frequentist approach to statistical inference - Bayesian inference considers the data to be known and the signal parameters to be uncertain while Frequentist inference considers thesignal parameters to be fixed and the data to be uncertain - Bayesian inference typically integrates over uncertainty (marginalization), while Frequentist analysis employs maximization. Despite these differences, the maximum likelihood analysis we have described can be used to estimate results from Bayesian inference by way of a Taylor expansion of the posterior distribution p(λ⃗|s) = p( s|λ⃗) p(λ⃗) /p( s). Expanding about the mode of the posterior distribution (also known as themaximum a posteriori probability (MAP) estimate, λ⃗_ MAP), we have the quadratic approximationp(λ⃗|s) ≃ (2π)^-D/2 ( det K)^1/2 e^-1/2 K_ijΔλ^i Δλ^j,where Δλ⃗ =λ⃗ -λ⃗_ MAP andK_ij= -∂_i∂_j log(p( s|λ⃗) p(λ⃗)) |_ MAP.When the likelihood is more informative than the prior,λ⃗_ MAP≈λ⃗_ ML and K is well approximated by the Fisher information matrix Γ, though even small contribution from the derivatives of the log prior can have a an important stabilizing effect on K.To illustrate the relationship between the maximum likelihood analysis and Bayesian inference we produced simulated data consisting of stationary, Gaussian white noise with variance σ^2 and a sinusoidal signal h(A,f_0, t_0,ϕ_0) = A cos( 2 π f_0 (t-t_0)+ϕ_0).We held the phase parameter ϕ_0 = π fixed in the analysis as otherwise there is a near perfect degeneracy between the time offset t_0 and the phase offset ϕ_0, which significantly complicates the analysis. The simulated data consisted of N= 10^4 evenly spaced samples spanning T= 100 seconds, with A=√(2), f_0=0.25 Hz, t_0=1 second and σ^2 = 100. The noise level was set to yield a signal-to-noise ratio of SNR = 10.A plot of the simulated data and signal are shown in the upper panel of Figure <ref>.For the Bayesian analysis we assumed uniform priors on the signal and noise model parameters (A,f_0, t_0, σ^2) across a range that was much wider than the predicted statistical errors so that posterior distribution and the likelihood were effectively identical. A Markov Chain Monte Carlo (MCMC) simulation was used to compute the mean and variance of the signal parameters and the waveform error, while (<ref>) and (<ref>) were used to estimate the parameter shifts at maximum likelihood and the variances. The MCMC and ML derived values for the parameter shifts and standard deviations are listed in Table <ref> for a particular noise realization. The marginalized posterior distributions for the parameters are compared to the predictions of the Gaussian approximation (<ref>) in Figure <ref>. The agreement between the ML and MCMC seen in this example was typical of what we found when repeating the analysis for different noise realizations. That is not to say that the Gaussian approximation will be this accurate in more realistic settings where the noise is more complicated and the parameters are highly correlated <cit.>, but it does provide useful order-of-magnitude estimates in most situations.The displacement of the parameters from their true values is related to the waveform error Δ h shown in the lower panel of Figure <ref> which displays the ML, MAP and mean waveform errors. Note that the mean waveform error is found by averaging the waveform errors, and not by using the mean parameter values to compute a waveform. The good agreement between the maximum likelihood and mean waveform residual hides a key difference between the frequentist and Bayesian analyses: in the Bayesian global fit the waveform uncertainties are marginalized over, while the frequentist analysis uses point estimates. Rather than subtracting a particular point estimate for each signal from the data, the Bayesian approach subtracts a range of estimates for each signal such that the residual is consistent with the noise model. This procedure is illustrated in Figure <ref> for the sinusoid signal model, where the waveform residuals from each iteration of the MCMC analysis are used to produce a histogram of the residual at each time sample. Also shown in Figure <ref> is the MAP point estimate for the waveform residual. Notice that the full posterior distribution for the waveform residuals has significant spread about the point estimate.§ MULTIPLE SOURCES The LISA data will contain many signals that partially overlap in both time and frequency. Extracting information about these signals necessitates finding a global solution for all signals that can be resolved - that is, signals that are both sufficiently loud and sufficiently distinct to be individually identified. The full LISA data stream can be written as s = H + n,where H =∑h_T denotes the sum total of all gravitational wave signals, which can be further separated into a resolved, H_R, and un-resolved H_U, component. The unresolved component is often referred to as “confusion noise”, the largest component of which is expected to come from white dwarf binariesin our galaxy. The ML analysis for single sources can be applied to multiple sources by replacing h with H_R, and by replacing n byn+H_U. The resolved signals will include a large number of bright galactic binaries, along with multiple supermassive black holes and EMRIs. The parameter estimation for the resolved systems will be impacted by the unresolved signals, which add to the effective noise level, and by the other resolved signals due to signal overlap. To simplify the discussion imagine that the resolved signals consist of bright galactic binaries H_G and a single massive black hole binary h̅. The parameter vector λ⃗ runs over the full set of galactic binary parameters (denoted by indices early in the alphabet, λ^a,λ^b… etc) and the black hole parameters (denoted by indices later in the alphabet λ^i,λ^j… etc). The full set of signal parameters for the galactic sources and the black hole are indicated by Greek indices. The Fisher information matrix for the combined solution, Γ_αβ can be broken into a block diagonal part B_αβ formed from a galactic-binary block, G_a b=(∂_aH_G | ∂_bH_G), and a black hole block B̅_ij=(∂_ih̅ | ∂_jh̅), and a mixed block M_aj= (∂_aH_G | ∂_jh̅). The waveform error for the black hole signal is thenΔh̅ = -∂_ih̅(n|∂_αH_R)(Γ^-1)^i α + … .The expectation value for the squared SNR of the black hole waveform residual is thenE[ (Δh̅|Δh̅) ] = B̅_ij(Γ^-1)^ij + … ,Assuming the mixture terms M_aj are small compared to the terms on the diagonal, the inverse of the full Fisher matrix can be expanded as(Γ^-1)^αβ =(B^-1)^αβ- (B^-1)^αμ M_μν(B^-1)^νβ+(B^-1)^αμM_μν(B^-1)^νγM_γη(B^-1)^ηβ + … .The black-hole block of the inverse, (Γ^-1)^ij, lacks the second term since (B^-1)^i μ M_μν(B^-1)^ν j = (B̅^-1)^i k M_k l(B̅^-1)^l j = 0. Therefore we haveE[ (Δh̅|Δh̅) ]=B̅_ij((B̅^-1)^ij + (B̅^-1)^ikM_k a(G^-1)^ab M_a l( B̅^-1)^lj) + … =D̅ + (B̅^-1)^ij(G^-1)^abM_j aM_b i + … .where D̅ is the dimension of the black hole model. We see that the black hole waveform residuals are inflated from the isolated source result by the mixture term(B̅^-1)^ij(G^-1)^abM_j aM_b i = E[ Δλ^iΔλ^j]E[ Δλ^aΔλ^b] (∂_jh̅|∂_aH_G)(∂_bH_G|∂_ih̅) ≈E[ Δλ^iΔλ^jΔλ^aΔλ^b](∂_jh̅|∂_aH_G)(∂_bH_G|∂_ih̅) =E [(Δh̅|ΔH_G)^2].The second line is obtained using Isserilis's theorem and dropping the cross terms E [ Δλ^iΔλ^a ] which would produce higher order corrections. Using E[ ΔH̃_G(f)ΔH̃^*_G(f')]= 1/2 S_Δ H_G(f) δ(f-f') we haveE [(Δh̅|ΔH_G)^2] = 4 ∫^∞_0 |Δh̅(f)|^2S_Δ H_G(f) /S^2_n,0(f) df .If we were to switch the roles of the black hole and the resolved galactic binaries we would find the the squared SNR of the galactic residuals were inflated by exactly the same amount:E[ (ΔH_G|ΔH_G) ] = D_G + E [(Δh̅|ΔH_G)^2] .On the other hand, the squared SNR of the full residual is equal to the total parameter dimension:E[ (ΔH_R|ΔH_R) ] = Γ_αβ(Γ^-1)^αβ = D̅ + D_G.These results can be reconciled by noting thatE[ (ΔH_R|ΔH_R) ] = E[ (ΔH_G|ΔH_G) ] + E[ (Δh̅|Δh̅) ] + 2 E[ ( Δh̅ | ΔH_G) ],and usingE[ ( Δh̅ | ΔH_G) ] = M_iaΓ_αβ(Γ^-1)^iα(Γ^-1)^aβ=-M_ia M_jb(B̅^-1)^ij(G^-1)^ab≈ - E [(Δh̅|ΔH_G)^2] .Thus we see that the extra residual for each source class is canceled by the cross-correlation of the residuals between the source classes. Note that the results for the SNR of the signal residuals are unchanged to the order we are considering when using the full noise model or the unperturbed noise model.Next we consider the impact on the black hole parameter estimation errors caused by fitting the bright galactic binaries. The variance in the parameter estimation errors can be estimated from the diagonal entries of inverse of the full Fisher information matrix(Γ^-1)^ii = (B̅^-1)^i i +(B̅^-1)^i kM_k a(G^-1)^a bM_b n(B̅^-1)^n i + … .The second term in the expansion comes from correlations between the black hole signal and the resolved galactic binaries as is positive definite since x^TGx≥ 0 for a positive-definite matrix. Thus, the simultaneous fitting of the galactic binary signals and the black hole signal tends to inflate the parameter estimation errors. Expanding to leading order the second term is given byB̅^i kM_k aG^a bM_b nB̅^n i = E[Δλ^aΔλ^b(∂_kh̅|∂_aH_R)_0 (∂_bH_R|∂_nh̅)_0](B̅_0^-1)^i k(B̅_0^-1)^n i, = E[ (h_,k|ΔH_G )_0 (ΔH_G|h_,n)_0] (B̅_0^-1)^i k(B̅_0^-1)^n i =4 (B̅_0^-1)^i k(B̅_0^-1)^n i(R∫_0^∞∂_kh̅^*∂_nh̅S_Δ H_G/S_n,0^2(f)df) ,where ∂_aH_R = ∂_aH_G as the derivatives are with respect to GB parameters. Using an estimate to the waveform errors of the resolved sources we can express this result in a more useful form: 4 (R∫_0^∞∂_kh̅^*∂_nh̅S_Δ H_G/S_n,0^2(f)df)≈2 D_(R∫_0^∞∂_kh̅^*∂_nh̅/S_n,0(f)dN/db df) ≤ 2 D_(R∫_0^∞∂_kh̅^*∂_nh̅/S_n,0(f) df)(dN/db)_.This implies that the covariance matrix inflates with the following upper bound: (Γ^-1)^ii≤B̅_0^ii( 1+2 D_(dN/db)_) ,where dN/db of sources resolved per frequency bin. In the next section we will obtain an estimate for the noise due to GB waveform errors which will allow us to obtain a more useful expression for this overlap term. A similar inflation of GB parameter variances results from the overlap with the BH signal. § GALACTIC CONFUSION NOISETo obtain an estimate of the Galactic confusion foreground we employ an iterative subtraction scheme. Previously, this scheme was performed with perfect removal of source waveform <cit.>, but clearly noise will lead to errors in the parameters estimation and signal subtraction, and a reduction of the confusion noise estimate.The revised procedure is as follows: A simulated data set is produced that includes a realization of the instrument noise and the sum of the strains due to the galactic binaries H from the galactic population model. A smooth fit to the power spectral density of the instrument noise and the signals is used as an initial estimate for the noise in each data channel. Next we identify sources which are loud (SNR > 7) relative to this noise estimate and subtract the best-fit waveform h(λ⃗)_ best fit = h_T-∂_ih_TΔλ^i from the data. A smooth fit to the power spectral density of the remaining signals and noise is computed, and signals above the SNR threshold for the updated noise estimate are identified and subtracted. As can be seen in Figure <ref>, the subtraction procedure quickly converges.The number of sources which can be resolved converges after just 5 or 6 iterations. The number density of sources, measured in terms of the number per a frequency bin, dN/db, is shown in Figure <ref> for a single channel and dual channel analysis. We see that more sources can be resolved when multiple data channels are used in the analysis. Attempts have been made to deal with the identification and subtraction of signals which overlap <cit.> and of how many sources per frequency band <cit.> can be resolved. Here we see that with two data channels and a 4-year mission the peak density is roughly one source resolved per ten frequency bins. In our simulations we made the simplifying assumption that the augment Fisher matrix for the galactic population is block diagonal. That is, we ignored correlations between galactic signals. This approximation is reasonable when dN/db is small, but may be questionable in the highest density regions and for the occasional systems that happen to have high overlap. We will re-visit this complication in a future study, as the parameter estimation errors grow significantly for highly overlapping systems <cit.>.Figure <ref> compares the Michelson-equivalent strain power spectral densities for the imperfect and perfect subtraction scheme. The dashed lines show galactic confusionnoise and the solid lines show the combined instrument and confusion noise.Note that the differences between the PSD arise where the most sources are resolved as one would expect (see Figure <ref>). In the dual A, E channel the PSD is lower as indicated by the noise levels specified by the reference frequency 2 mHz.We can estimate the power spectral density of the combined waveform residual,S_Δ H_R(f) = T/2E[ |ΔH̃_R(f)|^2] by applying (<ref>) to the full compliment of N resolved binaries:E[ρ_Δ H_R^2] = N D_ eff = 4 ∫_0^∞E[|ΔH̃_R(f)|^2]/S_n,0(f)df,Considering the contribution in a small frequency range Δ f centered at f we findS_Δ H_R(f) = D_ eff/2dN/db S_n,0(f).Figure <ref> compares this analytic estimate to the numerical value found from the iterative subtraction scheme. The prediction lines up quite well with a small deviation at the frequencies where the number of resolved sources per bin peaks. § DISCUSSION We have used the maximum likelihood approximation to derive a number of analytic results pertaining to the LISA global analysis problem. A simple toy model was used to demonstrate the relevance of these estimates to a full Bayesian analysis, though we cannot guarantee that the approximations will be as reliable when applied to LISA data analysis. We extended the standard maximum likelihood analysis to include noise modeling, and found that the estimated noise level is lowered whenever signals are subtracted from the data. We applied our general results to the simultaneous fitting of a black hole binary and a collection of galactic binaries and found that the errors in the black hole waveform recovery are increased, as wells as the variances of the source parameters. It is important to note that it is not the overlap of the signals which cause the inflation of variances, but rather, it is due to the waveform subtraction errors. We concluded by incorporating parameter estimation errors in the estimation of the galactic confusion noise, and derived a useful expression that can be used to predict the reduction in the confusion noise in terms of the number density of resolved signals. Equations (<ref>) and (<ref>) provide quick estimates for how parameter estimation errors are inflated by noise fitting and source confusion. §.§ AcknowledgmentsWe are grateful for the support provided by NASA grant NNX16AB98G.§ APPENDIX AHere we derive the more general signal plus noise model ML analysis results presented in section 4. Beginning with the log-likelihood obtained from equation (<ref>) log p(s|h(λ⃗),S_n(f;θ⃗)) =-1/2∫_0^∞ T log[π T S_n(f;θ⃗)]df - 1/2(s-h(λ⃗)|s-h(λ⃗))_θ⃗ ,where the noise-weighted inner product is now parameterized by θ⃗ as denoted by ( · | · )_θ⃗. We may expand about the true noise model S_n,0(f) and the true signal model h_T and maximized to obtain estimates ofΔθ⃗ and Δλ⃗. Expanding out the normalization constant and dropping terms which are constant with respect the maximization gives -1/2∫_0^∞log[2π T S_n(f;θ⃗)]df ≈-1/2Δθ^i∫_0^∞T S_n,i/S_n,0df-1/4Δθ^iΔθ^j∫_0^∞T S_n,ij/S_n,0df+1/4Δθ^iΔθ^j∫_0^∞T S_n,iS_n,j/S_n,0^2df ,where T is the observation period. An arbitrary noise-weighted inner product expanded out takes the form (a|b)_θ⃗≈ (a|b)_0 -Δθ^i(a|b)_i - 1/2Δθ^iΔθ^j(a|b)_ij +Δθ^iΔθ^j(a|b)_i;j ,where (a|b)_i4 R∫_0^∞ã^*b̃/S_n,0S_n,i/S_n,0df, (a|b)_ij4 R∫_0^∞ã^*b̃/S_n,0S_n,ij/S_n,0df, (a|b)_i;j4 R∫_0^∞ã^*b̃/S_n,0S_n,iS_n,j/S_n,0^2df.With these pieces in hand the chi-squared piece of the log-likelihood, dropping constants with respect to the maximization, is- 1/2(s-h(λ⃗)|s-h(λ⃗))_θ⃗≈ 1/2Δθ^i(n|n)_i+1/4Δθ^iΔθ^j(n|n)_ij-1/2Δθ^iΔθ^j(n|n)_i;j+Δθ^i(n|Δh)_i+1/2(n|Δh)_0+1/2Δθ^i (Δh|Δh)_i-1/2(Δh|Δh)_0Collecting terms, and maximizing results in the solutionΔλ^j= (n|∂_jh_T)_θ⃗(Γ_θ⃗^-1)^ij Δθ^j≈ [ ∫S_n,iS_n,j/S_n,0^2(T S_n,0- 4 ñ^*ñ/S_n,0)df - ∫S_n,ij/S_n,0(T S_n,0- 2 ñ^*ñ/S_n,0)df ]^-1×[∫S_n,i/S_n,0(T S_n,0- 2 ñ^*ñ/S_n,0)df - 2(n|Δh)_i - (Δh|Δh)_i],where Δh need only be kept to leading order. The Fisher matrix can be obtained similarly to the toy white noise problem presented in section <ref> Γ = ( [ (∂_ih_T|∂_jh_T)_0 + 1/2(n|∂_ijh_T)_0 (n|∂_jh_T)_i; ; (n|∂_ih_T)_j1/2∫S_n,ij/S_n,0( 2 ñ^*ñ-T S_n,0/S_n,0)df;+1/2∫S_n,iS_n,j/S_n,0^2( 4 ñ^*ñ - TS_n,0/S_n,0) df ]) .Inverting the Fisher matrix and considering the signal model variances (Γ^-1)^ii≈ (Γ^-1_0)^ii -1/2(Γ^-1_0)^im (n|∂_mnh_T)_0(Γ^-1_0)^ni+ 2 (Γ^-1_0)^im (n|∂_mh_T)_a[∫_0^∞S_n,ab/S_n,0( 2 ñ^*ñ-T S_n,0/S_n,0)df . . + ∫_0^∞S_n,aS_n,b/S_n,0^2( 4 ñ^*ñ - TS_n,0/S_n,0) df]^-1(n|∂_nh_T)_b(Γ^-1_0)^ni. § REFERENCES | http://arxiv.org/abs/1705.09421v1 | {
"authors": [
"Travis Robson",
"Neil Cornish"
],
"categories": [
"gr-qc"
],
"primary_category": "gr-qc",
"published": "20170526031814",
"title": "Global analysis for the LISA gravitational wave observatory"
} |
http://arxiv.org/abs/1705.09351v2 | {
"authors": [
"A. A. Mamun",
"C. Constantinou",
"M. Prakash"
],
"categories": [
"nucl-th"
],
"primary_category": "nucl-th",
"published": "20170525202410",
"title": "Pairing properties from random distributions of single-particle energy levels"
} |
|
Bulletin of Mathematical Biology * Algorithmexp instructioncounter Algorithm[1][] Algorithm expAlgorithm #1 Algorithm (Algorithminstructioncounter)instructioncounter Centre for Mathematical Biology, Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK. Division of Biomedical and Life Sciences, Faculty of Health and Medicine, Furness Building, Lancaster University, Bailrigg, Lancaster, LA1 4YG, UK. MRC Human Genetics Unit, MRC IGMM, Western General Hospital, University of Edinburgh, Edinburgh, EH4 2XU, UK. Current Address: Rosalind and Morris Goodman Cancer Research Centre, Department of Human Genetics, McGill University, 1160 Pine Avenue West, Montreal, Quebec, H3A 1A3, Canada.A multi-stage representation of cell proliferation as a Markov process Christian A. Yates1Corresponding author.E-mail: [email protected] www: www.kityates.com Matthew J. Ford3^,4 Richard L. Mort2 Received: date / Revised version: date =========================================================================================================================================== The stochastic simulation algorithm commonly known as Gillespie's algorithm (originally derived for modelling well-mixed systems of chemical reactions) is now used ubiquitously in the modelling of biological processes in which stochastic effects play an important role. In well-mixed scenarios at the sub-cellular level it is often reasonable to assume that times between successive reaction/interaction events are exponentially distributed and can be appropriately modelled as a Markov process and hence simulated by the Gillespie algorithm. However, Gillespie's algorithm is routinely applied to model biological systems for which it was never intended. In particular, processes in which cell proliferation is important (e.g. embryonic development, cancer formation) should not be simulated naively using the Gillespie algorithm since the history-dependent nature of the cell cycle breaks the Markov process. The variance in experimentally measured cell cycle times is far less than in an exponential cell cycle time distribution with the same mean.Here we suggest a method of modelling the cell cycle that restores the memoryless property to the system and is therefore consistent with simulation via the Gillespie algorithm. By breaking the cell cycle into a number of independent exponentially distributed stages we can restore the Markov property at the same time as more accurately approximating the appropriate cell cycle time distributions. The consequences of our revised mathematical model are explored analytically as far as possible. We demonstrate the importance of employing the correct cell cycle time distribution by recapitulating the results from two models incorporating cellular proliferation (one spatial and one non-spatial) and demonstrating that changing the cell cycle time distribution makes quantitative and qualitative differences to the outcome of the models. Our adaptation will allow modellers and experimentalists alike to appropriately represent cellular proliferation - vital to the accurate modelling of many biological processes - whilst still being able to take advantage of the power and efficiency of the popular Gillespie algorithm.A multi-stage representation of cell proliferation as a Markov process Christian A. Yates1Corresponding author.E-mail: [email protected] www: www.kityates.com Matthew J. Ford3^,4 Richard L. Mort2 Received: date / Revised version: date ===========================================================================================================================================§ INTRODUCTION In a well-mixed solution of chemicals undergoing reactions, non-reactive collisions occur far more often than reactive collisions allowing us to neglect the fast dynamics of motion. We can thus assume that the time between reactive collision events is exponentially distributed with rates which are a combinatorial function of the numbers of available reactants <cit.>. This premise is the basis of the <cit.> stochastic simulation algorithm[Although Gillespie was amongst the first to popularise the stochastic simulation algorithm and derived it from physical considerations, its conception dates back to the work of <cit.> and <cit.>. It was derived in a different form by <cit.>.]. The Gillespie algorithm has become a ubiquitous algorithm for the simulation of stochastic systems in the biological sciences, in particular in computational systems biology <cit.>.However, the Gillespie algorithm is often used inappropriately to represent processes for which the inter-event time is not exponentially distributed. One prevalent example of this is in the simulation of the cell cycle <cit.> (see figure <ref>).The assumption of memorylessness, and consequently exponentially-distributed cell cycle times, means that with high probability a daughter cell may divide immediately after the division event which created it. This is not biologically plausible since each cell is required to pass through the G_1, S, G_2 and M phases of the cell cycle before division, and these phases (in particular S-phase) are rate-limiting.As an illustrative example, <cit.> present a well-mixed stochastic model of small populations of cancer stem cells, which they use to suggest treatment strategies. Both analytical and simulated results pertaining to the survival of the tumour are based on the assumption that inter-division times are exponentially distributed.We demonstrate later in this paper that using the correct cell cycle time distributions (CCTDs) could alter their results leading to different suggested treatment strategies.Similarly, in a spatially extended context, <cit.> investigate the role of spatial correlations on individual-based models of cell migration, proliferation and death, designed to represent experimental assays of cell behaviour in culture. The individual-based models they employ are on-lattice, volume-exclusion processes in two and three dimensions in which cell migration, proliferation and death events are all considered to be exponentially distributed. Cell proliferation occurs when a cell is chosen to divide, placing a daughter cell at one of its neighbouring lattice sites. <cit.> demonstrate the effects of spatial correlations in these models. In particular, they find that changing the motility of the cells can alter their net rate of growth in comparison to the logistic growth predicted by a simple mean field assumption which neglects the effects of correlations. This effect is due to the fact that cell motility serves to break up correlations allowing more proliferation events to occur in comparison to the lower motility case. By explicitly considering the correlations between the occupancies of pairs of lattice sites, <cit.> derive a more accurate population-level model which better represents the growth in the number of cells over time for a diverse range of parameter values. We will investigate the effects of incorporating more realistic CCTDs on the outcomes of the model simulations.Non-Markovian simulation methods exist for events which do not have exponentially distributed inter-event times <cit.>. However,these algorithms are often difficult to understand and complex to encode since we are required to keep track of every cell individually. This presents a potential barrier to their use and consequently a barrier to the appropriate modelling of CCTDs.Given the ubiquity of the Gillespie algorithm, it would be significantly more beneficial if we could decompose the cell cycle into a series of exponentially distributed events which could be naturally encoded in the framework of the Gillespie algorithm. One potential solution to this problem is the use of the hypoexponential family of distributions. It has been suggested that these distributions can be used to accurately represent phases of the cell cycle (and, by closedness of the sums of these distributions, the cell cycle itself) <cit.>. Hypoexponential distributions are made up of a series of k independent exponential distributions, each with its own rate, λ_i, in series. If k is large then these models may face issues of parameter identifiability. Recently, <cit.> have suggested that a delayed hypoexponential distribution (consisting of three delayed exponential distributions in series) could be used to appropriately represent the cell cycle. These delayed exponential distributions represents the G_1, S and a combined G_2/M phases of the cell cycle.Their model is an extension of the seminal stochastic cell cycle model of <cit.> who use a single delayed exponential distribution to capture the variance in the cell cycle. Delayed hypoexponential distributions representing periods of the cell cycle have been justified by appealing to the work of <cit.>. <cit.> showed that the completion time for a large class of complex theoretical biochemical systems, including DNA synthesis and repair, protein translation and molecular transport, can be well approximated by either deterministic or exponential distributions. In this paper we consider two special cases of the general hypoexponential distribution: the Erlang and exponentially modified Erlang distribution which, in turn, are special cases of the Gamma and exponentially-modified Gamma distributions. For reference their PDFs P_E and P_EME, respectively, are given below: P_E(x)=λ^k x^k-1 e^-λ x/(k-1)!,P_EME=λ_1^kλ_2e^-λ_2 x/(k-1)!∫_0^x e^(λ_2-λ_1) tt^k-1 t. With suitable parameter choices, both distributions have been shown to provide good fits to large numbers of experimentally-derived CCTDs <cit.> (see Fig <ref> for one such example). As special cases of the hypoexponential distributions, these distributions also have the significant advantage that they can be simulated using the ubiquitous Gillespie stochastic simulation algorithm. This will allow for the appropriate representation of CCTDs in stochastic models of cell populations, in contrast to the inappropriate exponentially distributed times which are commonly used <cit.>. Additionally the two and three parameters (respectively) of the Erlang and exponentially modified Erlang distributions (respectively) simplify parameter identification in comparison to more highly parametrised distributions. These two choices (Erlang and exponentially modified Erlang distributions) are not the only non-monotone distributions which could be used to appropriately represent the cell cycle. However, they are the general, non-monotone, hypoexponential distributions with the fewest number of parameters (two for Erlang and three for exponentially modified Erlang). These features will aid parameter identifiability (few parameters) and crucially mean the distributions can be simulated using the Gillespie algorithm (hypoexponentiality), making these the most suitable distributions to consider. Figure <ref> demonstrates the improved agreement between the Erlang and exponentially modified Erlang distributions with the experimental data in comparison to the exponential distribution (c.f. figure <ref>). In each case the parameters of the distributions are fitted by minimising the sum of squared residuals, Σ, between the curve and the bars of the histogram[Note that even simpler methods of fitting are possible. In particular, given the first two (three) moments for the Erlang (exponentially modified Erlang) distribution, parameters can be identified by matching moments. This implies the knowledge of the whole distribution of cell cycle times is not necessary. For the Erlang distribution one would need only the mean and variance of cell cycle times. For the exponentially modified Erlang distribution the skewness would also be required.]. Clearly, for the exponential distribution (see figure <ref>), the shape of the curve is incorrect. Consequently the exponential distribution gives a poor representation of the true CCTD with a sum of squared residuals Σ=1.86× 10^-6. Evidently the Erlang distribution with parameters λ=0.0083 and k=12 gives a much better agreement to the experimental data (see figure <ref> figure:data_comparison_Erlang), with a minimised sum of squared residuals, Σ=1.23× 10^-7. Finally, the exponentially modified Erlang distribution with parameters λ_1=0.0251, λ_2=0.0019 and k=26 gives an even better agreement to the data[The three-parameter exponentially modified Erlang distribution may be poorly constrained for some data sets. That is to say there are several values of the parameters which give almost equally good fits to the data. Keeping the aim of carrying out stochastic simulation using the determined distribution in mind, we suggest a preference for acceptable parameter values with the smallest possible value of k. The larger the value of k the more “reaction channels” the Gillespie algorithm must account for which, if rate-limiting, will significantly reduce the algorithm's performance.] with a minimised sum of squared residuals, Σ=6.01× 10^-8. The exponentially modified Erlang distribution achieves a minimised sum of squared residuals which is around half that of the Erlang distribution. Never-the-less, both Erlang and exponentially modified Erlang are good candidates for fitting cell cycle time data and can both be simulated within the existing Gillespie framework, so will be considered here. In Section <ref> we begin by outlining a general hypoexponential model of the cell cycle and noting that many previous models of the cell cycle are special cases. By simplifying the model further we demonstrate that the Erlang and exponentially modified Erlang distributions are also special cases.In Section <ref>, we consider the special case of the Erlang distributed CCTD in more detail. Undertaking some simple analysis we derive the expected behaviour of the mean cell number in the case of Erlang CCTDs and demonstrate analytically that significant differences can arise in comparison to models in which exponentially distributed CCTDs are used. In Section <ref> we demonstrate the utility of our new CCTD representation in stochastic simulations of two biological models in which cellular proliferation is of critical importance. In each case we show, through simulation, that there are important quantitative and qualitative differences between models which represent cell cycle times appropriately and those which do not. We conclude in Section <ref> with a short discussion on the implications of our findings.§ MULTI-STAGE MODEL OF THE CELL CYCLE We divide the cell cycle (with mean length C) into k stages[Note, these stages do not necessarily correspond to the traditional (G_1, S, G_2 and M) phases of the cell cycle. Indeed, there is good evidence that at least one of the phases is not exponentially distributed <cit.>. Rather they are arbitrary division of the cell cycle which will allow us to recreate the correct CCTD.]. The time to progress through each of these stages is exponentially distributed with mean μ_i. We can represent the progression through these stages of the cell cycle as the following chain of `reactions'X_1λ_1→ X_2λ_2→…λ_k-1→ X_k λ_k→ 2 X_1,where λ_i=1/μ_i.The CCT under this model is hypoexponentially distributed. Although there is no simple closed form for the probability density function of the hypoexponential distribution we can find simple expressions for its mean and variance. The mean is given by the sum of the means of the exponentially distributed stage times ∑^k_i=1μ_i=C and the variance is the sum of the variances of these stage times, ∑^k_i=1μ^2_i. By increasing the number of exponentially distributed stages, whilst decreasing their mean duration (in order to maintain the correct mean CCT), we can arbitrarily decrease the variance of the CCT. Many multi-stage models of the cell cycle are special cases of this general model <cit.>.We can analyse the cell cycle reaction chain (<ref>) further by considering the associated probability master equation (PME). Let P(x_1,x_2,…,x_k,t) be shorthand for the probability that there are x_1 cells in stage one, x_2 in stage two and so on. The PME isP(x_1,x_2,…,x_k,t)t =∑^k-1_i=1P(x_1,…,x_i+1,x_i+1-1,…,x_k,t)(x_i+1)λ_i+P(x_1-2,x_2,…,x_k+1,t)(x_k+1)λ_k-∑^k_i=1P(x_1,…,x_i,x_i+1,…,x_k,t)x_iλ_i.By multiplying the PME by x_j and summing over the state space we can find the evolution of the mean number of cells, M_j=∑_xx_j P, in each stage, where ∑_x is shorthand for ∑^∞_x_1=1⋯∑^∞_x_k=1 and P is shorthand for P(x_1,x_2,…,x_k,t). Upon simplification we find the following evolution equations for the mean number of cells in each stageM_jt= 2λ_kM_k-λ_1 M_1,forj=1, λ_j-1M_j-1-λ_jM_j,forj≠ 1. § IDENTICAL RATES OF PROGRESSION The hypoexponential model's generality is also a significant drawback since it hampers parameter identifiability <cit.>. As such, we seek to reduce the number of free parameters in the model whilst maintaining its ability to accurately represent CCTDs. Several authors have suggested using the Gamma distribution to model CCTDs <cit.>. If we assume that all transition rates, λ_i, are identically equal to λ_1 (for i=1,…, k) in our general hypoexponential model, then the time to progress through the whole cell cycle is distributed according to the sum of k identically exponentially distributed random variables. It is straightforward to show (using moment generating functions or convolutions) that the CCTD, is Erlang distributed with scale parameter μ=C/k and shape parameter k. In analogy with the general hypoexponential case, if we decrease μ and simultaneously increase k so that μ k = C remains constant, the Erlang distribution approaches the Dirac delta function centred on C, demonstrating that we can still arbitrarily reduce the variance to match the distribution we are trying to model. §.§ Analysis of the CCDT with equal rates of progressionWe now analyse this CCTD model with identical rates of progression, noting that <cit.> studied this case extensively and we draw on some of his analyses below.Although for this special case it is possible to derive a closed form first order partial differential equation for the evolution of the generating function corresponding the the master equation (<ref>), solving the associated characteristic equations is analytically intractable for all but the simplest case (k=1) <cit.>. Instead we will focus on the mean behaviour of the cell population with which we can make some analytical progress.In the equal rates case we can sum the individual equations in system (<ref>) to give Mt=λ_1 M_k,where M=∑_i^k M_i. Consider the naive one stage (i.e. k=1) cell-cycle mode with mean cell cycle time C:X1/C→ 2X.The evolution of the mean number of cells is given by the special case of equation (<ref>): Mt=M/C.In the multi-stage model, under the assumption that all cells are evenly distributed between the stages (i.e. M_i=M/k), we can replace M_k with M/k in equation (<ref>) to give a closed equation for the evolution of the total number of cells which matches equation (<ref>):Mt=λ_1 M/k=M/C. However, the assumption on the even distributions of cells between stages is incorrect. This leads to differences not just, as might be expected, between the variation exhibited by the multi-stage and single-stage models, but also between their mean behaviour. In figure <ref> figure:stages_summed a clear difference between the k=1 and k=4 models is evident. The mean total cell number grows significantly more slowly in the k=4 case than the k=1 case. This is true for all models in which k>1. Intuitively, exponentially distributed CCTs imply that the most probable time for a cell to divide is the current time. Once a cell has divided it is immediately able to divide again with high probability allowing cells proliferating under the exponentially distributed CCT assumption to reinforce their numbers. This is in direct contrast to cells with Erlang distributed CCTs (with the same mean but k>1) which, with high probability, will wait for a period of time before dividing. In short, the larger variance of the exponentially distributed CCT population allows it to grow more rapidly. Introduced the scaled variables m_j=M_je^kt/C, for j=1,…, k. Under the assumption ofidentical transition rates, equation system (<ref>) can be reduced to a closed equation for the scaled mean number of cells in the k^th stage^k m_k/ t^k=2(k/C)^k m_k,and a set of k-1 ODEs which relate the number of cells in the other stages to m_km_j=(C/k)^k-j^k-j m_k/ t^k-j, for j=1,…, k-1.Under the given initial conditions, a single cell in the first stage and no cells in any other stages, we can solve these equations to find the unscaled mean number of cells in each stage:M_j=2^(1-j)/k/k∑_r=0^k-1z^(1-j)rexp((2^1/kz^r-1)k t/C),where z is the first k^th root of unity <cit.>. Although this expression looks complicated in some cases it is possible to express M_j in a simple closed form. For example, when k=4M_1= exp(-4t/C)/2{cosh(2^9/4t/C)+cos(2^9/4t/C)}, M_2= exp(-4t/C)/2^5/4{sinh(2^9/4t/C)+sin(2^9/4t/C)},M_3= exp(-4t/C)/2^3/2{cosh(2^9/4t/C)-cos(2^9/4t/C)}, M_4= exp(-4t/C)/2^7/4{sinh(2^9/4t/C)-sin(2^9/4t/C)}.A comparison between these analytical solutions and their numerically solved counterparts demonstrates their veracity in Figure <ref> figure:all_four_stages.By summing equation (<ref>) over all values of j=1,…,k we can also find an expression for the total number of cells in a population:M(t)=1/2k∑_r=0^k-12^1/kz^r/2^1/kz^r-1exp((2^1/k z^r -1)kt/C). Although these formulae (equations (<ref>)–(<ref>)) may be useful in specific cases where the closed form of the solution is readily accessible, their real utility is in shedding light on the asymptotic properties of the mean numbers of cells. In the limit that t gets large for finite k the dominant term in the summation in equation (<ref>) corresponds to the case r=0. Indeed for k≤ 28 the real parts of the other elements in the summation are negative and hence these terms decay <cit.>. Thus we havelim_t→∞M_j≈1/k2^(1-j)/kexp(t α_k/C),whereα_k= k(2^1/k-1).Summing equation (<ref>) over all k stages leads to the asymptotic behaviour of the cell population as a whole:lim_t→∞M(t)=2^1/k/2α_kexp(t α_k/C).Equation (<ref>) can be re-written aslim_t→∞M(t)=2^1/k/2α_k( e^α_k)^(t/C).For all k>1, not only is the base of the exponent t/C less then e (since α_k<1, for k>1), but the coefficient is less than unity <cit.>. This implies that, asymptotically, the expected total number of cells in a multi-stage model will always be less than the number expected in a single stage cell cycle model (which can be determined upon substituting k=1 in to (<ref>)).Note that in the limit as k→∞, α_k→ln 2.Thus, as might have been expected for the deterministic model resulting from the limit k→∞, the asymptotic population grows with base 2, doubling at regular intervals as the cells divide synchronously:lim_ k→∞lim_t→∞ M=lim_ k→∞2^1/k/2α_k· 2^kt/C .Surprisingly though, the coefficient of 2^k/C t does not tend to unity in equation (<ref>) as might have been expected. Thus the total population grows likelim_k→∞lim_t→∞ M≈ 0.721· 2^kt/C .Reversing the order of limits and taking the limit as k tends to infinity of equation (<ref>) for finite t gives the limitlim_k→∞M(t)=2^⌊ t/C ⌋,for non-integer value of t/C, where ⌊ x⌋ gives the integer part of x <cit.>. For integer values of t/C the limit is lim_k→∞M(t)=3/4 2^⌊ t/C ⌋,corresponding to the algebraic mean of the limits of equation (<ref>) as integers values are approached from the left and right hand sides. This “deterministic” doubling process is unsurprising since the waiting time distribution tends to a delta function in the k→∞ limit, implying that cell division is synchronous. §.§ Cells are not distributed proportional to stage length Returning to equations (<ref>) under the assumption of identical rates of progression through the stages, we can derive corresponding equations for the mean proportion of cells in each stage, M̂_j=M_j/M for j=1,…, k:M̂_jt=λ_1 (2M̂_k-M̂_1-M̂_1M̂_̂k̂),forj=1,λ_1 (M̂_j-1-M̂_j-M̂_jM̂_k),forj≠ 1.At steady state we have the following recurrence relations for the mean proportion of cells in each stage M̂^st_j=2M̂^st_k/1+M̂^st_k,forj=1,M̂^st_j-1/1+M̂^st_k,forj≠ 1.In particular, this implies that M̂^st_j<M̂^st_j-1 for j=2… k, so that, at steady state, as we progress through the stages we will have successively fewer cells in each stage on average (independent of the initial distribution of cells amongst different stages).By solving these recurrence formulae we can find exact expressions for the steady state proportions:M̂^st_j = (√(2))^k-j(√(2)-1) .In particular, note thatM̂^st_1/M̂^st_k→ 2 as k→∞.That is to say there are twice as many cells in the first stage as the last stage at steady state when the number of stages is large.These differences are potentially important for determining average CCTs experimentally. One popular method for determining cell cycle times is to label S-phase cells using two sequentially administered distinct DNA specific labels <cit.>. The administration of the labels is separated by a known time period. By counting cells labelled with one or both labels, and with reference to the known time period of separation, it is possible to calculate the mean duration of the S-phase. Once the proportion of cells in S-phase and the mean duration of S-phase have been determined it is also possible to calculate the mean cell cycle time for the population <cit.>. The method outlined above implicitly makes the assumption that the number of cells in a particular phase of the cell cycle is proportional to the length of that phase. For the multi-stage model we have demonstrated that in the large time limit this is unequivocally not the case. Equation (<ref>) can also be used to show that this phenomenon holds dynamically, although the mathematics is cumbersome. Instead we solve equations (<ref>) numerically. Numerical solution also allows us to investigate the more general hypoexponential CCTD model (<ref>) for which no analytical solutions are available. Figures <ref> figure:even_unnormalised and figure:even_normalised display the evolution of the mean numbers and proportions (respectively) of cells in each stage for equal stage progression rates, λ_1, and figures <ref> figure:random_unnormalised and figure:random_normalised display the equivalent for unequal progression rates. The number/proportion of cells in each stage is not proportional to the mean duration of the stage, μ_i, either at steady state (compare actual steady state proportions with the stars representing the normalised mean stage durations, μ_i/∑μ_i, in <ref> figure:even_normalised and <ref> figure:random_normalised) or dynamically. §.§ The exponentially modified Erlang distribution Although the identical-stage model, which gives rise to the Erlang distribution for CCTDs, is convenient from a mathematical perspective, it has been shown to have been outperformed by a number of other distributions <cit.>. In particular, by considering 77 independent CCT data sets, <cit.> has recently shown that one of the most appropriate distributions for representing CCTDs is the exponentially modified Gamma (EMG) distribution. For our purposes we will require that the shape parameter of the Gamma distribution is be integer valued so that the CCTD is actually an exponentially modified Erlang (EME). This will mean that we can continue to simulate CCTs using a series of exponentially distributed random variables (albeit one of them will have a different rate). Consequently this will allow us to continue to appropriately simulate processes in which cell division is important using the popular Gillespie algorithm. In order to generate EME distributed CCTs we modify our multi-stage cell cycle model as follows: X_1λ_1→ X_2λ_1→…λ_1→ X_k λ_1→ X_k+1λ_2→ 2 X_1.Note that, in system (<ref>), the rate of progression is identical through each of the initial k stages of cell cycle and that we have added an additional exponentially distributed stage at the end whose rate, λ_2, is distinct from the rate, λ_1, of the previous k stages.We can ascertain the probability density function for the EME distribution, P_EME(t), by convolving the Erlang (P_ER(t)) and exponential (P_E(t)) distributions as follows, P_EME(t)=P_E(t)∗ P_ER(t)=∫^t_0 λ_2exp(-(t-u)λ_2)·u^k-1exp(-λ_1 u)λ_1^k/(k-1)! u,where λ_2 is the rate of the exponential distribution with which we are convolving and, as before, λ_1 is the rate of progression through each of the k identical exponentially distributed stages which comprise the Erlang distribution. We can simplify expression (<ref>) to the following formulationP_EME(t)=λ_2 e^-tλ_2(λ_1/λ_1-λ_2)^k{1-Γ(k,L t)/(k-1)!},where L=λ_1-λ_2 and Γ(k,L t)=∫^∞_L tz^k-1e^-z z is the complementary incomplete gamma function.We note that it is almost as simple to simulate this more general distribution in the Gillespie algorithm using a series of exponentially distributed stages as it is to simulate the distribution with constant rates of progression between stages. Indeed the simulation of any hypoexponential CCTD is straightforward in the Gillespie algorithm. However, the addition of extra parameters hampers their identifiability when fitting to experimental data and as such we only suggest using the Erlang or exponentially modified Erlang distributions in models of the CCTD. In the following section we illustrate the importance of incorporating non-exponentially distributed CCTDs into stochastic simulations of cellular proliferation. For ease of understanding we concentrate purely on Erlang CCTDs, and note that the parameters are not based on fitted CCTDs but merely chosen for illustrative purposes.§ ILLUSTRATIVE EXAMPLESIn this section we recapitulate results from two different models which each assume exponentially distributed CCTs. The first is a well-mixed model of cancer stem cell proliferation and differentiation in a brain tumour. The second a spatially extended model of cell migration and proliferation mimicking a growth-to-confluence experimental assay. In each case we alter the CCTD in order to see what effect this has on the qualitative and quantitative results presented in the papers. For clarity we will restrict ourselves to Erlang distributed CCTs, but note that results are qualitatively similar for exponentially modified Erlang distributed CCTs. §.§ Cancer stem cell maintenance<cit.> investigate the role of sub-populations of cells within a brain tumour possessing stem cell-like properties and responsible for maintaining the tumour. In situations (e.g. post treatment) in which there are small numbers of stem cells they consider a stochastic model of cell proliferation and differentiation. Stem cells, S, can undergo symmetric division in which the daughter cells possess the same characteristics as the parent cells (see equation (<ref>)) and the stem cell population increases. They can also undergo asymmetric self renewal in which one stem cell and one progenitor cell, P, are produced (see equation (<ref>)) or symmetric proliferation in which two progenitor cells result from a stem cell division (see equation (<ref>)).Cell cycle times are exponentially distributed with rate ρ_s and fate choices (about which division type to undergo) are made at the point of division. With probability r_1 symmetric division occurs and with probability r_3 symmetric proliferation occurs. Consequently with probability r_2=1-r_1-r_3 asymmetric self renewal occurs. These divisions with their effective rates are captured in the reaction system (<ref>)–(<ref>):S ρ_s r_1→S+S,S ρ_s r_2→ S+P,S ρ_s r_3→ P+P. Under the assumption of exponential CCTDs, <cit.> write down and solve a simple probability master equation for the stem cell population. In particular, they consider the case in which r_1>r_3 which implies a positive net growth rate of the stem cell population. Under this assumption the mean number of cells in the stem cell population and variance can be shown to increase exponentially. Since the model is linear, by appealing to the central limit theorem, <cit.> argue that, for large enough cell populations, the exact mean field equations given by St=ρ_s(r_1-r_3)Swill approximate the stochastic dynamics well.In order to ensure a more realistic representation of the CCTD we introduce a multi-stage cell division process, as suggested above, so that the CCTDs are now Erlang distributed with the same mean ρ_s, but with shape parameter k (and thus scale parameter μ_1=1/(ρ_s k)). As, before, at each division event, a choice about the type of division to occur is made with the same probabilities (r_1,r_2, r_3) as previously specified. Although we still get exponential increases in the mean and variance, the rate of increase is significantly decreased (see Figs. <ref> figure:mean_cell_numbers and figure:variance_of_cell_numbers). Crucially this means that the deterministic mean-field model derived from the original process will significantly overestimate the number of cancer stem cells in the tumour which could have significant therapeutic implications. Under this model with r_1>r_3, if a tumour is not completely eradicated by treatment it is possible that it can return. It may, therefore be informative to know the probability that a tumour will reach a certain sizeby a particular time in order to plan appropriate follow-up therapeutic intervention. For example we may be interested to know the evolution of the probability that the tumour has reached 1000 stem cells in size (which we will denote p_1000(t)) so that we might calculate the most appropriate time to initiate the follow-up intervention. In figure <ref> figure:p_exceed we plot the evolution of p_1000(t) over time. It is clear, by t=100, that the probability of the tumour having grown to 1000 stem cells, p_1000 varies significantly depending on the value of k used in the model despite the cells having the same mean CCT[Note, that by varying k with a fixed cell cycle time we are implicitly varying λ, the rate of progression through each stage.]. The effects of varying CCTD can clearly be seen to influence the model outcome even in this relatively straightforward linear model of cellular proliferation. In more complex models, in which other species depend in a non-linear manner on the number of cells, the effects will no doubt be further exacerbated. The potential for therapeutic interventions to be based on stochastic mathematical models of cellular proliferation further emphasises the importance of modelling the CCTD correctly.§.§ Growth to confluence assays Next we investigate the effect of incorporating a more realistic model of cell proliferation on the behaviour of a spatially extended individual-level model of cell migration and proliferation <cit.>. As such, we alter the mechanism of cellular proliferation from the original, exponentially distributed division times to our more realistic multi-stage Erlang distributed division times and observe the effect this has on the growth of the cell population. In order to achieve this we break the proliferation process into k stages, the passage through each of which has an exponentially distributed waiting time (as described above). As before, we chose the parameter of each stages' waiting time to ensure we have the same mean proliferation attempt time as in the original model.In more detail, we consider a volume-exclusion process on a regular, square lattice in two dimensions with periodic boundary conditions. Each lattice site, of length h, can hold at most one cell. Each repeat realisation begins by initialising, particles uniformly at random across the L_x× L_y sites of the lattice. Agents can move between adjacent (in the von Neumann sense) lattice sites with rate P_m. Movement is unbiased, meaning that once a cell has been chosen to move it does so into one of its four neighbouring lattice sites with equal probability. If the site into which a cell attempts to move is already occupied then that movement event is aborted: the cell attempting movement remains at its current site. Agents undergo a proliferation stage change with rate P_p k (giving average rate P_p for unhindered progression through the k stages required for division) this results in the cell's current proliferation stage being incremented by one if the cell is currently in one of the first k-1 stages. If the cell is in the final stage (stage k) of proliferation and is selected to change stage then the cell attempts to place a daughter in one of its four neighbouring lattice sites with equal probability. If the chosen site is empty, the cell places a daughter in the empty site and the proliferation stages of both the parent and the daughter are reset to unity. However, if the cell attempts to place a daughter in a site which is already occupied then that proliferation event is aborted. In the multi-stage model of the cell cycle, we then have two choices:* the progression stage of the cell attempting proliferation is reset to unity; * the cell remains in the k^th stage.In the original model in which k=1 these two choices are identical. Under implementation <ref> cells would have the same average rate of division attempts as in the original model. However, it could be argued that implementation <ref> is more realistic as real cells do not reverse through the cell cycle if division is not favourable, but remain held at checkpoints <cit.>. We will investigate both possibilities. In order to clearly distinguish the effects of the different CCTDs we will not consider cell death in our simulations. For different values of P_p the population will naturally grow at different rates. As in <cit.> we will rescale time, t̅=P_p t, in order to make population evolutions comparable.Figure <ref> shows example snapshots of the domain occupancy at rescaled time t̅=10 for three different values of k=1,10,100. Panels figure:baker_Pp=1_Pd=0_RESET=1_k=1_t=10–figure:baker_Pp=1_Pd=0_RESET=1_k=100_t=10 represent implementation <ref> for k=1,10,100, respectively. Panels figure:baker_Pp=1_Pd=0_RESET=0_k=1_t=10–figure:baker_Pp=1_Pd=0_RESET=0_k=100_t=10 represent implementation <ref> for k=1,10,100, respectively. Spatial correlations in the occupancies of lattice sites (clusters) are clearly visible in all cases.In the multi-stage model, implementation <ref> generally leads to less dense colonies than implementation <ref> since cells do not attempt division as frequently. Under implementation <ref> (see figure <ref> figure:baker_Pp=1_Pd=0_RESET=0_k=10_t=10 and figure:baker_Pp=1_Pd=0_RESET=0_k=100_t=10) a clear proliferating rim of (grey) cells can be seen with the bulk of cells being kept at stage k (black). Under implementation <ref> every cell can be found in any stage of the cell cycle so it is hard to distinguish the proliferating rim (see figure <ref> figure:baker_Pp=1_Pd=0_RESET=1_k=10_t=10 and figure:baker_Pp=1_Pd=0_RESET=1_k=100_t=10). The difference between the two implementations however, is not due to aborted proliferation events in the bulk (away from the rim) but to the ability of cells at the proliferating rim to rapidly undergo a further division attempt after an aborted attempt under implementation <ref>. This suggests that the difference between the two implementations will only be apparent at high densities for which correlations have built up and significant numbers of division attempts are being aborted. For low density systems, in which very few particles are adjacent, the mean cell division attempt times are almost the same for all values of k independent of the implementation (<ref> or <ref>). However, the variance in the CCTDs for low density systems affects the rate of growth with larger values of k (less variance in the CCTD) generally leading to slower growth. This effect can be understood by considering equations (<ref>) and (<ref>) for a non-excluding population of cells, for which the finite time and asymptotic time behaviours, respectively, of cell populations with different values of k can be contrasted.In figure <ref> we contrast the evolution of the spatially-averaged density for three values of k=1, 10, 100 and three proliferation rates, P_p=0.05, 0.5, 1, under implementation <ref> (figure:baker_Pp=0.05_Pd=0_RESET=1_mean_density_evolution–figure:baker_Pp=1_Pd=0_RESET=1_mean_density_evolution) and implementation <ref> (figure:baker_Pp=0.05_Pd=0_RESET=0_mean_density_evolution–figure:baker_Pp=1_Pd=0_RESET=0_mean_density_evolution).Under implementation <ref> (see figure <ref> figure:baker_Pp=0.05_Pd=0_RESET=1_mean_density_evolution–figure:baker_Pp=1_Pd=0_RESET=1_mean_density_evolution), even as cell-cell correlations build up, multi-stage cells still proliferate more slowly than single-stage cells, since the mean division attempt time remains the same for all values of k. The increased variance of cells with fewer stages results in faster population growth.However, under the more realistic implementation <ref> (see figure <ref> figure:baker_Pp=0.05_Pd=0_RESET=0_mean_density_evolution–figure:baker_Pp=1_Pd=0_RESET=0_mean_density_evolution) cells with multi-stage cell cycles are able to re-attempt division after abortive division events more quickly than they otherwise could under the single-stage cell cycle model. Thus, effective average CCTs for cells with a multi-stage cell cycle at the proliferating rim of a cluster decreases in comparison to cells with a single-stage cell cycle. The faster thepairwise correlations build up, the more pronounced this effect becomes. Witha very low proliferation rate (in comparison to fixed motility — see figure <ref> figure:baker_Pp=0.05_Pd=0_RESET=0_mean_density_evolution) cell movement is effective at breaking up correlations meaning that large clusters do not form and that we only see the effect of decreasing mean CCT for larger values of k at late (scaled) times, when the density is higher. Contrastingly, when proliferation is high in comparison to motility (see figure <ref> figure:baker_Pp=1_Pd=0_RESET=0_mean_density_evolution) clusters can form quickly preventing the bulk of cells from proliferating earlier and allowing the cells with multi-stage cell cycles to divide faster on average at the proliferating rim of these clusters than their single-stage counterparts.It is also worth noting that the greater synchrony in cell division times for larger values of k (exemplified by equations (<ref>)–(<ref>) for the limit of infinite k) is in evidence in the jagged nature of the yellow curves (corresponding to k=100) in all six subfigures.§ DISCUSSIONCurrently, many stochastic models which incorporate cell proliferation employ the ubiquitous Gillespie stochastic simulation algorithm <cit.>. Unfortunately, in its basic form the Gillespie algorithm represents all events as exponentially distributed. Cell cycle times are not exponentially distributed and can not, therefore be represented by a single reaction event in the Gillespie algorithm. Modelling cell cycle times as a single exponentially distributed event can lead to significant alterations in model behaviour in comparison to more appropriate CCTDs. Consequently, we postulated a simple, general hypoexponentially distributed CCT which can be broken down into exponentially distributed stages allowing for straightforward simulation with the popular Gillespie algorithm. To account for ease of parameter identification we suggested two special cases of this more general model which have been shown to do an excellent job of recapitulating CCTDs <cit.>.We postulate that the general hypoexponential distribution <cit.> or even the more specific Erlang <cit.> or exponentially modified Erlang <cit.> inter-event distribution time models could be used to allow the simplified simulation of complex biochemical and biophysical processes (e.g. enzymatic reactions <cit.>, allosteric transitions in ion channels <cit.>, the movement of molecular motors <cit.>, DNA unwinding <cit.>) using the Gillespie algorithm. More generally, non-Markovian processes for which only the inter-event distribution, rather than the mechanism which generates this distribution, is important might be simulated efficiently using our proposed mechanism <cit.>.We employed our improved model of cell cycle proliferation times on two recent models of real biological processes <cit.>. In each case we found that the incorporation of multiple stages to the cell cycle led to significant differences in the population size in comparison to the original exponentially distributed CCT model. We suggest that these difference will hold more generally throughout stochastic models in which CCTs are currently modelled as exponential. In particular, we intend to investigate the effects of our modified CCTD on the speed of invasion of a population of migrating and proliferating cells.The application here of hypoexponentially distributed CCTs built up from a number of intermediary exponential stages assumes that the CCT is not correlated between direct descendants or within a given generation. Whilst there are scenarios in which there is no evidence for a correlation in CCT between related cells <cit.> there are other situations in which this assumption is clearly invalid <cit.>. It is possible that some of these correlation effects can be attributed to the environment in which the cells are proliferating. However, in NIH 3T3 cells a clear correlation has been observed between daughter cells of a given mitotic event compared to more distant relatives; implying a heritable predisposition <cit.>. Therefore, one obvious extension to this work would be to incorporate the effects of correlations in cell and phase times to better reflect the biological heterogeneity of a given system. § APPENDIX A - MATERIALS AND METHODS In order to determine cell cycle times in cell culture, NIH 3T3 Flp-In cells <cit.> were seeded at various densities in phenol-red free dulbeccos modified eagle medium (DMEM) containing 10% fetal calf serum, 1% Penicillin/Streptomycin and 100 µg/mlHygromycin B on glass bottomed 24-well culture plates (Greiner bio-one, UK). The next day, time-lapse imaging was performed on subconfluent wells with a 20× objective using a Nikon A1R inverted confocal microscope in a heated chamber supplied with 5% CO_2 in air. The time elapsed between mitotic events was measured using the Fiji <cit.> distribution of ImageJ -an open source image analysis package based on NIH Image <cit.>.§ APPENDIX B - PSUEDOCODE FOR THE MULTI-STAGE MODEL OF THE CELL CYCLEOne of the original (and most popular) implementations of the mathematically exact SSA is known as the direct method <cit.>. Here we present pseudocode for the direct method implementation of the simple multi-stage model of the cell cycle (corresponding to Erlang distributed CCTs) in a well mixed context. Let X(t) be a vector of length k which contains the number of cells in each stage at time t. A time interval τ, until the next stage advancement event, is generated. Along with it, an index j, is chosen which determines from which stage a cell will advance from time t+τ. The changes in the numbers of cells caused by the stage advancement are implemented, the propensity functions (progression rate, λ, multiplied by the number of cells in each stage) are altered accordingly and the time is updated, ready for the next (τ,j) pair to be selected. A method for the implementation of this algorithm is given below: * Initialize the time t=t_0 and the number of cells in each stage, X(t_0)=x_0. *Evaluate the propensity functions, a_j(X(t))=λ X_j(t), (for j=1,…,k) associated with the advancement of cells from their respective stages, and their sum a_0(X(t))=∑^k_j=1a_j(X(t)).* Generate two random numbers rand_1 and rand_2 uniformly distributed in (0,1).* Use rand_1 to generate a time increment, τ, an exponentially distributed random variable with mean 1/a_0(X(t)). i.e.τ=1/a_0ln(1/rand_1).* Use rand_2 to generate index of the next stage advancement event, j, with probability a_j(X(t))/a_0(X(t)), in proportion with its propensity function. i.e. find j such that[Note in step <ref>, that in the case j=1 we assume ∑_j'=1^0=0.]∑_j'=1^j-1a_j'(X(t))<a_0(X(t))· rand_2<∑_j'=1^ja_j'(X(t)).* Update the time, t=t+τ, and the state vector to reflect the advancement of one cell from the chosen stage, X_j=X_j-1if j = 1,…,kandX_j+1=X_j+1+1ifj≠ k, orX_1=X_1+2 if j = k. * If t<t_final, the desired stopping time, then go to step (<ref>). 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Stochastic simulation in systems biology. Comput. Struct. Biotechnol. J., 120 (20):0 14–25, 2014.[Turner et al.(2009)Turner, Stinchcombe, Kohandel, Singh, and Sivaloganathan]turner2009cbc C. Turner, A.R. Stinchcombe, M. Kohandel, S. Singh, and S. Sivaloganathan. Characterization of brain cancer stem cells: a mathematical approach. Cell proliferation, 420 (4):0 529–540, 2009.[Weber et al.(2014)Weber, Jaehnert, Schichor, Or-Guil, and Carneiro]weber2014qlv T.S. Weber, I. Jaehnert, C. Schichor, M. Or-Guil, and J. Carneiro. Quantifying the length and variance of the eukaryotic cell cycle phases by a stochastic model and dual nucleoside pulse labelling. PLoS. Comput. Biol., 100 (7):0 e1003616, 2014.[Wimber and Quastler(1963)]wimber1963tdl D.E. Wimber and H. Quastler. A ^14c- and ^3h-thymidine double labeling technique in the study of cell proliferation in Tradescantia root tips. Experimental Cell Research, 300 (1):0 8–22, 1963.[Zaider and Minerbo(2000)]zaider2000tcp M. Zaider and G.N. Minerbo. Tumour control probability: a formulation applicable to any temporal protocol of dose delivery. Phys. Med. Biol., 450 (2):0 279, 2000.[Zhou and Zhuang(2007)]zhou2007kas Y. Zhou and X. Zhuang. Kinetic analysis of sequential multistep reactions. J. Phys. Chem. B, 1110 (48):0 13600–13610, 2007.[Zilman et al.(2010)Zilman, Ganusov, and Perelson]zilman2010sml A. Zilman, V.V. Ganusov, and A.S. Perelson. Stochastic models of lymphocyte proliferation and death. PLoS one, 50 (9):0 e12775, 2010. | http://arxiv.org/abs/1705.09718v3 | {
"authors": [
"Christian A Yates",
"Matthew J Ford",
"Richard L Mort"
],
"categories": [
"q-bio.QM"
],
"primary_category": "q-bio.QM",
"published": "20170526205710",
"title": "A multi-stage representation of cell proliferation as a Markov process"
} |
Multiscale interaction between a large scale magnetic island and small scale turbulence M J Choi^1, J Kim^1, J-M Kwon^1, H K Park^1,2, Y In^1, W Lee^1, K D Lee^1, G S Yun^3, J Lee^2, M Kim^2, W-H Ko^1, J H Lee^1, Y S Park^4, Y-S Na^5, N C Luhmann Jr^6, B H Park^1 December 30, 2023 =================================================================================================================================================================================== Automatically learning features, especially robust features, has attracted much attention in the machine learning community. In this paper, we propose a new method to learn non-linear robust features by taking advantage ofthe data manifold structure. We first follow the commonly used trick of the trade, that is learning robust features with artificially corrupted data, which are training samples with manually injected noise. Following the idea of the auto-encoder, we first assume features should contain much information to well reconstruct the input from its corrupted copies. However, merely reconstructing clean input from its noisy copies could make data manifold in the feature space noisy. To address this problem, we propose a new method, called Incremental Auto-Encoders, to iteratively denoise the extracted features.We assume the noisy manifold structure is caused by a diffusion process. Consequently, we reverse this specific diffusion process to further contract this noisy manifold, which results in an incremental optimization of model parameters . Furthermore, we show these learned non-linear features can be stacked into a hierarchy of features. Experimental results on real-world datasets demonstrate the proposed method can achieve better classification performances. § INTRODUCTIONFeature extraction, transforming the original input features to new feature space, has attractedmuch attention in machine learning community, especially when data are represented by high dimensional feature vectors. Many linear (e.g. PCA, LDA, etc.) and non-linear feature learning methods (e.g. sparse coding, dictionary learning, etc.) have been proposed to address this problem during the past few years <cit.>. Recent years, learning robust features is getting more and more attention from researchers in various areas, especially in the deep learning community <cit.>.In general, considering the types of training sets, robust feature learning methods can be roughly classified into two groups. Algorithms in one group learn features from natural noisy datasets. Whereas, methods in the other group are given clean training datasets.In order to extract robust features, they learn with artificially corrupted data, which are training samples with manually injected noise.For example, to learn robust features, handwritten digits dataset are manually injected with various noises, such as random binary background noise or image background noise <cit.>.In the deep learning literature, DAE (Denoising Auto-encoder) <cit.>, composed of an encoding and a decoding function,is one of the best known building blocks for constructing a hierarchy of non-linear features. Features are made robust by reconstructing the clean input from its artificially corrupted copies via a decoding function. To make features invulnerable to different noises, AMC-SSDA combines multiple DAEs by a set of weights <cit.>. Although the performances of these methods are prominent in many cases, their efficiency can also be improved since from the view of manifold learning, the high dimensional data are nearly lying on a low dimensional manifold. These methods have not taken fully considerations about the manifold structure.To leverage the manifold structure,some methods have been proposed. Typical linear feature learning methods are: LPP <cit.>, LLE <cit.> and Isomap <cit.>. They learn linear features by preserving the local relationships within the data set and uncovering its essential manifold structure. There are also some other non-linear feature learning algorithms, such as SNE <cit.>, t-SNE <cit.>, etc. Commonly, these methods use various neighborhood graphs to characterize the manifold structure.Differently, in the deep learning community, CAE (Contractive Auto-encoder) uses a contractive penalty term to force the learned features to capture the local direction of the non-linear manifold <cit.>. Compared with linear feature learning approaches, non-linear methods have proven to perform better in many cases. In addition, extracting robust features with the consideration of non-linear manifolds structures has also attracted much attention <cit.>. However, these works learn the same dimensional features as the high dimensional input. In practice, not all features are relevant and important to the learning task, many of which are often redundant.In light of these works, in this paper we propose a new method which can learn non-linear and robust features from manifold-embedded datasets. Similar to the above work, we first follow the well-known trick of the trade to learn with artificially corrupted data for extracting robust features. We assume extracted features should contain much information. Thus they can well reconstruct the clean input from its corresponding corrupted samples via a decoding function.To get more reliable features, we then using a denoising method based on the following assumption: local structures of data in different features space should be consistent. From the view of manifold learning, artificially corrupting data makes the embedded manifold of input noisy. Merely minimizing the reconstruction error can not guarantee manifold in the new feature space being noiseless.Thus manifolds are inconsistent with that of the input. To address this problem, we iteratively refine the learned features using a Laplacian-based method. We assume the noisy manifold is formed by a diffusion process on the Laplacian graph of data. We then reverse this diffusion process to denoise hidden features. Each step, representations are denoised towards the manifold. Step by step, the manifold structure of data becomes more and more refined. We further show that these non-linear features can be stacked to yield multiple levels of representations. Experimental results on several real world datasets illustrate that the proposed method can achieve better performance.The remainder of the paper is organized as follows. We introduce the related work in Section 2. Then the proposed methodis presented in Section 3, followed by the optimization of the proposed method in Section 4. Following the experimental results in Section 5,we conclude the paper in Section 6.§ RELATED WORKIn recent years, automatically feature learning has received increasing attention from machine learning community, especially in the deep learning community, due to its wide applications in practice. There are a rich body of work on feature learning in the literature. We provide a review to the most related methods in this section.Auto-encoder. The auto-encoder is one of the most popular methods for learning informative non-linear features. It assumes these extracted features should contain as much information of input as possible and well reconstruct its input. To extract non-linear features, it exploits a direct parameterized function f(), called encoder, to output hidden representations, defined as follows.= f() = s_e(_1+ _1)where _1 ∈ℝ^K × D is the weight matrix and _1 is the bias vector. is the K-dimensional feature vector.In the meanwhile, another function g(), called decoder, is defined to map from feature space back into the input space, producing a reconstruction . It is parameterized as:= g() = s_d(_2+ _2)where _2 ∈ℝ^K × Dand _2 ∈ℝ^D are the weight matrix and bias vector respectively. s_e and s_d are the activation functions, whose typical choices are sigmoid, tanh, rectified linear.The set of parameters θ = {_1, _2, _1, _2} are learned simultaneously on the task of minimizing the reconstruction error over the whole training dataset = {_1, _2, ..., _n}∈ℝ^D × n, which correspond to the following optimization function:θ^⋆ = min_θ1/n∑_i = 1^nl(_i, g(f(_i))),where l is the reconstruction loss, whose typical choices are cross-entropy lossand the squared error loss.Traditionally, auto-encoder is used as a dimensionality reduction technique, which can learn equivalent or more useful features than what are obtained with simple linear PCA. Recently, a more successful use of auto-encoder is to learn over-complete features, yielding more rich hidden representations. However, this renders the problem that the basic auto-encoder can learn an identity mapping with perfectly reconstructing its input and without extracting more meaningful features. To tackle this problem, various methods with different criteria have been proposed, such as sparse auto-encoder <cit.>, RBM <cit.> and so on. Among all the various constraints, robustness of features is most favored.Denoising Auto-encoder. One popular method to impose the robustness constraint is denoising auto-encoder (DAE). Except for remaining much information of input, it assumes good hidden features should well reconstruct its clean input from the corrupted copies, which avoids the uninteresting solutions of auto-encoder. From the geometric structure of input, which assumes high dimensional data are concentrated on a low dimensional manifold , DAE maps far away corrupted data to small regions close to the intrinsic data manifold. Formally, it is trained by the following function:θ^⋆ = min_θ1/nm∑_i = 1^n ∑_j = 1^m l(_i, g(f(_i_j))),where each sample _i is reconstructed from its m corrupted copies _i_j = ρ(_i). Typical choices for the corrupting function ρ are additive isotropic Gaussian noise, salt and pepper noise and masking noise.Comparing to the traditional auto-encoder, these learned features are qualitatively better in classification performance. Exploiting DAE as a building block, several other methods have been proposed, such as AMC-SDAE <cit.>, spDAE <cit.>, mDAE <cit.> and so on.However, DAE still subjects to some drawbacks. Based on the manifold hypothesis, hidden representations correspond to an intrinsic coordinate system on the manifold structure. Variations in the input should be reflected in the learned representation. Whereas, since DAE makes the whole mapping robust instead of , this assumption is not guaranteed. In addition, just mapping back corrupted samples to a nearby region makes the intrinsic manifold structure divergent. It fails to maintain the local structure when multiple manifolds exists in training data, which is often the case.Contractive Auto-encoder. Another method to learn robust features is contractive auto-encoder (CAE). From a different perspective, it assumes features should be contractive along the orthogonal direction to the manifold. Its goal is achieved by adding a contractive penalty term directly on the hidden features to the basis auto-encoder. Hidden features are made insensitive to small changes of input by the Frobenius norm of the encoder's Jocabian. It is trained by minimizing the following objective function:θ^⋆ = min_θ∑_i=1^nl(_i, g(f(_i))) + λ ||(_i)||^2_Fwhere ∈ℝ^K × D is the encoder's Jacobian matrix and λ is the trade-off parameter.Comparing with DAE, CAE captures the local changes of the data manifold in the hidden representation. However, the contractive penalty term merely encourages robustness to infinitesimal changes of input. Thus, when data is corrupted by a large noise, it could fail. This problem is further considered by <cit.>, which penalizes all higher order derivatives.§ INAE: INCREMENTAL AUTO-ENCODERS §.§ Problem modelingFrom the manifold hypothesis, hidden representations correspond to an intrinsic coordinate system on the embedded manifold .Variations along the manifold in the input space should be well captured or reflected in the learned representations. However, merely reconstructing clean input from itself or its noisy copies could make manifoldin the feature space noisy. As a result, intrinsic manifolds between original space and the hidden feature space are not consistent. To converge the noisy manifold, DAE uses a denoising criterion while CAE proposes a contractive penalty. Differing from them, we refine the manifold structure by reversing a diffusion process, which results in an incremental optimization of the model parameter. §.§ Reverse the diffusion process to contract the noisy manifoldThe data manifold in the extracted feature space is noisy, which is obtained by learning with artificially corrupted data, i.e. training samples with manually injected noise. We assume the divergent manifold is caused bya diffusion process from the intrinsic manifold .Consequently, we propose to reverse the specific diffusion process to refine the manifold structure .Formally, given the noisy hidden features = {_1, _2, ..., _nm} , we reverse the diffusion process iteratively by the following equation:∂_t= -γ,where γ is the diffusion constant and t indicates the t-th iteration. = ( - )∈ℝ^N × N is the Laplacian matrix of , whereis the similarity matrix ofand _ii = ∑_j=1^N _ij. Along with the increase in the number of iteration,is inching closer to the data manifold .Using an implicit Euler-scheme,is updated by the following function,^t+1 = ( + δ t γ )^-1^t,where δ t is the time-step and can be chosen arbitrarily.We assume step by step, goes closer to . §.§ Adaptively construct the neighboring graphDuring the process of reversing the diffusion process, its generator, i.e. the graph Laplacian of the neighborhood graph is a key factor.Similarly, it does matter how the neighborhood graph is constructed. A good neighborhood graph can potentially preserve the locality of data manifold. Here we use two alternative strategies to construct the neighboring graph .The first one is the popular method k-nn, which chooses k nearest neighbors for data _i.k-nn performs pretty well in most cases. However, several problems still arises with k-nn, especially when data are concentrated on a non-linear manifold.(1) k-nn assumes data are distributed over a Gaussian distribution. This is often violated by real world data which are concentrated over a complex non-linear manifold. Most Euclidean nearest neighbors are chosen from different data manifolds. (2) When clusters have unbalanced number of training samples, it is not proper to set the same value of k for different clusters.For each data, it would be better to choose itsnearest neighbors automatically according to the intrinsic manifold.Thus, an alternative method is proposed.First, we utilize k-nn to choose relatively large number of neighbors for each data point . These neighbors, denoted as _G(), come from not only the same manifold as , but also different manifolds of other classes. Then we explore a sparse subspace learning method <cit.>to further select these neighbors. The sparse subspace clustering method has turned out to be very effective for discovering data manifold in high dimensional space. It assumes each datais a linear combination of its neighbors within the same cluster. By optimizing a ℓ_1 minimization problem, samples with non-zero coefficients are adaptively selected as neighbors.Thus,the nearest neighbors are finally selected by optimizing the following function:min_ ||- _G() || + λ ||||_1,where ∈ℝ^k is the corresponding sparse coefficient. Points with the non-zeros coefficients, are treated as the neighbors of , denoted as _L(). Combining these two steps can not only select effective neighbors from the same manifold, but remove the neighbors lying in the different manifolds, as the experimental results demonstrate.Thus, the similarity matrixcan be constructed as follows:_ij = {[d(_i, _j) if_j ∈_L(_i);0 else ].where d(_i, _j) measures the similarity between _i and _j, which can be chosen as Gaussian kernel exp^-||_i - _j||^2/2 σ^2 or cosine distance _i^T _j/||_i||||_j|| <cit.>. §.§Impose insensitivity to input noiseTo further learn robust features, we follow the same idea in DAE, i.e. features shouldwell reconstruct clean input from its corrupted copies . In each step, we assume the hidden feature should: (1) contain much information of the input and well reconstructfrom its corrupted versions . (2) approach the intrinsic manifold gradually, i.e. manifold ofis being gradually denoised.Thus, we formulate the proposed method, incremental auto-encoder (InAE) is obtained by combing Eq.<ref> with Eq.<ref>. In each step, features are learned by optimizing the following objective function.θ^⋆_t+1 = min _θ_t+11/nm∑_i = 1^n ∑_j = 1^m l(_i , g(f(_i_j))),^t+1 = ( + δ t γ )^-1^t,where _i_j is the j-th noisy copy of _i and ^t = f(_1^t + _1^t).From the objective function Eq.<ref>, we see: (1) the proposed method explicitly constrains the extracted features, which are prompted to capture the variations of the input; (2) Differing from CAE-like methods, the noise magnitude is not confined to infinitesimal. Thus robustness of features is guaranteed.§ OPTIMIZATION OF THE OBJECTIVE FUNCTIONTo train this model, we rewrite Eq.<ref> as a general regularized function. Following the idea in <cit.>, Eq.<ref> is equivalent to the solution of the minimization of the following regularization problem:Φ(^t+1) = ||^t+1 - ^t||^2_F + (δ t) tr(^t+1^t+1^T),tr(·) computes the trace value.Thus, the objective function in each step becomes:θ^⋆_t+1= min _θ_t+11/nm∑_i = 1^n ∑_j = 1^m l(_i , g(f(_i_j)))+ αΦ(^t+1),where α is the trade-off parameter between the reconstruction error and the process of reversing diffusion process. θ_t+1 = {_1^t+1, _2^t+1, _1^t+1, _2^t+1} contains all the parameters in (t+1)-th iteration.By analyzing the two penalty terms in Eq.<ref>, we see (1) two consecutive updates ofare forced to change smoothly. In other words,comes gradually closer to the manifold , which evades the oscillation phenomenon when optimizing the objective function. (2) close-by points in the original space is rendered to be close in the new feature space. Sincetr(^t+1^t+1^T) = ∑_i,j^N ||^t+1_i - ^t+1_j||^2 _ij,if _i and _j are close, i.e. _ij is large, _i and _j should be close as well. Specifically, the local structure in the data can be maintained.Different from the traditional auto-encoder, the proposed method is trained by an incremental optimization procedure, resulting in a series of parameter updates:^0 →θ_1→^1 →θ_2 …^T-1→θ_T,where T denotes the number of update. Each parameter θ_t+1 is better than the last update θ_t. To obtain each parameter θ, Eq.<ref> is optimized by stochastic gradient descent. Here, we just give a simple description of the first derivative of the last penalty. For clarity, we omit the subscript t+1 and denote the whole number ofas N.First, we compute the derivative w.r.t. each element _1^ij.∂ tr(^T)/∂_1^ij = ∂∑_m,n^N||_m - _n||^2_mn/∂_1^ij = ∂∑_m,n^N _mn(_m^T_m + _n^T_n - 2_m^T_n)/∂_1^ij = 2∑_m^N _mm∂_m^T_m/∂_1^ij - 2∑_m,n^N _mn∂_m^T_n/∂_1^ij As _m is a non-linear function of _m = _1 _m + _1, using the chain rule, Eq.(<ref>) is written as:4∑_m^N(_mm_m_j - ∑_n^N _mn_n_j)∂_m_j/∂_m_j_m_i,where _m_j is the j-th element in _m.Therefore, we get the derivative w.r.t. _1 as follows:∂ tr(^T)/∂_1 = 4 ∑_m^N {_m[(_mm_m- ∑_n^N_mn_n)^T ∘ s'(_m)]^T}where ∘ is the Hadamard product. In this paper, we use the sigmoid function in the encoder, where s'(x) = s(x)(1-s(x)).In summary, the whole training algorithm is described in Algorithm. <ref> §.§ Multiple levels of representationSimilar to the auto-encoder, we treat the proposed method as a building block of forming the deep architecture.Stacking several layers to initialize a deep network works in much the same way as stacking auto-encoders. We stack multiple layers by feeding the outputof l^th layer as input into the (l+1)^th layer. Once one layer is trained, the encoding function f is used to generate uncorrupted input for the next layer.Once a deep architecture has thus been built, its highest level output representation can be used as input to a stand-alone supervised learning algorithm. Experimental results show that the high level representations achieve better performance.§ EXPERIMENTAL RESULTSIn this section, we separate the experiments into model validation on synthetic data and performance comparison on benchmark datasets, showing the prominent locality preserving and discriminative performance of our proposed method. §.§ Model validation on synthetic dataDataset. To validate our proposed method, we generate a moon-like dataset consisting of 2 clusters, each of which is generated from a 2-D function and embedded into a 9-D space with an isotropic Gaussian noise ϵ∼ N(,σ).Evaluation metric. We evaluate our method on the locality preserving ability in two indicators N_ratio and C_ratio, and on discriminative power in classification error.Since the number of neighbours is adaptive, we choose the first k neighbours corresponding to the number in k-nn algorithm. For each data _i, we introduce N_ratio as the ratio of the number of selected neighbours on same manifold of _i to k and C_ratio indicates the percentage of sum of the coefficients of selected neighbours on same manifold. And large values mean good locality preserving ability. The error is compared when k is 30 and iterations number t is 5.Experimental results. In order to show the performance of the proposed InAE, we give simple illustrations in Fig.<ref>.Fig.<ref> (a) and (b) illustrate results on indicators N_ratio and C_ratio of the two strategy for constructing neighboring graph, as described in Section <ref>. We see in (a) when k is small, k-nn and our method perform almost equally well,since the data is relatively large. The difference in N_ratio grows larger as the number of neighbours is larger. Similar result of C_ratio in (b) further convinces the locality preserving ability of our method.Fig.<ref> (c) and (d) show the errors respect to dimensionality and noise scale on several numbers of iterations t. We notice from (c) that higher level representation is more discriminative, and our model is more suitable when the dimension is higher. And (d) shows when Gaussian noise scale σ is small, higher level representations are already able to obtain high classification performance. However, large σ destroys data severely, causing high classification error.Fig.<ref> gives an intuitive denoised results , with t increases,become cleaner and preserve their original manifold at the same time. From Fig.<ref> and Fig.<ref>, we see just 2 or 3 iterations is sufficient to greatly improve the classification accuracy of hidden features. The discrimination power is strengthened at the cost of small computational complexity. §.§ Performance on benchmark datasets Dataset. After the validation of our model on synthetic data, we compare our method with state-of-the-art algorithms on several popular benchmark datasets, i.e. the handwritten digits MNIST dataset and its variants, and CIFAR-10 dataset. The variants of MNIST are generated by imposing various challenging factors <cit.>.The CIFAR-10 dataset consists of 6000 32×32 color images in 10 classes. Evaluation metric. Besides the traditional classification error metric on synthetic data, we employ another measure eig(^-1_w_b), which denotes the maximum eigenvalue of ^-1_w_b, where _w and _b are the with-in-class and between-class variance of , respectively. This measure is inspired from Fisher LDA, which assumes that a discriminative feature should make data in different classes far away, while data in same class close to each other. Large value indirectly indicates the discriminative power of the hidden representations, and better classification performance is expected.Experimental results. We compare our method against the following algorithms on feature extraction: AE (traditional auto-encoder), DAE-b (denosing auto-encoders with masking-out noise), CAE (contractive auto-encoders) and RBM (restricted Boltzmann machine). We use a linear SVM on the raw image pixels as baseline. For all methods but RBM, we use untied weights (i.e. _1≠_2) in each layer and train them using Stochastic Gradient Descent. RBM is trained using Contrastive Divergence, of which the hyper-parameter are chosen by a grid search on a validation set.We represent the classification error and eig(^-1_w_b) on MNIST and CIFAR-10 datasets in Tab.<ref>, and error rate on MNIST variant datasets in Tab.<ref>. Our proposed method achieves best performance in almost all datasets, and Tab.<ref> proves that the metric eig(^-1_w_b) is positively correlated with classification performance. We draw a conclusion that distance between samples provides features valuable information for subsequent tasks. Since the manifold structure of data is important for good performance, you may try to well utilize it before we have a chance. Moreover, Tab.<ref> demonstrates that stacking multiple layers significantly improves performance. § CONCLUSIONSIn this paper, we proposed a novel robust feature learning method by utilizing the Laplacian structure of training data. To learn robust features, it follows the well known trick in machine learning and learns with artificially corrupted data, which are training samples with manually injected noise. First, we assume features should contain much information and well reconstruct the clean input. However, since the data manifold is injected by some noise, this structure can not be consistent in the new feature space, if features are learned merely based on minimization of the reconstruction error. To address this problem, we model the noisy manifold is the result of a diffusion process on the Laplacian graph of training data. Then we reverse this specific diffusion process to denoise the manifold. Each time the diffusion process is reversed, the manifold is refined. This results in an incremental optimization of model parameters.In addition, a new strategy of constructing the neighboring graph of data is introduced. We find that the Incremental Auto-Encoder is capable of contracting the noisy manifold in the feature space.Experimental results on real-world datasets suggest that the Incremental Auto-Encoder performs better than other comparing methods.named | http://arxiv.org/abs/1705.09476v1 | {
"authors": [
"Yanan Li",
"Donghui Wang"
],
"categories": [
"cs.LG",
"cs.CV"
],
"primary_category": "cs.LG",
"published": "20170526083041",
"title": "Learning Robust Features with Incremental Auto-Encoders"
} |
#1 =0.8pt =2.5pt[][email protected][][email protected] School of Physics, Sun Yat-Sen University, 510275, Guangzhou, ChinaMuOn-decay MEdium baseline NeuTrino beam experiment (MOMENT) is a next-generation accelerator neutrino experiment, which can be used to probe new physics beyond Standard Model. We try to simulate neutrino oscillations confronting with Charged-Current and Non-Standard neutrino Interactions(CC-NSIs) at MOMENT. These NSIs could alter neutrino production and detection processes and interfere with neutrino oscillation channels. We separate a perturbative discussion of oscillation channels at near and far detectors, and analyze parameter correlations with the impact of CC-NSIs. Taking δ_cp and θ_23 as an example, we find that CC-NSIs can induce bias in precision measurements of standard oscillation parameters. In addition, a combination of near and far detectors using Gd-doped water cherenkov technology at MOMENT is able to provide good constraints of CC-NSIs happening to the neutrino production and detection processes. 13.15.+g, 14.60.Pq, 14.60.StStudy of Non-Standard Charged-Current Interactions at the MOMENT experiment Yibing Zhang December 30, 2023 =========================================================================== § INTRODUCTION In the past decades, we have seen enormous progress from neutrino oscillation experiments using solar, atmospheric, accelerator and reactor neutrinos <cit.>.In the framework of three neutrino oscillations, there are six physics parameters including three mixing angles θ_12, θ_13, θ_23, one Dirac CP phase δ_cp and two mass squared splittings Δ m^2_31, Δ m^2_21. According to a global analysis of these neutrino oscillation experiments <cit.>, mixing angles θ_12, θ_13 & θ_23 and mass square differences Δ m^2_21 & |Δ m_31^2| have so far been well measured.The mixing angle θ_23, however, has not been determined with enough precision to disentangle whether the mixing angle θ_23 is 45^∘, while many discrete models point to a maximal mixing θ_23=45^∘ with regard to a μ-τ symmetry.In addition, a deviation from θ_23=45^∘ causes an octant degeneracy problem in certain neutrino oscillation channels <cit.>.Nonetheless, the Dirac CP phase describing the difference between matter and anti-matter as well as the sign of Δ m^2_31 (normal mass hierarchy:Δ m^2_31>0; inverted mass hierarchy: Δ m^2_31<0) have not been well constrained yet. Though recent results from T2K <cit.> and NOνA <cit.> disfavor the inverted mass hierarchy at a low confidence level and give hints of δ_CP≈ -90^∘, we expect more data to draw a solid conclusion or further call for the next-generation experiments such as accelerator neutrino oscillation experiments like DUNE <cit.> and T2HK <cit.>, the medium-baseline reactor experiments <cit.> like JUNO <cit.> and RENO-50 <cit.>, atmospheric neutrino experiments like INO <cit.>, PINGU <cit.> and KM3Net <cit.>.Neutrinos are massless in Standard Model (SM) and phenomenon of neutrino oscillations is new physics beyond SM.It is required to generate massive neutrinos by extending SM, such popular candidates as seesaw models, supersymmetry models, extra-dimension models and the like. With more particle contents in new physics models, it might contain the sub-leading effects induced by non-standard neutrino interactions (NSIs) in nature. Effective operators towards this direction have been adopted to link neutrino mass models and NSIs <cit.>. Though there are viable models for sizable NSIs associated with neutral-current interactions <cit.>, it is still a tough task to model CC-NSIs surviving the overwhelming constraints from precision measurements of charged lepton properties. Several studies have been conducted on NSIs from the experimental and model-building point of view <cit.>. A review of NSIs is given in detail in <cit.>. With the help of an effective field theory, we can generally integrate out the mediator/propagator in the Feynman diagram and keep four fermions contact with each other. New physics scale is then embedded into the effective coupling constant ϵ_α^'β^'^αβ where α/β or α^'β^' are the related fermion flavours. In theory the higher the new physics scale, the harder it is to reach a small effective coupling constant. We have reached an era of precision measurements of neutrino mixing parameters after an establishment of neutrino oscillation. It is promising for us to develop better neutrino detectors to search for sub-leading NSIs in the current and next-generation neutrino oscillation experiments as a complementary to the new physics search with the high intensity machine at the collider. The MuOn-decay MEdium baseline NeuTrino beam experiment (MOMENT) is a next-generation accelerator neutrino experiment proposed for discovery of leptonic CP violation <cit.>. The atmospheric neutrino flux is a serious hindrance to the study of CP violation at MOMENT. Backgrounds caused by the atmospheric neutrinos exceeds the oscillation signal events significantly at O(100) MeV. Neutrino beams from such a continuous proton accelerator provide high luminosity fluxes but result in a loss of timing information which is traditionally used to suppress atmospheric neutrino backgrounds in the accelerator neutrino oscillation experiment with the pulsed proton beam facility. A new detector technology, however, might overcome the barrier and offer precision tests of θ_23. The new detection technology might also lead to a discovery of the CP violating phase in the framework of 3-flavour neutrino oscillations <cit.>, which complement the study at T2K and NOνA to solve the degeneracy problem and exclude CP conserved phase at a relatively high confidence level. In addition, a comprehensive study of the bounds on NSI parameters has been carried out. The bounds on NSI parameters governing the neutrino productions and detections are about one order of magnitude stronger than those related to neutrino propagation in matter, taking the current bounds on ϵ^ud and ϵ^μ e as an example <cit.>:|ϵ^μ e|< ( [ 0.0250.030.03; 0.0250.030.03; 0.0250.030.03; ]) 90%C.L. |ϵ^u d|< ( [ 0.041 0.025 0.041; 0.026 0.078 0.013;0.12 0.0130.13; ]) 90%C.L.A special case study was performed at the Daya Bay reactor neutrino experiment to constrain NSI parameters where neutrinos are produced by beta decays and detected by inverse beta decays <cit.>. The relevant ee sector of ϵ^μ e got an upper limit of O(10^-3). With the help of a perturbation theory, neutrino oscillation probabilities in the presence of source/detector and matter NSIs can be found in the reference <cit.>, which is motivating further study and optimization of new experimental proposals to pin down the current bounds. The first glimpse of NSI effects during neutrino propagation in matter at MOMENT has been shown in the reference <cit.>. It has discussed the sensitivity of neutral-current NSIs by means of accelerator neutrino oscillations in matter. However, the impact of source and detector NSIs associated with charged-current interactions has not been discussed. Within a theoretical model predicting new neutrino interactions, it is natural and fair for neutrinos to carry new charged-current and neutral-current interactions during the production process, the propagation process and the detection process. Furthermore, the current neutrino experiment T2K and NOνA are based on the superbeam neutrino production where neutrinos come from pion decays. We have to stress that NSIs associated with muon decays are very different from those happening at pion decays if we take the new physics into account. Therefore, it is necessary to bring source/detector NSIs for a complete analysis at MOMENT where neutrinos are produced by muon decays.In this work, we explore the charged current NSIs effects at MOMENT. We focus on the precision measurement of standard neutrino mixing parameters and constraints of NSI parameters in the presence of non-standard charged-current interactions at the source and detector. The paper is organized as follows: we discuss neutrino oscillation channels at a short and long distance in Section <ref>. In Section <ref>, we describe our implementations of MOMENT and details in simulation. In Section <ref>, we show the impacts of NSIs on precision measurements of standard neutrino parameters and present the correlations and constraints of NSI parameters within production and detection at MOMENT, and compare the expected results with current bounds. The summary follows in Section <ref>.§ DISCUSSION OF NEUTRINO OSCILLATION CHANNELS The formalism of NSI is a general way of studying the impacts of new physics in neutrino oscillations.Without dealing with the matter NSIs due to the short baseline, we start with the neutrino production and detection processes involving non-standard interactions. These processes are often related to the charged lepton and called the charged-current-like NSIs.The neutrinos at the MOMENT experiment are produced by the muon decay processes μ^-→ e^-+ν̅_e+ν_μ and μ^+→ e^+ +ν_e +ν̅_μ and are detected mainly through quasielastic charged-current interactions: ν_ℓ+n→ p+ℓ^- and ν̅_ℓ+p→ n + ℓ^+ (Here ℓ denotes e or μ) in the neutrino detector. The CC-NSIs imposed on the production and detection are two different types: the NSIs involved in the muon decay production process are related to charged leptons, while the NSIs involved in the detection process are associated with quarks. For simplicity, we have restricted theoperators to (V-A)(V-A) Lorentz structure and neglected NSIs including right handed neutrinos, where the process is helicity suppressed. One may ask why we ignore other potential Lorentz structures, since the most general way of constructing the four-fermion interactions could come with the current of (V± A)(V± A), (S± P)(S± P) and TT, where V stands for vector couplings, A for axial-vector couplings, S for scalar couplings, P for pseudo-scalar couplings and T for the tensor couplings. There have been several attempts in the literature towards the chirality discussion of NSIs (see e.g. <cit.>). Except (V-A)(V-A), other structures are either helicity suppressed or very small due to their contributions by higher order corrections. Therefore, the most interesting CC-NSIs could be parameterized as the effective four-fermion Lagrangians at the detector: ℒ_CC-NSI^d = G_F/√(2)ϵ_αβ^d[ν̅_βγ^μ(1-γ^5)ℓ_α][f̅γ_μ(1-γ^5)f']+h.c.where the superscript “d” represents the interactions at the detector, G_F is the Fermi constant, f (f') represents d (u) quark, α (α = e, μ, τ) is neutrino index and β (β = e, μ) is lepton index. The NSIs at the detector are parametrized by ϵ_αβ^d which give the strength of NSIs relative to G_F.The detector projects the neutrino wave function not only onto the standard weak eigenstates, but onto a combination of them:⟨ν_β^d| = ⟨ν_β|+∑_α=e,μ,τϵ^d_αβ⟨ν_α| In a muon-decay accelerator neutrino experiment, the neutrinos are produced by the muon decay process μ^+→ e^+ +ν_e+ν̅_μ and the charge conjugatedprocess in the Standard Model. The effective lagrangian involving NSIs in production processes can be expressed as: ℒ_CC-NSI^s = G_F/√(2)ϵ_γδ^s[ν̅_δγ^μ(1-γ^5)ℓ_γ][f̅γ_μ(1-γ^5)f^']+h.c.where the superscript “s” represents the interactions at the source and two fermion fields “f”and “f^'” stand for a neutrino or a charged lepton in order to avoid the confusion of the neutrino flavour index and the index related to the γ matrix. For simplicity, we assume that the dominant NSI processes can interfere coherently with standard oscillations. Here we only consider two separate cases (take the μ^+ decay as an example): (1) μ^+→ e^+ + ν_α + ν̅_μ for NSIs ϵ^s_e α with any flavour (α=e,μ,τ) by fixing ν̅_μ, where we assume f=μ, f^'=ν_μ, ℓ_γ=e.(2) μ^+→ e^+ + ν_e + ν̅_α for NSIs ϵ^s_μα with any flavour (α=e,μ,τ) by fixing ν_e, where we assume f=e, f'=ν_e, ℓ_γ=μ.Otherwise, the incoherent process μ^+→ e^+ + ν_α+ν̅_β (α,β=e,μ,τ) might occur with arbitrary choice of ν_α and ν̅_β <cit.>. However, those incoherent contributions to the probabilities are very small since they are suppressed by at least an order of |ϵ|^2 where the SM weak interactions are completely replaced by two NSI vertices. Similarly, the neutrino flavour states produced at the source can be written as superpositions of pure flavour eigenstates:|ν_δ^s⟩ = |ν_δ⟩+∑_γ=e,μ,τϵ^s_δγ|ν_γ⟩Thus, the oscillation probability is given by:[ P(ν_δ^s →ν_β^d) = |⟨ν_β^d |e^-i ℋ L|ν_δ^s⟩|^2; = |(1+ϵ^d)_ηβ(e^-i ℋ L)_ηλ(1+ϵ^s)_δλ|^2; =|[(1+ϵ^d)^T e^-i ℋ L(1+ϵ^s)^T]_βδ|^2 ]Here the Hamiltonian takes the form of ℋ=Udiag(E+m_1^2/2E, E+m_2^2/2E, E+m_3^2/2E)U^† and U is the PMNS mixing matrix relating the neutrino flavour eigenstates to mass eigenstates |ν_α⟩ =∑_i U^*_α i|ν_i⟩. The ϵ^s and ϵ^d are the charged-current NSI matrices for the production and detection, respectively. There are 18 NSI real parameters in total because each complex element ϵ_αβ^s/d consists of the amplitude |ϵ_αβ^s/d| and the phase ϕ_αβ^s/d.In principle, accelerator neutrinos pass through the earth matter until reaching a detector and matter effects change the oscillation probability. However, MOMENT is a medium baseline experiment and the matter effects are relatively small. Meanwhile, the main topic in the current study is delivered to NSIs happening at the production and detection processes rather than NSIs in matter. Therefore, we will only, for sake of simplicity, display the probabilities perturbatively in vacuum for related appearance and disappearance channels and try to extract useful information for physics performance study. Of course, matter effects are taken into account simultaneously in the complete simulation for physics performance. Since near and far detectors will be used in the simulation later, we will discuss the oscillation channels at a short and far distance separately. §.§ Oscillation channels at a near detectorHere a near detector means detecting neutrinos at a distance of O(100) meters. In the standard oscillation frame without non-standard interactions, ν_μ(ν̅_μ) and ν_e(ν̅_e) can not develop neutrino oscillation patterns in such a short distance and their probabilities are equal to 1. However, NSIs are able to generate zero-distance effects so that the disappearance probabilities are allowed to be larger than 1, equal to 1, or smaller than 1. After dropping the terms O(ϵ^2), we approximate the probabilities asnear_pee_antipee and near_pmm_antipmm:P^ND_ν_e^s→ν_e^d(P^ND_ν̅_e^s→ν̅_e^d) ≈ 1 + 2|ϵ_ee^s|cosϕ_ee^s+2|ϵ_ee^d|cosϕ_ee^d P^ND_ν_μ^s→ν_μ^d(P^ND_ν̅_μ^s→ν̅_μ^d) ≈ 1 + 2|ϵ_μμ^s|cosϕ_μμ^s+2|ϵ_μμ^d|cosϕ_μμ^dIt is easy to see that P(ν̅_e^s →ν̅_e^d)/P(ν̅_μ^s →ν̅_μ^d) deviates from unity with some constant terms in the presence of relevant NSI parameters ϵ_ee^s and ϵ_ee^d (ϵ_μμ^s and ϵ_μμ^d). If neutrinos are produced with charged lepton decays and detected by identifying the same charged leptons, the contribution of ϵ_ee^s to the probability is equivalent to ϵ_ee^d and then the sensitivity to these two parameters should be the same at the near detector.Similarly, the appearance channels ν_μ→ν_e (ν_e→ν_μ) must remain zero in the standard 3-flavor neutrino scenario. After dropping the O(ϵ^3) and O(ϵ^4) terms, the expressions for ν_e and ν_μ appearanceprobabilities including CC-NSIs can be written as near_pme_antipme and near_pem_antipem, respectively:P_ν_μ^s →ν_e^d^ND(P_ν̅_μ^s →ν̅_e^d^ND) ≈ |ϵ_μ e^s|^2 + |ϵ_μ e^d|^2 + 2|ϵ_μ e^s| |ϵ_μ e^d|cos(ϕ_μ e^s - ϕ_μ e^d) P_ν_e^s →ν_μ^d^ND(P_ν̅_e^s →ν̅_μ^d^ND) ≈ |ϵ_eμ^s|^2 + |ϵ_eμ^d|^2 + 2|ϵ_eμ^s| |ϵ_eμ^d|cos(ϕ_eμ^s - ϕ_eμ^d)Here the ν_e(ν_μ) appearance probability would depend on ϵ_μ e^s and ϵ_μ e^d(ϵ_eμ^s and ϵ_eμ^d) after we introduce the NSIs at the neutrino source and detector. The probability of each oscillation channel and its conjugate partner shares the same form at the near detector.In fact, there are only four effective channels even though eight neutrino oscillation channels can get involved in the MOMENT experiment. It is a discovery of new physics to observe zero-distance effects at near detectors for disappearance or appearance channels.§.§ Oscillation channels at a far detector Oscillation patterns get more complicated as soon as we consider channels suitable for the far detector at MOMENT. In the standard framework describing three neutrino mixings, the probability ofν_μ→ν_e channel is calculated by a simple change of sign of the sinδ term in a T-reversed channel of ν_e→ν_μ. Due to CC-NSIs, ν_μ→ν_e and ν_e→ν_μ probabilities are not so obvious any more. We perturbatively derive the explicit expressions of their probabilities in vaccum as given in far_pme_invacuum and far_pem_invacuum, considering α=Δ m_21^2/Δ m_31^2≈0.03, s_13=sinθ_13≈0.15 and NSI parameters as small numbers. In order to clearly show the impacts of NSIs, we can split the P_ν_μ→ν_e^FD (P_ν_e→ν_μ^FD) into a sum of three terms: the standard oscillation termP_ν_μ→ν_e^SM (P_ν_e →ν_μ^SM), the dominant order of 𝒪(ϵ s_13) NSI oscillatory term P_ν_μ→ν_e^NSI(ϵ s_13) (P_ν_e →ν_μ^NSI(ϵ s_13)) and the sub-dominant order of 𝒪(αϵ) NSI oscillatory term P_ν_μ→ν_e^NSI(αϵ) (P_ν_e →ν_μ^NSI(αϵ)).For an oscillation channel of ν_μ→ν_e, the probability can be written as:P_ν_μ→ν_e^FD = P_ν_μ→ν_e^SM+ P_ν_μ→ν_e^NSI(ϵ s_13)+P_ν_μ→ν_e^NSI(αϵ)+𝒪(α^3)+𝒪(α^2s_13)+𝒪(α s_13^2)+𝒪(s_13^3)+𝒪(ϵα^2) +𝒪(ϵ s_13^2)+𝒪(ϵ^2), withP_ν_μ→ν_e^SM≈ s_2×13^2 s_23^2 sin^2Δ_31+αΔ_31s_2×12s_2×23s_13[sin(2Δ_31)cosδ - 2sinδsin^2Δ_31]+α^2Δ_31^2c_23^2s_2×12^2,P_ν_μ→ν_e^NSI(ϵ s_13)≈-2s_2×13s_23[|ϵ_μ e^s|cos(δ+ϕ_μ e^s)+c_2×23|ϵ_μ e^d|cos(δ+ϕ_μ e^d)-s_2×23|ϵ_τ e^d|cos(δ+ϕ_τ e^d)] sin^2Δ_31 -s_2×13 s_23[|ϵ_μ e^s|sin(δ+ϕ_μ e^s)+|ϵ^d_μ e|sin(δ+ϕ^d_μ e)]sin(2Δ_31),P_ν_μ→ν_e^NSI(αϵ)≈+2αΔ_31 |ϵ_μ e^d| s_2×12 c_13 c_23 s_23^2 cosϕ_μ e^d sin(2Δ_31)+2αΔ_31 |ϵ_τ e^d| c_13 c_23^2 s_2×12 s_23cosϕ_τ e^d sin(2Δ_31) -2αΔ_31 |ϵ_μ e^d| s_2×12 c_13 c_23sinϕ_μ e^d (1-2 s_23^2 sin^2Δ_31)+4αΔ_31 |ϵ_τ e^d| c_13 c_23^2 s_2×12 s_23sinϕ_τ e^d sin^2Δ_31 -2αΔ_31 |ϵ_μ e^s| c_13 s_2×12 c_23sinϕ_μ e^s.In a similar way, the probability of ν_e→ν_μ can be expressed as:P_ν_e →ν_μ^FD = P_ν_e →ν_μ^SM+ P_ν_e →ν_μ^NSI(ϵ s_13)+P_ν_e →ν_μ^NSI(αϵ)+𝒪(α^3)+𝒪(α^2s_13)+𝒪(α s_13^2)+𝒪(s_13^3)+𝒪(ϵα^2) +𝒪(ϵ s_13^2)+𝒪(ϵ^2), withP_ν_e →ν_μ^SM≈ s_2×13^2 s_23^2 sin^2Δ_31+αΔ_31s_2×12s_2×23s_13[sin(2Δ_31)cosδ+2sinδsin^2Δ_31]+α^2Δ_31^2c_23^2s_2×12^2,P_ν_e →ν_μ^NSI(ϵ s_13)≈-2s_2×13s_23[|ϵ_eμ^d|cos(δ-ϕ_eμ^d)+c_2×23|ϵ_eμ^s|cos(δ-ϕ_eμ^s)-s_2×23|ϵ_eτ^s|cos(δ-ϕ_eτ^s)]sin^2Δ_31 +s_2×13s_23[|ϵ_eμ^s|sin(δ-ϕ_eμ^s)+|ϵ_eμ^d|sin(δ-ϕ_eμ^d)]sin(2Δ_31),P_ν_e →ν_μ^NSI(αϵ)≈ +2αΔ_31 |ϵ_eμ^s| s_2×12 c_13 c_23 s_23^2 cosϕ_eμ^s sin(2Δ_31)+2αΔ_31 |ϵ_eτ^s| c_13 c_23^2 s_2×12 s_23cosϕ_eτ^s sin(2Δ_31) -2αΔ_31 |ϵ_eμ^s| s_2×12 c_13 c_23sinϕ_eμ^s (1-2s_23^2 sin^2Δ_31)+4αΔ_31 |ϵ_eτ^s| c_13 c_23^2 s_2×12 s_23sinϕ_eτ^s sin^2Δ_31 -2αΔ_31 |ϵ_eμ^d| c_13 s_2×12 c_23sin(ϕ_eμ^d). Here s_ij=sinθ_ij, c_ij=cosθ_ij, s_2× ij=sin2θ_ij, c_2× ij=cos2θ_ij, Δ_31=Δ m_31^2 L/4 E, α=Δ m_21^2/Δ m_31^2 and ϵ_αβ^s/d=|ϵ_αβ^s/d| e^iϕ_αβ^s/d. Our derived oscillation probabilities are consistent with the reference <cit.>. We can immediately read the impacts of NSI parameters on appearance channels in order:* It is clear that ν_μ→ν_e (ν_e→ν_μ) is affected by the dominant NSI parameters ϵ_μ e^s, ϵ_μ e^d and ϵ_τ e^d (ϵ_eμ^s, ϵ_eτ^s and ϵ_eμ^d). When the dominant term of 𝒪(s_13ϵ) is considered, turning on ϵ_μ e^s and ϵ_τ e^d (ϵ_eμ^d and ϵ_eτ^s) is approximately equivalent to enlarging or compressing the amplitude of sin^2Δ_31. Meanwhile, the term of ϵ_μ e^d (ϵ_eμ^s) will change the maximal position of its oscillation probability. * When θ_23 approaches 45 degrees, s_2×23≈1 and c_2×23≈0. Thus ϵ_μ e^d (ϵ_eμ^s) would lose the effects on the term of sin^2Δ_31 related to 𝒪(ϵ s_13) for the ν_μ→ν_e (ν_e→ν_μ) channel.* Since these NSI parameters are entangled with standard mixing parameters, NSIs would interfere with precision measurements of the standard CP-violating phase δ and θ_23 which is manifested by related terms in far_pme_invacuum and far_pem_invacuum. These two channels are sensitive to the octant of θ_23 beacuse the leading order of far_pme_invacuum and far_pem_invacuum depend on sin^2θ_23. The presence of NSIs may induce a wrong determination of θ_23. Fig. <ref> shows a specific situation of the degeneracy caused by NSIs at the probability level. Thecontinuous line and the dashed line show the case of θ_23=41.6^∘ and θ_23=48.4^∘ without NSI, respectively. We can see a clear separation between two scenarios. However, introducing NSIs can shift the amplitude of the probability and fake the contribution of θ_23. For instance, the dotted line shows the special case with |ϵ_μ e^s|=0.02 and ϕ_μ e^s=π/2. It is clear that the dotted oscillation curve almost coincides with the continuous curve. This indicates that we may get the wrong measurement of θ_23, if there is indeed CC-NSIs but we ignore them by only fitting the data within the standard neutrino oscillation framework. * An assumption of real NSI parameters with all NSI phases ϕ=0 could leads to a substantial simplification in each channel, since those terms proportional to sinϕ_αβ^s/d would vanish. One can observe that the sensitivity to ϵ_τ e^d (ϵ_eτ^s) from the oscillation channel ν_μ→ν_e (ν_e→ν_μ) would be tiny for δ = 3π/2, π/2. Moreover, we can obtain a linear correlation of ϵ_μ e^s and ϵ_μ e^d (or ϵ_eμ^s and ϵ_eμ^d). In the case of δ=0/π, sinδ term will vanish and we can get the linear correlation of ϵ_μ e^s, ϵ_μ e^d and ϵ_τ e^d (ϵ_eμ^s, ϵ_eμ^d and ϵ_eτ^s). Futhermore, if θ_23 is close to 45 degrees, the contributions from ϵ_μ e^d (ϵ_eμ^s) tend to disappear. In brief, the sensitivities to the related CC-NSI parameters at a far detector vary with the standard parameters θ_23 and δ.Similar to the discussion above, ν_μ and ν_e disappearance probabilities are given as far_pmm_antipmm and far_pee_antipee:P_ν_μ^s→ν_μ^d^FD≈ 1 -s_2×23^2sin^2Δ_31 +2|ϵ_μμ^s|cosϕ_μμ^s+2|ϵ_μμ^d|cosϕ_μμ^d +s_2×23(|ϵ_μτ^s|sinϕ_μτ^s+|ϵ_τμ^d|sinϕ_τμ^d)sin(2Δ_31) -2s_2×23^2(|ϵ_μμ^s|cosϕ_μμ^s+|ϵ_μμ^d|cosϕ_μμ^d)sin^2Δ_31 -2c_2×23s_2×23(|ϵ_μτ^s|cosϕ_μτ^s+|ϵ_τμ^d|cos_τμ^d)sin^2Δ_31 P_ν_e^s→ν_e^d^FD≈ 1-4s_13^2sin^2Δ_31+2|ϵ_ee^s|cosϕ_ee^s+2|ϵ_ee^d|cosϕ_ee^d -2s_13s_23[|ϵ_eμ^s|sin(δ-ϕ_eμ^s)-|ϵ_μ e^d|sin(δ+ϕ_μ e^d)]sin(2Δ_31) -2s_13c_23[|ϵ_eτ^s|sin(δ-ϕ_eτ^s)-|ϵ_τ e^d|sin(δ+ϕ_τ e^d)]sin(2Δ_31)-4s_13s_23[|ϵ_eμ^s|cos(δ-ϕ_eμ^s)+|ϵ_μ e^d|cos(δ+ϕ_μ e^d)]sin^2Δ_31 -4s_13c_23[|ϵ_eτ^s|cos(δ-ϕ_eτ^s)+|ϵ_τ e^d|cos(δ+ϕ_τ e^d)]sin^2Δ_31Apart from standard neutrino oscillations, major contributions come from the terms proportional to ϵ_μμ^s/d (ϵ_ee^s/d) rather than other NSIs in the ν_μ (ν_e) disappearance channel. We expect better constraints on ϵ_μμ^s/d (ϵ_ee^s/d).The ν_μ→ν_μ (ν̅_μ→ν̅_μ) is an important channel to measure θ_23, which is expected to judge whether θ_23 is maximal or not. The channel of ν_e→ν_e (ν̅_e→ν̅_e) is good at precision measurements of θ_13. Without NSIs, ν_e (ν̅_e) disappearance channel has no dependence of the standard CP phase. After introducing NSI parameters, however, even the standard CP-violating phase would appear in the ν_e (ν̅_e) disappearance probability. Under the assumption of ϕ=0, we will have a rather simplified correlation of standard neutrino oscillation and NSI parameters. A few comments for this special case are given as follows:* The oscillation channel ν_μ→ν_μ (ν̅_μ→ν̅_μ) is affected by |ϵ_μμ^s/d|, |ϵ_μτ^s| and |ϵ_τμ^d|. Apart from standard neutrino oscillation terms, the major contributions could be associated with the constant terms of |ϵ_μμ^s/d|. When θ_23 is getting closer to 45 degrees, the coefficient of |ϵ_μμ^s/d| should be much larger than the coefficient of |ϵ_μτ^s| (|ϵ_τμ^d|). Therefore, this channel is more sensitive to |ϵ_μμ^s/d| rather than |ϵ_μτ^s| and |ϵ_τμ^d|. * Parameters ϵ_ee^s/d, ϵ_eμ^s, ϵ_μ e^d, ϵ_eτ^s and ϵ_τ e^d affect the ν_e (ν̅_e) disappearance probability. ϵ_ee^s/d have the most important impacts on the channel. The sensitivities to ϵ_eμ^s, ϵ_μ e^d, ϵ_eτ^s and ϵ_τ e^dare smaller and depend on δ. On the contrary, these four parameters can be well constrained in appearance channels.In Fig. <ref>, we make a comparison of the probabilities P(ν_μ→ν_e) and P(ν_e→ν_μ) with/without NSIs. We turn on non-vanishing parameters of |ϵ_μ e^s|=|ϵ_μ e^d|=|ϵ_τ e^d|=0.03 for an illustration. The amplitude of oscillation pattern is shifted significantly by these NSI impacts as we point out from the perturbative approximation. We display the probability curve of standard three-flavour neutrino oscillation and the probability bands originating from the variations of NSI parameters. The shaded region in Fig. <ref> shows changes of oscillation probabilities when the standard CP-violating phase is 3π/2 and the NSI phases vary in [-π, π]. It is easy to see the difference between P(ν_μ→ν_e) and P(ν_e→ν_μ) without NSI effects, where P(ν_μ→ν_e)-P(ν_e→ν_μ) becomes negative for δ∈(0, π) and P(ν_μ→ν_e) is larger than P(ν_e→ν_μ) for δ∈(π,2π); with δ=0/π, there is no difference between P(ν_μ→ν_e) and P(ν_e→ν_μ). Once we turn on NSI parameters, however, the relationship between the probabilities of these two channels will be changed significantly as can be seen from the shaded regions. After we introduce the characteristics of MOMENT and simulation details in the next section, we will be further convinced by an analysis of event rates as shown in events. § CHARACTERISTICS OF MOMENT AND THE DETAILS OF SIMULATIONContinuous high-energy proton beam at 1.5 GeV with a power of 15 MW will be produced in the accelerator facility. A choice of target stations with fluidised tungsten is still under optimization to generate charged mesons most of which are pions and kaons. We can expect 1.1×10^24 Proton On Target (POT) per year. A magnetic solenoid will be deployed to make the pion beam focused and selected. The curvature of the solenoid helps with selecting muons from pion decays, followed by a straight tunnel to prepare neutrinos from muon decays. The neutrino fluxes are provided by the accelerator working group in MOMENT for our physics performance study <cit.>.We intend to extract more information from eight oscillation channels using the muon-decay neutrino beams in the simulation study: ν_e→ν_e, ν_e→ν_μ, ν_μ→ν_e, ν_μ→ν_μ and their conjugate partners. Since we have to conduct flavour and charge identifications to distinguish secondary particles, we consider the new technology using Gd-doping water to separate both Cherenkov and coincident signals from capture of thermal neutrons. Muon taggings can be efficiently obtained by daughter electrons together with pulse shape discrimination of waveforms. We follow the detector description from a sophisticated study in the CERN-MEMPHYS project <cit.> and update the related new technology with regard to Gd-doping water. The major backgrounds for MOMENT come from atmospheric neutrinos. We believe that they can be suppressed by the beam direction and proper modelling background spectra within the beam-off period, which is to be extensively studied in detector simulations in future study. Charged-current interactions are used to identify neutrino signals: ν_e + n → p + e^- ν̅_μ + p → n + μ^+ ν̅_e + p → n + e^+ ν_μ + n → p + μ^-A few remarks for signals and backgrounds are given as follows:* Gd doping into pure water could be used to discriminate electron neutrinos and antineutrinos by whether there is a capture of the scattered thermal neutron or not. Neutron capture on Gd emits the 8 MeV gamma rays. The ν̅_e (ν̅_μ) signal is reconstructed by tagging the neutron in coincidence with the positron to suppress most of backgrounds associated with single events. While water Cherenkov detection is not significantly changed, ν_e signals come from the Cherenkov ring created by ν_e elastic scattering with electrons. * In a Water Cherenkov detector, electron and muon-flavour neutrinos are well separated by event reconstructions, where the former type creates electron showers and the latter type leads to muon tracks. Sometimes, low-energy muons decay and cause flavour misidentifications. Here we ignore the flavour misidentification in the simulation. * The imperfection of detectors leads to misidentifications of charge-current interactions for neutrino signals. Here we suppose their effects are negligible. * It is also possible for neutral-current interactions with accidental single events to be identified as the coincident signal. A neutron knocks a nucleus off of an oxygen, resulting in excited states and photons from de-excitations will mimick ν̅_e signals oscillated from accelerator neutrino beams. We expect them to be rather small and assign an extremely small background over signal ratio in our simulation.Table <ref> lists the simulation details about the neutrino detector. A baseline of 150 km is assigned in the current proposal based on the neutrino beam energy range <cit.>. We assume a near detector with a fiducial mass of 100 t and a far detector with a fiducial mass of 500 kton. The running time is 5 years for each polarity. In the massive water cherenkov detector, we follow electron and muon selection efficiencies given in Ref. <cit.>. With regard to the normalization error on signals and the normalization error on backgrounds, we assume they are at the level of 5%. As for the atmospheric backgrounds, they could be suppressed via sending the neutrino beam in short bunches with a suppression factor of 2.2×10^-3 <cit.>.The cross section for quasi-elastic interactions is taken from the reference <cit.>. The values of the standard neutrino oscillation parameters are taken from the latest nu-fit results <cit.>. Table <ref> shows the central values and their uncertainties in the present work. Unless otherwise mentioned, we expect a determination of mass hierarchy without NSIs before running MOMENT and assume the normal mass hierarchy in our simulation without a loss of generality, i.e. Δ m_31^2>0. A list of assumptions for near and far detectors are given in Table <ref>. We present numerical results by simulating the neutrino oscillation signals and backgrounds using GLoBES <cit.>. Similar to the probability-level analysis given in probband, we present event rates of ν_μ→ν_e and ν_e→ν_μ channels versus the neutrino energy in events. The event spectra are shifted significantly after we consider CC-NSI effects. The shaded region highlights the large variation from the new CP phases caused by CC-NSIs even if we fix their strength of NSI couplings. It is then straightforward to discuss physics performance for the MOMENT experiment with simulated event spectra. We calculate the event rates by defining the true values (central values) for standard oscillation parameters and fit with/without NSI impacts to extract useful information. We compute the χ^2 using the following approach:χ^2=[∑_j^channel∑_i^bin|N_ij(ρ_true,ϵ_true)-N_ij(ρ_test,ϵ_test,s)|^2/N_ij(ρ_true,ϵ_true)+∑_α(ρ_α-ρ_α^true)^2/σ^2_ρ_α+∑_β(s_β-s_β^true)^2/σ^2_s_β]_min,where the index j denotes the channel number and i denotes the bin number, ρ_true and ρ_test represent the standard vectors of true and test values, respectively, s is the vector of systematics related to the neutrino beam and detector,ϵ_true and ϵ_test are the non-standard vectors of true values and test values, and N_ij is the expected event for the j-th channel and i-th bin. The second term corresponds to the contribution to χ^2 from the external inputs which are based on results from previous experiments. σ_ρ_α is the external error (input error) in GLoBES imposed on the central values. Similarly, the third term represents the treatment of systematic errors implemented on the χ^2.§ PHYSICS PERFORMANCE OF MOMENT§.§ Impacts on precision measurements of standard mixing parameters by CC-NSIs The CKM mixing matrix is well measured in the quark sector at the sub-percent level <cit.>, while mixing parameters in the lepton sector are far away from such a precision. It is very likely for the next-generation experiment like MOMENT to achieve the goal of doing precision measurements. In Fig. <ref>, we have showed that CC-NSIs may induce a bias in precision measurements of θ_23 at the probability level. In this section, we take the Dirac CP-violating phase δ_cp and θ_23 as an illustration to show the impacts from CC-NSIs after the simulation. The true value of θ_23 is taken as 41.6^∘. We choose two δ values with δ=π/2 and δ=3π/2 to simulate all oscillation channels at MOMENT and fit the neutrino spectra with/without NSIs. Fig. <ref> demonstrates the numerical results. The true values of the standard oscillation parameters are shown by a red point in each panel. In all sub-figures (a), (b) and (c), we have considered uncertainties of standard mixing parameters.Panels (a1) and (a2) show the determination of δ_cp and θ_23 in the case of the standard neutrino oscillation without NSIs. By running MOMENT, we can determine the mixing angle θ_23 with an error bar of one degree at the 3σ confidence level, while the precision for δ_cp is good enough. In sub-figure (b) and (c), NSIs happening at the source and detector are turned on. All the corresponding CC-NSI phases can vary within (0, 2π). In the panel (b1) and (b2), we only consider the CC-NSIs (ϵ_eμ^s, ϵ_eτ^s and ϵ_eμ^d, and their marginalization ranges are allowed within the current bounds given in bounds1 and bounds2.) which are related to ν_μ (ν̅_μ) appearance channels. The panels of (b1) and (b2) show the enlarged uncertainties in parameter fittings. Especially, a degeneracy pops up in the measurement of θ_23-δ_cp for the case of δ=3π/2, while it is still safe for the case of δ=π/2. Furthermore, we go to panels (c1)/(c2) by turning on those CC-NSIs related to the ν_e (ν̅_e) and ν_μ (ν̅_μ) appearance channels (ϵ_e μ^s, ϵ_e τ^s, ϵ_e μ^d, ϵ_μ e^s, ϵ_μ e^d, ϵ_τ e^d and their marginalization ranges are within current boundsgiven in bounds1 and bounds2). As we have discussed in Section <ref>, the ν_e appearance channel is affected by the parameters: ϵ_μ e^s, ϵ_μ e^d, ϵ_τ e^d, while the ν_μ appearance channel is mainly determined by the parameters ϵ_eμ^s, ϵ_eτ^s, and ϵ_eμ^d. This feature can be understood after a closer look at the far_pem_invacuum. If δ is equal to π/2, the peak of the probability in ν_e→ν_μ channel is much larger than the case of δ=3π/2. In turn, the event rate in the detector for δ=π/2 is much higher. The corresponding fitted results are much better. Therefore, CC-NSI parameters destroy the precise determinations of standard mixing parameters. As can be seen from panels (c1) and (c2), the degeneracy even shows up at high confidence levels when we consider all the relevant NSIparameters. We might get into the wrong best-fit region if we neglect the CC-NSIs from the new physics. A combination of different neutrino oscillation experiments might resolve such an ambiguity and finish the task of precision measurements of neutrino mixing parameters to the same level as quark mixing parameters. Or we need a more powerful machine, such as a neutrino factory. §.§ Correlations and constraints of NSI parameters We have introduced the NSIs by integrating the potential heavy propagator from the new physics scale based on the effective theory. Each NSI parameter has a magnitude which tells us the strength of new couplings and its associated phase to bridge the CP violating story. Therefore, it is convenient to adopt two methods: one is to take the NSI parameters as real, which corresponds to the strength of the coupling constant or switch off the NSI-induced CP violation phases; the other is to keep general assumptions given complex NSI parameters. In this section we discuss the constraints on source and detector NSIs from the far and near detector, respectively. For the former case, Table <ref> demonstrates the sensitivity of MOMENT in constraining the NSI parameters using the single-parameter-fit at 90% C.L. The results are obtained by using all the neutrino and antineutrino oscillation channels with both near and far detectors. In a comparision of the current bounds and the expected limits from our simulation, we find that the MOMENT experiment with running time of 5+5 years has a potential to improve the constraints for the CC NSIs. In Table <ref>, one can observe that most of the bounds for NSIs are improved except for ϵ_eτ^s, ϵ_μτ^s, ϵ_τ e^d and ϵ_τμ^d which are marked by red fonts. These results also confirm our observations in Section <ref>:* NSI parameters ϵ_ee^s/d and ϵ_μμ^s/d, at the leading order of NSI terms in the disappearance channels can be well constrained with a combination of near and far detectors.* ϵ_eτ^s, ϵ_μτ^s, ϵ_τ e^d and ϵ_τμ^d cannot be well constrained by the near detector due to their negligible effects. Their constraints by the far detector are even weaker than the current bounds since their contributions to the dominant order of 𝒪(ϵ s_13) in far_pme_invacuum and far_pem_invacuum would be zero when δ=270 degrees.* The constraints on ϵ_eμ^s, ϵ_μ e^s, ϵ_eμ^d, ϵ_μ e^d are mainly from the appearance channels. A combination of near and far detectors have stronger constraints on ϵ_eμ^s and ϵ_eμ^d than ϵ_μ e^s and ϵ_μ e^d. Fig. <ref> can be used to explain this property: the channel ν_e→ν_μ has more events compared to the channel ν_μ→ν_e. In addition, it should be noted that ν_e and ν̅_e disappearance channels are also sensitive to ϵ_eμ^s and ϵ_μ e^d. In the following part, we will focus on the exclusion curve of the amplitude versus its corresponding phase for each NSI parameter. Based on the previous discussions in Section. <ref>, we neglect certain parameters which have trivial influence on the probabilities. We generate the event spectra with the true central values of standard oscillation parameters given in Table <ref>. Then we turn on one NSI parameter and scan its amplitude and phase to fit the data. At the near detector, we pay more attention to these parameters: ϵ_ee^s/d, ϵ_μμ^s/d, ϵ_μ e^s/d and ϵ_eμ^s/d. At the far detector, parameters ϵ_ee^s/d,ϵ_μμ^s/d, ϵ_μτ^s, ϵ_τμ^d, ϵ_μ e^s, ϵ_μ e^d, ϵ_τ e^d, ϵ_eμ^s, ϵ_eτ^s and ϵ_eμ^d are taken into account. The results at a near/far detector are presented in Fig. <ref> and Fig. <ref>, respectively. In Fig. <ref>, we show the excluded parameter space without any color at a near detector. One can observe that ν_μ disappearance channel almost has the same performance with the ν_e disappearance channel. When ϕ_ee^s/d (ϕ_μμ^s/d) equals to±π or zero, the corresponding amplitude |ϵ_ee^s/d| (|ϵ_μμ^s/d|) has the best limit. When these phases are equal to ±π/2, the sensitivity to the amplitude disappears. For the ν_e (ν_μ) appearance channels, the constraint to |ϵ_μ e^s/d| (|ϵ_eμ^s/d|) is almost irrelevant to the phase ϕ_μ e^s/d (ϕ_eμ^s/d) supposing we only consider the source NSI parameter ϵ_μ e^s(ϵ_e μ^s) or detector NSI parameter ϵ_μ e^d(ϵ_e μ^d). This is because the term containing the phases ϕ_μ e^s and ϕ_μ e^d plays an important role in two amplitudes as can be seen from near_pme_antipme and near_pem_antipem. Appearance channels can well constrain the magnitude of related CC-NSI parameters while they barely have impacts on their phases. On the other hand, disappearance channels exclude a large parameter space allowed in the current experimental bounds. There is a strong correlation between the coupling strength and its phases in neutrino oscillation experiments.In Fig. <ref>, we switch to the sensitivities of CC-NSI parameters at a far detector. The colorful regions are allowed after we run a far detector at the MOMENT experiment. Compared to the current bounds marked by the dashed red lines, we obtain good constraints on most of NSI parameters at MOMENT, especially for ϵ_μ e^s and ϵ_e μ^d. Here we list the features about the exclusion curves at the far detector for MOMENT: * The oscillation channels ν_μ→ν_e (ν̅_μ→ν̅_e) and ν_e→ν_μ (ν̅_e→ν̅_μ) are T-conjugate inverse of each other, leading to the symmetrybetween their NSIs: ϵ_μ e^s and ϵ_eμ^d have equal contributions in the probability of P^FD_ν_μ→ν_e and P^FD_ν_e→ν_μ, respectively. Similarly, the pairof ϵ_τ e^d and ϵ_eτ^s and the pair of ϵ_μ e^d and ϵ_eμ^s follow the same way. Therefore, these pairs of parameters have similarbehaviour in Fig. <ref>, which can be manifested by far_pme_2 and far_pem_2 in Section. <ref>. The dependence of constraints for |ϵ_μ e^s| (|ϵ_eμ^d|) on the corresponding phase is not strong, because |ϵ_μ e^s| (|ϵ_eμ^d|) depends on both terms of sin(δ+ϕ_μ e^s) (or sin(δ-ϕ_eμ^d)) and cos(δ+ϕ_μ e^s) (or cos(δ-ϕ_eμ^d)), which complement each other when varying the phase ϕ_μ e^s (ϕ_eμ^d). Although |ϵ_eμ^d| (|ϵ_eμ^s|) also depends on both terms of cos(δ+ϕ_μ e^d) (or cos(δ-ϕ_eμ^s)) and sin(δ+ϕ_μ e^d) (or sin(δ-ϕ_eμ^s)), the former term is suppressed by the coefficient c_2×23. Thus, the exclusion curve of |ϵ_μ e^d| -ϕ_μ e^d (|ϵ_eμ^s|-ϕ_eμ^s) mainly varies with the sine term. However, the constraint on |ϵ_τ e^d| (|ϵ_eτ^s|) only depends on the term of cos(δ+ϕ_τ e^d) (or cos(δ-ϕ_eτ^s)) so that the limit to |ϵ_τ e^d| (|ϵ_eτ^s|) will beextremely weak for ϕ_τ e^d (ϕ_eτ^s)=0, ±π and be the best for ϕ_τ e^d (ϕ_eτ^s)=±π/2.* Since we can take advantage of the ν_e (ν̅_e) channel in MOMENT, obtaining the sensitivities of ϵ_eτ^s, ϵ_eμ^s and ϵ_eμ^d would be accessible. It is noted thatthese parameters can not be constrained well in superbeam experiments.* The sensitivities to ϵ_eμ^s and ϵ_eτ^s are mainly extracted from the ν_μ and ν̅_μ appearance channels. It should be noted, however, that the ν_e and ν̅_e disappearance channels can help to enhance the constraints of them. Similarly, with contributions from ν_e and ν̅_e disappearance channels, the sensitivities to ϵ_μ e^d and ϵ_τ e^d will be improved.* The constraints to NSI parameters ϵ_ee^s, ϵ_μμ^s, ϵ_μτ^s, ϵ_τμ^d are extracted from the disappearance channels. The exclusion curves for ϵ_ee^s/d and ϵ_μμ^s/d are similar to each other at the near detector, since they are entangled with the same term in oscillation probabilities. In addition, there are some symmetric relationships between the source and detector NSI parameters, such as the pair of ϵ_μτ^s and ϵ_τμ^d, the pair of ϵ_ee^s and ϵ_ee^d, the pair of ϵ_μμ^s and ϵ_μμ^d. This is not surprising at all, since the pair of effective coupling constants will be the same for neutrinos produced and detected related to the same charged leptons. The far detector has a good sensitivity to NSI parameters, especially for ϵ_eμ^d and ϵ_μ e^s. The numerical results of the correlations between the amplitudes and phases can be interpreted with the previous probability-level discussions. Almost all NSI-induced phases change the exclusion limits severely except the e-mu sector. Meanwhile, limits on other sectors are not as good as those on the e-mu stamped CC-NSIs. Therefore, MOMENT using muon-decay beams has its unique capability of improving the constraints on ϵ_eμ^d and ϵ_μ e^s. § SUMMARY AND CONCLUSIONS New Physics beyond SM might cause non-standard neutrino interactions and leave imprints on the neutrino oscillation. The next-generation accelerator neutrino experiment MOMENT intends to produce the powerful neutrino beam with an energy of O(100) MeV by muon decays and leaves plenty of room for detector selections and physics study. At this energy range, quasi-elastic neutrino interactions dominate the detection process and backgrounds from π^0 are highly suppressed. Compared with traditional superbeams from charged meson decays where intrinsic backgrounds have to be alleviated by the off-axis technology like T2K and NOνA, beams from muon decays are cleaner neutrino sources and good at a detection of new physics. CC-NSIs happening at neutrino productions and detections point to the new phenomenon, where a neutrino produced or detected together with the charged lepton will not necessarily share the same flavour, and flavour conversion is present already at the interaction level and “oscillations" can occur at zero distance. With the capability of flavour and charge identifications, we have an opportunity to use eight appearance and disappearance oscillation channels in the physics study. We have chosen the advanced neutrino detector using the Gd-doped Water Cherenkov technology and studied neutrino oscillations confronting with CC-NSIs at the MOMENT experiment. In order to understand the relevant behavior from NSIs, we have perturbatively derived oscillation probabilities including CC-NSIs at a short and far distance, and tried to analyze parameter correlations of standard neutrino mixing parameters with NSI parameters. We have investigated impacts of the charged current NSIs at the neutrino oscillation probabilities, selected the following dominanting CC-NSI parameters ϵ_ee^s/d, ϵ_μμ^s/d, ϵ_μ e^s/d and ϵ_eμ^s/d for the near detector, and concentrated on ϵ_ee^s/d, ϵ_μμ^s/d, ϵ_μτ^s, ϵ_τμ^d, ϵ_μ e^s, ϵ_μ e^d, ϵ_τ e^d, ϵ_eμ^s, ϵ_eτ^s and ϵ_eμ^d for the far detector. A near detector at MOMENT is good at detecting the zero-distance effects induced by NSIs while the oscillation pattern would have not been developed in the standard neutrino oscillation paradigm. With near and far detectors, we have found that CC-NSIs can induce bias in precision measurements of standard mixing parameters. Taking δ_cp and θ_23 as an example, we have found degeneracies after introducing CC-NSI parameters. With a non-maximal θ_23, its degeneracy with the standard CP phase δ_CP gets much worse if CC-NSIs appear in the neutrino production and detection processes. The current bounds on NSI parameters governing the neutrino productions and detections are about one order of magnitude stronger than those related to neutrino propagation in matter. Our study has shown that a combination of near and far detectors at MOMENT is able to provide lower bounds on CC-NSIs where a factor of about two can be envisaged for most of parameters compared with the current experimental bounds, as shown in Table <ref>. We have found strong correlations of NSIs and constrained NSI parameters using a combination of near and far detectors at MOMENT.The feasibility of physics performance at MOMENT strongly depends on inputs of the accelerator facility and the advanced neutrino detection technology. In the future, results will be further improved by tuning the beam energy and optimizing the baseline. We hope that our study will boost the the research and development activities for MOMENT.§ ACKNOWLEDGEMENTThis work is supported by the start-up funding from SYSU, the National Natural Science Foundation of China under Grant No. 11505301 and the Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase) under Grant No.U1501501. YBZ appreciates valuable discussions with Steven Wong and Amir Khan. We thank Jiajun Liao and David Vanegas Forero's suggestions and comments on the preliminary draft. We would like to thank the accelerator working group of MOMENT for fruitful discussions concerned with neutrino fluxes. Furthermore, we are grateful to IHEP colleagues for collaborations, especially Jingyu Tang, Yu-Feng Li, Miao He and Nikos Vassilopoulos. Dr. Neill Raper is highly appreciated for his patient proofreading of our manuscript. | http://arxiv.org/abs/1705.09500v3 | {
"authors": [
"Jian Tang",
"Yibing Zhang"
],
"categories": [
"hep-ph",
"hep-ex"
],
"primary_category": "hep-ph",
"published": "20170526093812",
"title": "Study of Non-Standard Charged-Current Interactions at the MOMENT experiment"
} |
College of Physics and Materials Science, Tianjin Normal University, Tianjin 300387, ChinaSchool of Physics, Nankai University, Tianjin 300071, [email protected] School of Physics, Nankai University, Tianjin 300071, China We study inter-chain pair tunnelling dynamics based on an exact two-particle solution for a two-leg ladder. We show that the Hermitian Hamiltonian shares a common two-particle eigenstate with a corresponding non-Hermitian Hubbard Hamiltonian in which the non-Hermiticity arises from an on-site interaction of imaginary strength. Our results provides that the dynamic processes of two-particle collision and across-legs tunnelling are well described by the effective non-Hermitian Hubbard Hamiltonian based on the eigenstate equivalence. We also find that any common eigenstate is always associated with the emergence of spectral singularity in the non-Hermitian Hubbard model. This result is valid for both Bose and Fermi systems and provides a clear physical implication of the non-Hermitian Hubbard model. 03.65.-w, 11.30.Er, 71.10.Fd Non-Hermitian description of the dynamics of inter-chain pair tunnelling Z. Song December 30, 2023 ========================================================================§ INTRODUCTIONComplex parameter in a Hamiltonian, such as imaginary potential, has been investigated under the framework of non-Hermitian quantum mechanics <cit.>. The usefulness of the complex parameter can be explored by establishing a correspondence between a non-Hermitian system and a real physical system in an analytically exact manner. The discovery of a parity-time (𝒫𝒯) symmetric non-Hermitian Hamiltonian having an entirely real quantum-mechanical energy spectrum <cit.> stimulated the efforts of establishing 𝒫𝒯 symmetric quantum theory as a complex extension of conventional quantum mechanics <cit.>. This complex extension has profound theoretical and methodological implications in many other subjects, ranging from quantum field theory and mathematical physics <cit.>, to solid state <cit.> and atomic physics <cit.>.One way of extracting the physical meaning of a pseudo-Hermitian Hamiltonian with a real spectrum is to seek its Hermitian counterparts <cit.>. There exists another Hermitian Hamiltonian that shares the complete or partial spectrum when the spectrum of a pseudo-Hermitian Hamiltonian is real. The metric-operator theory outlined in Ref. <cit.> provides a mapping between a pseudo-Hermitian Hamiltonian and an equivalent Hermitian counterpart. However, the obtained equivalent Hermitian Hamiltonian is usually quite complicated <cit.>, and it is difficult to determine whether it describes real physics or is just an unrealistic mathematical object. An alternative way to establish the connection between a pseudo-Hermitian Hamiltonian and a physical system is considering the equivalence of eigenstates <cit.>. A Hermitian scattering center at resonant transmission shares the same wave function with the corresponding non-Hermitian tight-binding lattice consisting of the Hermitian scattering center with two additional 𝒫𝒯-symmetric on-site complex potentials.In this paper, we extend this approach to interacting particle systems. In condensed matter physics, inter-chain (inter-layer) pair tunnelling is a popular process, and is an important component for the mechanism of superconductivity <cit.>. We consider a two-leg system with inter-chain pair tunnelling. Based on the exact two-particle solution, we show that if the two-particle dynamics mainly refers to a specific invariant subspace, then the corresponding two-particle dynamics can be described by an effective non-Hermitian Hubbard system with an imaginary on-site interaction. For a given initial state, the strength of the imaginary on-site interaction is determined by the relative velocity of the two particles. When we consider the two-particle dynamics associated with the probability gain in one leg of the Hermitian system, a set of corresponding non-Hermitian Hamiltonians are related to the spectral singularities. Therefore, the dynamical correspondence is sensitive to the selection of the initial state. The particle-creation dynamics can be realized by considering the time-reversal process of it, which corresponds to the annihilation of two particles. On the other hand, the two-particle tunnelling associated with decrease of the probability in the other leg can be well described by a non-Hermitian Hubbard model with the definite pair dissipation. Especially, when the relative group velocity matches the strength of pair tunnelling, the two-particle probability will exhibit a completely transfer from one leg to the other, which corresponds to pair annihilation in the effective non-Hermitian system. From this point of view, we unveil the connection between the interacting Hermitian and non-Hermitian systems in the context of wavepacket dynamics.This paper is organized as follows. In Sec. <ref> we introduce the model Hamiltonians and their symmetry. In Sec. <ref>, we present the equivalence between the Hermitian Hamiltonian with inter-chain pair tunnelling and non-Hermitian Hamiltonians with an imaginary on site interaction. Sec. <ref> and Sec. <ref> are devoted to construct the connection between two types of the systems through wavepacket dynamics. Section <ref> provides the summary and discussion.§ MODEL HAMILTONIANS We address a physically meaningful non-Hermitian Hamiltonian by associating pair tunnelling with an imaginary on-site interaction in a non-Hermitian Hubbard model. As an illustration, we consider two simple models described by a Hermitian and a non-Hermitian Hamiltonian.The Hermitian Hamiltonian can be written as followsH=H_A+H_B+H_AB,andH_ρ = -κ∑_j=1^N( a_ρ ,j^†a_ρ ,j+1+H.c.) , (ρ =A,B), H_AB = -J/2∑_j=1^N( a_A,j^†a_A,j^†a_B,ja_B,j+H.c.) .Obviously, it represents a tight-binding system consisting of a two-leg ladder, with each leg H_ρ ( ρ =A,B) having dimension N. The two legs are coupled through a pair tunnelling term H_AB, which operates on the motion of multi particles. The Hamiltonian possesses two symmetries. One is the 𝒫 symmetry: here 𝒫 represents the space-reflection operator (or parity operator), and the effect of the parity operator is 𝒫a_A,j^† 𝒫^-1= a_B,j^†. The other is the particle-number symmetry, which ensures probability conservation and leads to the following commutation relationN_ρ,H]≠ 0,but [∑_ρN _ρ,H]=0,where N_ρ=∑_ia_ρ ,i^†a_ρ ,i ( ρ =A,B) are the particle-number operators for the upper and lower legs, respectively. The probability is conserved in the entire system H, but breaks in subsystems H_A and H_B. The inter-chain pair tunnelling admits a peculiar symmetry, ( -1) ^N_ρ,H]=0,i.e., the conservation of particle-number parity. Another related system is a non-Hermitian system composed by two independent Hubbard chains, which can be expressed asℋ=ℋ_A+ℋ_B,andℋ_ρ=-κ∑_i=1^N( a_ρ ,i^†a_ρ ,i+1+H.c.) +iU_ρ/2∑_in_ρ ,i( n_ρ ,i-1) ,where ρ =A,B. The non-Hermiticity of ℋ_ρ arises from the complex on-site interaction iU_ρ.We note that ℋ has the same symmetries as H does, i.e., [ ℋ_ρ,∑_in_ρ ,i] =0, [ℋ ,∑_ρ ,in_ρ ,i]=0, except [ ℋ_A,ℋ _B] =0. This allows us to construct the eigenstates of two models in the same invariant subspaces. For instance, particle-preserving symmetry leads to the two-particle invariant subspace, which can be further decomposed into two invariant subspaces with basis sets {a_A,i^†a_B,j^†| 0⟩} and {a_ρ ,i^†a_ρ ,j^†| 0⟩}, respectively. In the next section, we will investigate the connection between the two-particle solutions of these two Hamiltonians. In Fig. <ref>, we schematically illustrate the system H and ℋ.§ PAIR TUNNELLING AND SPECTRAL SINGULARITY Now we turn to study the two-particle eigenstates of H and ℋ, from which we expect to establish the connection between two models. We focus on the solutions in the invariant subspace spanned by {a_ρ ,i^†a_ρ ,j^†| 0⟩}, i.e., both particles are either in chain A or B. The derivation in Appendix <ref> shows that for each given {K,k} with K∈ [ -π ,π] ,k∈[ 0,π], there are two degenerate eigenstates of H with energy ε _K( k) =-4κcos( K/2) cos k.And the associated eigenstates can be written as|ψ _K,k^±⟩ =∑_r⩾ 0,ρ =A,Bf_K,k^ρ ,±( r) |ϕ _r^ρ( K) ⟩ , and|ϕ _0^ρ( K) ⟩ = 1/2√( N)∑_je^iKja_ρ ,j^†a_ρ ,j^†|vac⟩ ,|ϕ _r^ρ( K) ⟩ = e^iKr/2 /√(N)∑_je^iKja_ρ ,j^†a_ρ ,j+r^†|vac⟩,( r>1) ,where |ϕ _0^ρ( K) ⟩ and |ϕ _r^ρ( K) ⟩ are translational invariant bases. The corresponding wavefunctions f_K,k^ρ ,±( r) can be expressed explicitly asf_K,k^A,+( r)= f_K,k^B,-( r)= {[ e^-ikr+η _K,ke^ikr, r>0;( 1+η _K,k) /√(2), r=0 ] . , f_K,k^B,+( r)= f_K,k^A,-( r)= {[ ξ _K,ke^ikr,r>0; ( 1+ξ _K,k) /√(2),r=0 ] . ,whereη _K,k = λ _K,k^2-J^2/λ _K,k^2+J^2, ξ _K,k=-2iλ _K,kJ/λ _K,k^2+J^2,λ _K,k = 4κcos( K/2) sin k.We note that K represents the central momentum vector of two particles, while k represents the relative momentum between the two particles. In this sense, the eigenstates |ψ _K,k^±⟩ are associated with the dynamic process in which two particles collide with each other in one leg and then tunnel into the other leg.Similarly, we can construct the eigenstates of ℋ having the same form in Eq. (<ref>) based on the result shown in Appendix <ref>,|χ _K,k^±⟩ =∑_r⩾ 0,ρ =A,Bg_K,k^ρ ,±( r) |ϕ _r^ρ( K) ⟩ ,whereg_K,k^A,+( r)= g_K,k^A,-( r)= {[ e^-ikr+μ _K,ke^ikr, r>0;( 1+μ _K,k) /√(2), r=0 ] . ,g_K,k^B,+( r)= -g_K,k^B,-( r)= {[ e^-ikr+ν _K,ke^ikr, r>0;( 1+ν _K,k) /√(2), r=0 ] . ,and the parameters areμ _K,k=λ _K,k+U_A/λ _K,k-U_A,ν _K,k=λ _K,k+U_B/λ _K,k-U_B.It is easy to check that when the following conditions are satisfiedU_A=-J^2/λ _K,k,U_B=λ _K,k,we could obtain|ψ _K,k^+⟩ =|χ _K,k^+⟩ .Note that the eigenstates |χ _K,k^±⟩ are the functions of U_A and U_B. The equivalence condition (<ref>) denotes that the U_A and U_B are { K,k} dependent. Thus one requires two indices to label the eigenstate as |χ _K,k^±( U_A,U_B) ⟩. For the sake of convenience, we neglect the ( U_A,U_B) of |χ _K,k^±( U_A,U_B) ⟩. If we exchange the values of U_A and U_BU_A=λ _K,k,U_B=-J^2/λ _K,k,we have|ψ _K,k^-⟩ =|χ _K,k^+⟩ ,which arises from the parity symmetry of both H and ℋ. This indicates that the two Hamiltonians have common eigenstates, revealing the connection between a Hermitian and a non-Hermitian Hamiltonian. This connection has the following features: (i) We find that iU_A and iU_B are { K,k} dependent and for a given { K,k}, they are all imaginary but with different signs, representing a complementarity pair gain and loss. Further investigation in the next section will show that this ensures the conservation of particles in thewhole system. (ii) As an independent non-Hermitian Hubbard chain with on-site strength iU_ρ, the derivation in Appendix <ref> shows that whenU_ρ=λ _K,k this Hamiltonian has a spectral singularity at point { K,k}. (iii) Furthermore, we find that in the case of J^2=λ _K,k^2, two independent non-Hermitian Hubbard chains have a spectral singularity simultaneously at point { K,k}. The mechanism of the occurrence of the spectral singularity and the corresponding physical implications will be addressed in the next section.§ TUNNELLING DYNAMICS Considering two local particles in one of two legs, which have no overlap with each other, the tunnelling term would have zero effect on the dynamics. But when the two particles meet, particle transfer occurs between two legs. The pair transmission probability depends on many factors as discussed in the following. In this section, we will investigate the dynamics of two-wavepackets collision based on the above formalism. We start our investigation from the time evolution of an initial state as|Φ( 0) ⟩ =|Φ _A,a⟩|Φ _A,b⟩ ,which represents two separable boson wavepackets a and b. Here|Φ _ρ ,γ⟩ =1/√(Ω) ∑_je^-α ^2( j-N_γ) ^2e^ik_γja_ρ ,j^†|Vac⟩ ,with γ =a, b, and ρ =A, B represents a Gaussian wavepacket, which has a width 2√(ln 2)/α, a central position N_γ in chain ρ and a group velocity υ _γ=-2κsin k_γ. The condition that N_a-N_b≫ 1/α ensures that two initial bosons cannot overlap, and thus having no pair tunnelling. Straightforward derivation shows that|Φ( 0) ⟩ = 1/2∑_σ =±( |Φ _A,a⟩|Φ _A,b⟩ +σ|Φ _B,a⟩|Φ _B,b⟩)= 1/√(2Ω _1)∑_Ke^-( K-2k_c) ^2/4α ^2 × e^-iN_c( K-2k_c) |ψ _K^±( r_c,q_c) ⟩ ,where|ψ _K^±( r_c,q_c) ⟩ =1/ √(Ω _2)∑_re^-α ^2( r-r_c) ^2/2e^iq_cr/2|ϕ _r^±( K) ⟩ ,and Ω _1,2 is the normalized factor. Here we have used the following transformationsN_c = 1/2( N_a+N_b) ,r_c=N_b-N_a, k_c = 1/2( k_a+k_b) ,q_c=k_b-k_a, l= j+r,and identities2[ ( j-N_a) ^2+( l-N_b) ^2] = [ ( j+l) -( N_a+N_b) ] ^2 +[ ( l-j) -( N_b-N_a) ] ^2,2( k_aj+k_bl)= ( k_a+k_b) ( j+l) +( k_b-k_a) ( l-j) . We note that the component of state |Φ( 0) ⟩ on each invariant subspace represents an incident wavepacket along the chains described by H_eq^K,± with a width 2√(ln 2)/α, a central position r_c=N_b-N_a and a group velocity υ =-4κcos( K/2) sin( q_c/2). It is worth pointing out that as α≪ 1, the initial state is distributed mainly in the invariant subspace K=2k_c, where the wavepacket moves with the group velocity υ _r= -4κcos( k_c) sin( q_c/2) = υ _b-υ _a. Then the time evolution of state |Φ( t) ⟩ can be derived by the evolution of wavepacket in two chains H_eq^K,±, which eventually can be obtained from the solution in Eq. (<ref>). Furthermore, according to the solution, the evolved state of |ψ _K^±( r_c,q_c) ⟩ can be expressed approximately in the formof e^iβ( r_c^') R_2k_c,q_c/2^±|ψ _K^±( r_c^',-q_c) ⟩, which represents a reflected wavepacket in the equivalent semi-infinite chain H_eq^K,±. The expressions of R_2k_c,q_c/2^± and H_eq^K,± are given in the Appendix <ref>. Here β( r_c^'), as a function of the position of the reflected wavepacket, is an overall phase and is independent of J. We assume that the collision occurs at instant t_0, the evolved state at time t≫ t_0 has the form of|Φ( t) ⟩ = ∑_σ =±Ω ^-1e^iβ( | N_a^'-N_b^'|) R_2k_c,q_c/2^σ ×∑_j,le^-α ^2( l-N_b^') ^2e^-α ^2( j-N_a^') ^2 × e^ik_bje^ik_al( a_A,j^†a_A,l^†+σ a_B,j^†a_B,l^†) |Vac ⟩ .which also represents two separable wavepackets at N_a^' and N_b^', respectively. Comparing Eqs. (<ref>) and (<ref>), it is straightforward to figure out that the two-particle wavepackets behave as classical particles, which swap the momenta with each other after collision. For simplicity, we denote an incident single-particle wavepacket as |λ ,p,A⟩, where λ =L, R indicates the particle coming from the left or right of the collision zone, and p is the central momentum. In this context, we give the asymptotic expression for the collision process in the following: at time t≪ t_0, we have|L,k_a,A⟩|R ,k_b,A⟩ =1/√(2)( |F ^+⟩ +|F^-⟩) ,where|F^±⟩ =1/√(2)( |L,k_a,A⟩|R ,k_b,A⟩±|L,k_a,B⟩|R,k_b,B⟩) .and after collision, at time t≫ t_0, the wavepackets exchange their momenta, which admits|L,k_a,A⟩|R ,k_b,A⟩±|L,k_a,B⟩|R,k_b,B⟩⟼ R_2k_c,q_c/2^±( |L ,k_b,A⟩|R,k_a,A⟩.. ±|L,k_b,B⟩|R ,k_a,B⟩) .By neglecting the J-independent overall phase, therefore we have|L,k_a,A⟩|R ,k_b,A⟩⟼ cosΔ _2k_c,q_c/2|L ,k_b,A⟩|R,k_a,A⟩ +isinΔ _2k_c,q_c/2|L,k_b,B⟩|R,k_a,B⟩ ,whereR_2k_c,q_c/2^± = e^± iΔ _2k_c,q_c/2, Δ _2k_c,q_c/2 = 2tan ^-1( -J/λ _2k_c,q_c/2) ,as discussed in Appendix <ref>. Evidently, Eq. (<ref>) shows that after collision, one part of two wavepackets in leg A, which corresponds to the first term in Eq. (<ref>), is reflected as two identical classical particles. Meanwhile another part, which corresponds to the second term in Eq. (<ref>), tunnels into leg B.Considering a special case with υ _r=J, i.e., the pair-tunnelling amplitude is equal to the relative group velocity, we have Δ _2k_c,q_c/2=π /2, and this leads to|L,k_a,A⟩|R ,k_b,A⟩⟼ i|L,k_b,B ⟩|R,k_a,B⟩ .Clearly, this represents the process that two separable wavepackets on leg A tunnel into leg B completely.§ NON-HERMITIAN DYNAMICS From the above discussions regarding the dynamics of across-leg tunneling, we see that the two-particle probability transfers from one leg to another. The two-particle probability in one leg is not conserved. Thus, a natural question to ask is whether there exists an effective non-Hermitian Hamiltonian for characterizing such a dynamics. To this end, we first present the connections between Hermitian Hamiltonian H and ℋ in a compact form. There are N( N+1) eigenstates of H in the invariant subspace spanned by {a_ρ ,i^†a_ρ ,j^†| Vac⟩}. Each of the eigenstates {|ψ _K,k^±⟩} corresponds to a specific eigenstate |χ _K,k^+⟩ of the non-Hermitian Hubbard chain with the ( K,k)-dependent interaction iU_ρ as in Eqs. (<ref>) and (<ref>). We note that the eigenstates of H and ( K,k)-dependent Hamiltonian ℋ are related to the index ( K,k). In the following, we take a single index η to represent ( K,k) . For the system with 2N sites, all possible ( K,k) is denoted as η =1,2,...,N(N+1)/2. The eigenstates of H is denoted as |ψ̅_l⟩ (l∈ 1,N(N+1)]) with|ψ̅_η⟩≡|ψ _K,k^+⟩ , |ψ̅_η +N(N+1)/2⟩≡|ψ _K,k^-⟩ .Accordingly, the ( K,k)-dependent Hamiltonian ℋ is denoted as ℋ_l withℋ_η≡ℋ( K,k) , for U_A=-J^2/λ _K,k,U_B=λ _K,k,ℋ_η +N(N+1)/2≡ℋ( K,k) , for U_A=λ _K,k,U_B=-J^2/λ _K,k. The eigenstate of ℋ_l is denoted as |χ̅_l,l^'⟩ with|χ̅_η ,η ^'⟩≡|χ _K^',k^'^+⟩ , |χ̅_η ,η ^'+N(N+1)/2⟩≡|χ _K^',k^'^-⟩ , for ℋ( K,k)with U_A=-J^2/λ _K,k,U_B=λ _K,k,|χ̅_η +N(N+1)/2,η ^'⟩≡|χ _K^',k^'^-⟩ , |χ̅_η +N(N+1)/2,η ^'+N(N+1)/2⟩≡|χ _K^',k^'^+⟩ , for ℋ( K,k)with U_A=λ _K,k, U_B=-J^2/λ _K,k.Note that the eigenstate |χ̅_l,l^'⟩ possesses two subscripts. The first one indicates the ( K,k)-dependent on-site interactions U_A and U_B, and the second one denotes the center and relative momenta ( K,k) of the eigenstate for a given U_A and U_B. In Fig. <ref>(a), we illustrate the |χ̅_l,l^'⟩ via ket matrix. The states in lth row represents the complete set of eigenstates of ℋ_l. Based on this notation, the Schrodinger equations become compactH|ψ̅_l⟩ = E_l|ψ̅ _l⟩ ,ℋ_l|χ̅_l,l^'⟩ = ε _l,l^'|χ̅_l,l^'⟩ .Note that ε _l,l^' is related to the scattering solution of two particles, which possesses the form of ε _l,l^'=-4κcos (K^'/2)cos k^', where ( K^',k^') denotes possible center and relative momentuma. These eigenstates have simple relations|ψ̅_l⟩ =|χ̅ _l,l⟩ ,E_l=ε _l,l=-4κcos (K/2)cos k,which indicate that the diagonal states {|χ̅ _l,l⟩} of Fig. <ref>(a) is the complete set of eigenstates of H. Here, |ψ̅_η⟩ (|ψ̅_η +N(N+1)/2⟩) represents that the two particles collide with each other in leg A (B) and then tunnel into leg B (A).When considering the dynamical correspondence in the non-Hermitian Hubbard system, there exists two kinds of dynamical processes corresponding to |ψ̅_η⟩ and |ψ̅ _η +N(N+1)/2⟩: (i) |χ̅_η ,η^A⟩≡( |χ̅_η ,η⟩ +|χ̅_η ,η +N(N+1)/2⟩) /√(2) denotes the two-particle collision process in leg A accompanied by the decrease of the two-particle probability while |χ̅_η ,η^B⟩≡( |χ̅_η ,η⟩ -|χ̅ _η ,η +N(N+1)/2⟩) /√(2) represents a process related to the increase of two-particle probability in leg B. (ii) |χ̅_η +N(N+1)/2,η^B⟩≡( |χ̅_η +N(N+1)/2,η⟩ -|χ̅_η +N(N+1)/2,η +N(N+1)/2⟩) /√(2) represents the two-particle collision process in leg B accompanied by the decrease of the two-particle probability, and |χ̅_η +N(N+1)/2,η^A⟩≡( |χ̅_η +N(N+1)/2,η⟩ +|χ̅_η +N(N+1)/2,η +N(N+1)/2⟩) /√(2) denotes a process associated with the increase of two-particle probability in leg A. For a collision process along leg ρ ( ρ =A,B) in Hermitian systems, there are N( N+1) /2 related non-Hermitian Hamiltonians. Therefore one cannot obtain a Hubbard chain with a certain value of iU_ρ to describe the dynamics along one of two legs. However, for an initial state distributed mainly in the vicinity of |ψ̅ _l_0⟩, the correspondence of dynamics can be characterized by the eigenstates around l_0th row in which the value of l_0 is determined by the central and relative momenta ( K_0,k_0) of the considered initial state. This corresponds to a block around certain ket |χ̅_l_0,l_0⟩, which can be shown in Fig. <ref>(b). For the sake of simplicity and convenience, we confine the discussions to the case of l_0∈[ 1,( N+1) N/2]. The conclusion still holds for the case of l_0∈[ ( N+1) N/2+1,( N+1) N], in which the U_A and U_B exchange their values. To seek an effective non-Hermitian Hamiltonian to characterize such a dynamics, we first consider the collision dynamics in leg A, which is accompanied by the decrease of two-particle probability. If the involved wavefunctions changes slowly around ( |χ̅_l_0,l_0⟩ +|χ̅ _l_0,l_0+N( N+1) /2⟩) /√(2), then one can use an effective non-Hermitian Hamiltonian ℋ_A( K_0,k_0) with a definite U_A=-J^2/λ _K_0,k_0 as an approximation to describe such a dynamics of leg A in the Hermitian system. To this end, we take the derivative of the function μ _K,k with respect to K and k,( ∂μ _K,k/∂ K) _K_0,k_0 = -Λsin( K_0/2) sin k_0, ( ∂μ _K,k/∂ k) _K_0,k_0 = 2Λcos( K_0/2) cos k_0,where Λ =8κ J^2λ _K_0,k_0/( λ _K_0,k_0^2+J^2) ^2. The optimal condition can be achieved when ( ∂μ _K,k/∂ k) _K_0,k_0=0, and ( ∂μ _K,k/∂ K) _K_0,k_0=0. This can be realized through adjusting the relative group velocity of the initial two wavepackets. The condition also indicates that all the rows in such a block are identical approximately as shown in Fig. <ref>(b). Then the diagonal states of the block can be replaced by that in a row with green shadowed. On the other hand, for the dynamics along leg B, each of the eigenstates in the vicinity of |χ̅_l_0,l_0^B ⟩ =( |χ̅_l_0,l_0⟩ -|χ̅_l_0,l_0+N( N+1) /2⟩) /√(2) corresponds to a spectral singularity of the non-Hermitian Hamiltonian ℋ_B( K,k). This leads to the coefficients λ _K,k=U_B ( ν _K,k=∞). In order to avoid this divergence, we can rewrite the expression of Eq. (<ref>) in the formg_K,k^B,+( r) = -g_K,k^B,-( r) = {[ ς _ke^-ikr+ζ _ke^ikr, r>0; 2λ _K,k/√(2), r=0 ] . .where ς _k=λ _K,k-U_B, and ζ _k=λ _K,k+U_B. Here we want to point out that the relative magnitude between the amplitudes of right-going wave e^ikr and left-going wave e^-ikr is meaningful, since we focus on the scattering solution in the limit of N→∞. In this sense, the form of the wavefunctions g_K,k^A,σ( r) and g_K,k^B,σ( r) ( σ =±) are not unique. After multiplying K-k dependent constant, the renormalized scattering solutions are still the corresponding eigenstates of ℋ_A and ℋ_B. In the definition of Eq. (<ref>), the existence of the spectral singularity in system can be determined by either μ _K,k=0 ( ν _K,k=0) or μ _K,k=∞ ( ν _K,k=∞), which is associated with the pair-annihilation or pair-creation process. This corresponds to the case of ζ _k=0 or ς _k=0 in Eq. (<ref>). To obtain the effective non-Hermitian Hamiltonian, we focus on the variation of ς _k in the vicinity of |χ̅ _l_0,l_0^B⟩ in the ket matrix. As we have done in leg A, we take the partial derivative of the ς _k with respect to K and k, respectively, which yields( ∂ς _k/∂ K) _K_0,k_0 = 0,( ∂ς _k/∂ k) _K_0,k_0 = 0.This indicates that one can replace the diagonal states with row states for any given momentuma ( K_0,k_0) in the ket matrix as shown in Fig. <ref>(b). Thus, an effective Hamiltonian with definite U_B=λ _K_0,k_0 can be employed to simulate the dynamics of leg B in the Hermitian system. Here we want to stress that there is no tunneling between non-Hermitian Hamiltonians ℋ_A and ℋ_B. Thus, we cannot employ an effective non-Hermitian Hamiltonian ℋ with definite strengths of the pair dissipation and gain to describe the tunneling dynamics between two legs. The dynamical correspondence of leg B can be obtained through another method, which will be detailed in the following.In parallel, we can investigate the dynamics of a two-wavepacket collision by analyzing the time evolution of the initial state |Φ( 0) ⟩ in the effective non-Hermitian system ℋ with U_A=-J^2/υ _r. Similarly, we can obtain the asymptotic expression for the collision process as|L,k_a,A⟩|R ,k_b,A⟩⟼cosΔ _2k_c,q_c/2|L,k_b,A⟩|R,k_a,A⟩ ,which has the same form as the wave function in leg A of Eq. (<ref>). This indicates that the effective non-Hermitian Hamiltonianℋ can describe the wavepacket dynamics in subsystem (leg A) of a Hermitian system H. Naturally, when the strength of the imaginary on-site interaction U_A is equal to the relative group velocity υ _r (Δ _2k_c,q_c/2=π /2), the two particles will exhibit a behaviour of pair annihilation in leg A and will never tunnel into leg B. This process is schematically illustrated in Fig.<ref>(b). Note that ℋ cannot describe the wavepacket dynamics in leg B, because there is no tunnelling between leg A and Bin ℋ. However, the final state |L ,k_b,B⟩|R,k_a,B⟩ in leg B as in Eq. (<ref>) can be prepared by using non-Hermitian Hamiltonian H_B in another way. To this end, we require an initial state to simulate the creation of a pair of particles. Moreover, the modulus of the initial state should tend to be 0, owing to the fact that no one can create a pair of particles out of nothing. Then the initial state driven by the ℋ_B will evolve to |L ,k_b,B⟩|R,k_a,B⟩accompanied by the increase of two-particle probability. However the selection of such an initial state is too cumbersome. There is a lot of states with near-zero-modulus value. The different types of the initial states will exhibit distinct dynamical behaviors. In other words, the dynamics of the system is sensitive to the initial state. Therefore, the elaborately selection of the initial state is a crucial step to successfully mimic the dynamics of leg B in Hermitian system. Fortunately, we can chose the initial state by considering the time-reversal process of the dynamics of leg B, which corresponds to the annihilation of two wavepackets |L,k_b,B⟩|R ,k_a,B⟩. This can be realized through adjusting the on-site interaction U_B and relative group velocity υ _r based on the result obtained in leg A. In this sense, the final near-zero-modulus state can be selected as an initial state of the particles creation process. And the corresponding driven Hamiltonian can also be obtained by taking the time-reversal operation on the related non-Hermitian Hubbard Hamiltonian with pair annihilation.In order to further validate the conclusion obtained above, we compare the local-state dynamics in two such systems by numerical simulation. To do this, we introduce the quantity |⟨Φ( t) | n_j|Φ( t) ⟩| ^2 to characterize the shape and probability distribution of the two wavepackets in Fig. <ref>. For the Hermitian case as shown in Fig. <ref>(a) and <ref>(b), one can see that when the two wavepackets enter into the interaction region, the probability of the two wavepacket transfers from leg A to B due to the pair tunnelling J. The process of the decrease of the two-wavepacket probability in leg Acan also be approximately described through the two-wavepacket dynamics in an effective non-Hermitian Hubbard Hamiltonian ℋwith U_A=-J^2/υ _r, as is shown in Fig. <ref> (c).§ SUMMARYIn summary, we have studied the inter-chain pair tunnelling dynamics based on the exact two-particle solution of a two-leg ladder. It is shown that the Hermitian Hamiltonian shares a common two-particle eigenstate with a corresponding non-Hermitian Hubbard model, in which the non-Hermiticity arises from an imaginary on-site interaction. Such a common state is associated with the spectral singularity of the equivalent non-Hermitian system. The dynamical correspondence is dependent on the selection of the initial state. For the dynamics accompanied with the increase of the two-particle probability, such an initial state can be obtained through a time-reversal process of the annihilation of two wavepackets. On the other hand, the reduction of the two-particle probability in the other leg of the Hermitian system can be well characterized by the effective non-Hermitian Hubbard model with the definite strength of pair dissipation, which is also determined by the relative and center momentuma of the initial state. In addition, we have also found that the two particles display perfect transfer from one leg to the other when υ _r=J, which corresponds to the pair annihilation in the effective non-Hermitian Hubbard system with the strength of the imaginary on-site interaction υ _r=U_ρ. This result is valid for both Bose and Fermi systems and provides a clear physical implication of the non-Hermitian Hubbard model.§ APPENDIX§.§ Solution of the two-leg ladder In this section, we derive the solution of the Hamiltonian shown in Eq. (<ref>) in a two-particle invariant subspace. Here, we take the periodic boundary condition that a_ρ ,j=a_ρ ,j+N. Due to the symmetry in Eq. (<ref>), which preserves the parity of particle number in each leg, the 𝒫 symmetry, and the translational symmetry, the basis spanning the subspace can be constructed as|φ _0^±( K,r) ⟩ =1/2 √(N)∑_je^iKj( a_A,j^†a_A,j^†.. ± a_B,j^†a_B,j^†) |vac ⟩ ,|φ _r^±( K,r) ⟩ =1/ √(2N)e^iKr/2∑_je^iKj( a_A,j^†a_A,j+r^†. . ± a_B,j^†a_B,j+r^†) |vac⟩, ( r>1) ,where K=2nπ /N, n∈[ -N/2,N/2] is the momentum vector, and ± denote two degenerate subspaces originating from the 𝒫 symmetry. A two-particle eigenstate has the form of|ψ _K,k^±⟩ =∑_rF_K,k^±( r) |φ _r^±( K) ⟩,with the condition F_K,k^±( -1) =0, where the two degenerate wave functions F_K,k^±( r) satisfy the Schr ödinger equationsQ_r^KF_K,k^±( r+1) +Q_r-1^KF_K,k^±( r-1) +± Jδ _r,0+( -1) ^nQ_r^Kδ _r,N_0-ε _K]F_K,k^±( r) =0,with the eigen energy ε _K in the invariant subspace indexed by K. Here the factors are Q_r^K=-2√(2)κcos( K/2) for r=0 and -2κcos( K/2) for r≠ 0, respectively. It indicates that the eigen problem of two-particle matrix can be reduced to a single-particle governed by the equivalent HamiltoniansH_eq^K,±=± J| 0⟩⟨ 0| +∑_i=1^∞( Q_i^K| i⟩⟨ i+1| +H.c.) ,which clearly represents a semi-infinite chain with the ending on-site potential J. We are concerned with only the scattering solution by the 0 th end. The Bethe ansatz solutions have the formF_K,k^±( r) =e^-ikr+R^±e^ikr.Substituting F_K,k^±( r) into Eq. (<ref>), we haveε _K( k) =-4κcos( K/2) cos k,k∈[ 0,π] ,andR_K,k^±=iλ _K,k± J/iλ _K,k∓ J=e^± iΔ _K,k,withλ _K,k = 4κcos( K/2) sin k,Δ _K,k = 2tan ^-1( -J/λ _K,k) .For convenience in the application of wavepacket dynamics, we rewrite the solutions in the form|ψ _K,k^±⟩ =∑_r,ρf_K,k^ρ ,±( r) |ϕ _r^ρ( K) ⟩ ,where ρ =A,B and|ϕ _0^ρ( K) ⟩ =1/2√( N)∑_je^iKja_ρ ,j^†a_ρ ,j^†|vac⟩ ,|ϕ _r^ρ( K) ⟩ =1/√(N )e^iKr/2∑_je^iKja_ρ ,j^†a_ρ ,j+r^†|vac⟩,( r>1) .The corresponding wavefunctions f_K,k^ρ ,±( r) can be expressed asf_K,k^A,+( r)= f_K,k^B,-( r)= {[ e^-ikr+λ _K,k^2-J^2/λ _K,k^2+J^2e^ikr,r>0; ( 1+λ _K,k^2-J^2/λ _K,k^2+J^2) / √(2),r=0 ] . ,andf_K,k^B,+( r)= f_K,k^A,-( r)= {[ -2iλ _K,kJ/λ _K,k^2+J^2e^ikr,r>0; -√(2)iλ _K,kJ/( λ _K,k^2+J^2) ,r=0 ] . .§.§ Solution of the non-Hermitian Hubbard model Similarly, considering the Hamiltonian ℋ, we find that it admits all the symmetries we used for solving the eigen problem of H. Then a two-particle state for ℋ_ρ is written as|κ _K^ρ⟩ =∑_rG_K,k^ρ( r) |ϕ _r^ρ( K) ⟩,( G_K,k^ρ( -1) =0)where wave functions G_K,k^ρ( r) satisfy the Schrö dinger equationsQ_r^KG_K,k^ρ( r+1) +Q_r-1^KG_K,k^ρ( r-1) + iU_ρδ _r,0+( -1) ^nQ_r^Kδ _r,N_0-ϵ _K]G_K,k^ρ( r) =0,with the eigen energy ϵ _K in the invariant subspace indexed by K. We are concerned with only the scattering solution by the 0th end. In this sense, G_K,k^ρ can be obtained from the two equivalent Hamiltonians in two subspacesℋ_eq^K,ρ=iU_ρ| 0⟩⟨ 0| +∑_i=0^∞( Q_i^K| i⟩⟨ i+1| +H.c.) .By the same procedures, we haveG_K,k^ρ( r) ={[ e^-ikj+λ _K,k+U_ρ/λ _K,k-U_ρe^ikj,r>0; ( 1+λ _K,k+U_ρ/λ _K,k-U_ρ) / √(2),r=0 ] . .with eigen energy ϵ _K( k) =-4κcos( K/2) cos k, k∈[ 0,π]. Furthemore, we can rewrite the solution in the formg_K,k^±( r) =[ G_K,k^A( r) ± G_K,k^B( r) ] /√(2).§.§ Spectral singularity of Hubbard chain We note that wave function G_K,k^ρ( r)only depends on ρ via U_ρ. This is because the two chains A and B are independent. 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"authors": [
"X. Z. Zhang",
"L. Jin",
"Z. Song"
],
"categories": [
"cond-mat.str-el"
],
"primary_category": "cond-mat.str-el",
"published": "20170526091910",
"title": "Non-Hermitian description of the dynamics of inter-chain pair tunnelling"
} |
GXNOR-Net: Training deep neural networks with ternary weights and activations without full-precision memory under a unified discretization framework Lei Deng^*, Peng Jiao^*, Jing Pei^†,Zhenzhi Wu andGuoqi Li^†(Please cite us with:L. Deng, et al. GXNOR-Net: Training deep neural networks with ternary weights and activations without full-precision memory under a unified discretization framework. Neural Networks 100, 49-58(2018).)L. Deng,P. Jiao, J. Pei, Z. Wu and G. Liare with theDepartmentof Precision Instrument, Center for Brain Inspired Computing Research, Tsinghua University, Beijing, China, 100084. L. Deng is also with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, CA 93106, USA.Emails: [email protected] (L. Deng), [email protected] (P. Jiao), [email protected] (J. Pei),[email protected] (Z. Wu) and [email protected] (G. Li).^*L. Deng and P. Jiao contribute equally to this work. ^†The corresponding authors: Guoqi Li and Jing Pei.December 30, 2023 =======================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================================Although deep neural networks (DNNs) are being a revolutionary power to open up theAI era, the notoriously huge hardware overhead haschallenged their applications. Recently, severalbinary and ternary networks, in whichthe costly multiply-accumulate operations can be replaced by accumulations or evenbinary logic operations, make the on-chip training of DNNs quite promising. Therefore there is a pressing need to build anarchitecturethat could subsume these networksundera unified framework thatachieves bothhigher performance and less overhead. To this end, two fundamental issues are yet to be addressed. The first one is how to implement the back propagation when neuronal activations are discrete.The second one ishow to removethe full-precision hidden weights in the training phaseto break the bottlenecks of memory/computation consumption.To addressthe first issue, we presentamulti-step neuronalactivation discretization method andaderivative approximation techniquethat enable the implementingthe back propagation algorithm on discrete DNNs. While for the second issue, we propose a discrete state transition (DST)methodology to constrain the weights in a discrete space without saving the hidden weights.Through this way, webuild a unified framework thatsubsumesthe binary or ternary networks asits special cases, andunder which a heuristic algorithm is provided at the website https://github.com/AcrossV/Gated-XNOR. More particularly, we find that when boththe weights and activationsbecometernaryvalues, the DNNs can be reduced to sparse binary networks, termed as gated XNORnetworks(GXNOR-Nets) since onlythe event of non-zero weight and non-zero activation enables the control gateto start the XNOR logic operations in the original binary networks. This promises the event-driven hardware design for efficient mobile intelligence. We achieve advanced performance compared with state-of-the-art algorithms. Furthermore, the computationalsparsity andthe number of statesin thediscrete space can be flexibly modified to make it suitable for various hardware platforms. Keywords: GXNOR-Net, Discrete State Transition, Ternary Neural Networks, Sparse Binary Networks § INTRODUCTION Deep neural networks (DNNs) are rapidly developing with the use of big data sets, powerful models/tricks and GPUs, and have been widelyapplied in various fields <cit.>-<cit.>, such as vision, speech, natural language, Go game, multimodel tasks, etc. However, the huge hardware overhead is also notorious, such asenormous memory/computation resources and high power consumption, which has greatly challenged their applications. As we know, most of the DNNscomputing overheadsresultfrom the costly multiplication of real-valued synaptic weight and real-valued neuronal activation, as well as the accumulation operations. Therefore, a few compression methods and binary/ternary networks emerge in recent years, which aim to put DNNs on efficient devices. The former ones <cit.>-<cit.> reduce the network parameters and connections, but most of them do not change the full-precision multiplications and accumulations. The latter ones <cit.>-<cit.>replace the original computationsby only accumulations or evenbinary logic operations.In particular, the binary weight networks (BWNs) <cit.>-<cit.> and ternary weight networks (TWNs) <cit.> <cit.> constrain the synaptic weights to the binary space {-1,1} or the ternary space {-1,0,1}, respectively. In this way, the multiplication operations can be removed. The binary neural networks (BNNs) <cit.> <cit.> constrain both the synaptic weights and the neuronal activations to the binary space {-1,1}, which can directly replace the multiply-accumulate operations by binary logic operations, i.e. XNOR. So this kind of networks is also calledthe XNOR networks. Even with these most advanced models, thereareissues that remainunsolved.Firstly, the reported networks are based on specially designed discretization and training methods,and there is a pressing need to build anarchitecturethat could subsume these networks underaunified framework thatachieves bothhigher performance and less overhead.To this end,how to implement the back propagation for online training algorithmswhen the activations are constrained in a discrete space is yet to beaddressed. On the other side, in all these networks we have to save the full-precision hidden weights in the training phase, which causes frequent data exchange between the external memory for parameter storage and internal buffer for forward and backward computation. In this paper, we propose adiscretization framework: (1) A multi-step discretization function that constrains the neuronal activations in a discrete space, and a method to implement the back propagationby introducing an approximated derivative for thenon-differentiable activation function; (2) A discrete state transition (DST)methodologywith aprobabilistic projectionoperator which constrains the synaptic weights in a discrete space without the storage of full-precision hidden weights in the whole training phase.Undersuch a discretizationframework, a heuristic algorithm is provided at the website https://github.com/AcrossV/Gated-XNOR, where the state number of weights and activations are reconfigurable to make it suitable for various hardware platforms. In theextreme case, both the weights and activations can beconstrained in the ternary space {-1,0,1} to form ternary neural networks (TNNs). For a multiplication operation, when one of the weight and activation is zero or both of them are zeros, the corresponding computation unit is resting, until the non-zero weight and non-zero activation enable and wake up the required computation unit. In other words, the computation trigger determined by the weight and activation acts as a control signal/gateor an event to start the computation. Therefore,in contrast to the existing XNOR networks, the TNNs proposed in this paper can be treated as gated XNOR networks (GXNOR-Nets). We test this network model over MNIST, CIFAR10 and SVHN datasets, and achieve comparable performance with state-of-the-artalgorithms. The efficient hardware architecture is designed and compared with conventional ones. Furthermore, the sparsity of the neuronal activations can be flexibly modified to improve the recognition performance and hardware efficiency. In short, the GXNOR-Net promises the ultra efficient hardware for future mobile intelligence based on the reduced memory and computation, especially for the event-driven running paradigm. We define several abbreviated terms that will be used in the following sections: (1)CWS: continuous weight space; (2)DWS: discrete weight space; (3)TWS: ternary weight space; (4)BWS: binary weight space; (5)CAS: continuous activation space; (6)DAS: discrete activation space; (7)TAS: ternary weight space; (8)BAS: binary activation space; (9)DST: discrete state transition. § UNIFIED DISCRETIZATION FRAMEWORKWITH MULTI-LEVEL STATES OFSYNAPTIC WEIGHTS AND NEURONAL ACTIVATIONS IN DNNSSuppose that there areK training samples given by{(x^(1), y^(1)), ... (x^(κ), y^(κ)), ...,(x^(K), y^(K))} where y^(κ) is the label of the κth samplex^(κ).In this work, we are going to proposea generaldeep architectureto efficientlytrain DNNsin which both the synaptic weights and neuronal activations arerestrictedin a discrete spaceZ_Ndefined as[ Z_N={z^n_N| z^n_N=(n/2^N-1-1), n=0, 1, ... ,2^N} ]whereN isa given non-negative integer, i.e., N=0, 1, 2, ... and Δ z_N=1/2^N-1 isthe distance between adjacent states.Remark 1.Note that different values of N inZ_Ndenote different discrete spaces.Specifically, when N=0, Z_N={-1, 1} belongs tothebinary space and Δ z_0=2.When N=1, Z_N={-1, 0,1} belongs totheternaryspace and Δ z_1=1.Alsoas seen in (<ref>),thestates inZ_Nareconstrained in the interval [-1, 1], and without loss of generality,the rangecan be easily extended to [-H, H] by multiplying a scaling factor H. In the following subsections,we first investigate theproblem formulation forGXNOR-Nets, i.e,Z_N is constrained in the ternary space {-1,0,1}. Later we will investigatehow to implement backpropagation in DNNswith ternary synaptic weights and neuronal activations. Finallya unified discretization framework by extending the weights and activations to multi-level states will be presented.§.§ Problem formulation forGXNOR-NetBy constrainingboth thesynaptic weightsand neuronal activationsto binarystates { -1, 1 }for the computationin both forward and backward passes, the complicatedfloat multiplications and accumulationschange to be very simple logic operationssuch as XNOR. However, different from XNOR networks, GXNOR-Net can be regarded as a sparse binary network due to the existence of the zero state,in which the number of zero state reflects the networks'sparsity. Only when both the pre-neuronal activation and synaptic weight are non-zero, the forward computation is required, marked as red as seen inFig. <ref>.This indicates thatmost of the computation resources can be switched off to reducepower consumption. The enable signal determined by the corresponding weight and activation acts as a control gate for the computation. Therefore, such a networkis calledthe gated XNOR network (GXNOR-Net). Actually, the sparsity is also leveraged by other neural networks, such as in <cit.> <cit.>. Suppose that there are L+1layers in a GXNOR-Net where both the synaptic weights and neuronal activations arerestrictedin a discrete spaceZ_1={-1,0,1} except thezerothinput layer and the activations of the Lth layer.As shown in Fig. <ref>, the lastlayer, i.e., the Lth layer is followed by a L_2-SVM output layer with the standard hinge loss, which has been shown to perform better thansoftmax on several benchmarks <cit.><cit.>. DenoteY^l_ias the activation of neuron i in layer l given byY^l_i=φ(∑_j W^l_ijY^l-1_j) for1 ≤ l ≤ L-1, where φ(.) denotes anactivation function and W^l_ij represents the synaptic weight between neuron j in layer l-1 and neuron i in layer l.For the κth training sample,Y_i^0representsthe ith element of theinput vector of x^(κ), i.e.,Y_i^0=x_i^(κ)∈ R. For theLth layerof GXNOR-Netconnected with the L2-SVM output layer,the neuronal activation Y_i^L∈ R. The optimization model ofGXNOR-Net is formulated as follows[argmin_W,Y E(W,Y);s.t. W^l_ij∈{-1, 0, 1}, Y^l_i∈{-1, 0, 1} l=1, 2 ...., L-1; Y^l_i=φ(∑_j W^l_ijY^l-1_j), l=1, 2 ...., L-1; Y^l_i=∑_j W^l_ijY^l-1_j, l=L;Y_i^l=x_i^(κ), l=0, κ=1, 2, ... , K ]Here E(W,Y) representsthe cost functiondepending onall synaptic weights (denoted as W) and neuronal activations (denoted as Y) in all layersofthe GXNOR-Net.For the convenience of presentation, wedenote the discrete spacewhendescribingthe synaptic weight andthe neuronal activation asthe DWS andDAS, respectively. Then,thespecial ternary spaceforsynaptic weight and neuronal activationbecomethe respective TWS andTAS. BothTWS and TAS are theternary space Z_1={-1, 0, 1} defined in (1).The objective is tominimizethe cost function E(.)in GXNOR-Netsby constrainingall the synaptic weights andneuronal activationsin TWS and TASforboth forward and backward passes. In theforwardpass,we will firstinvestigatehow to discretize theneuronal activations by introducinga quantizedactivationfunction. In the backward pass,we willdiscuss how to implement the back propagation with ternary neuronal activations through approximatingthe derivativeof the non-differentiable activationfunction. After that, theDST methodology for weight updateaiming to solve (<ref>) will be presented. §.§ Ternary neuronal activation discretizationin theforwardpass We introduce a quantization function φ_r(x)todiscretizethe neuronal activationsY^l (1≤ l ≤ L-1) by setting[ Y_i^l=φ_r (∑_j W_ij^lY_j^l-1) ]where φ_r(x) =1, 0,-1,In Fig. <ref>, itis seen thatφ_r(x) quantizes the neuronal activation tothe TAS Z_1and r>0 is a windowparameter which controls the excitability of theneuron and the sparsity of the computation. §.§ Back propagation with ternary neuronal activations through approximatingthe derivativeof the quantizedactivationfunctionAfter theternary neuronal activation discretization in the forward pass, model (<ref>)has nowbeen simplified to the following optimization model[argmin_W E(W);s.t. W^l_ij∈{-1, 0, 1}, l=1,2,...,L; Y^l_i=φ_r(∑_j W^l_ijY^l-1_j), l=1, 2 ...., L-1; Y^l_i=∑_j W^l_ijY^l-1_j, l=L; Y_i^0=x_i^(κ), κ=1, 2, ... , K ] As mentioned in the Introduction section, in order to implement the back propagationin thebackwardpasswherethe neuronal activations are discrete, we need to obtain the derivative of the quantization function φ_r(x) in(<ref>).However, it is well known thatφ_r(x) is not continuous and non-differentiable, as shown in Fig. <ref>(a) and (b). This makes it difficult to implement the back propagation inGXNOR-Net in this case.Toaddress this issue, we approximate the derivative ofφ_r(x) with respect to xas follows ∂φ_r(x)/∂ x = 1/2a,0,otherswhere a is a small positive parameterrepresenting the steep degree of the derivative in the neighbourhood of x. In real applications, there are many other ways to approximate thederivative. For example, ∂φ_r(x)/∂ x can also be approximated as ∂φ_r(x)/∂ x =- 1/a^2(|x|-(r+a)), 1/a^2(|x|-(r-a)),0,others for a small given parameter a. The above two approximated methods are shown in Fig. <ref>(c) and (d), respectively. It is seen thatwhen a → 0, ∂φ_r(x)/∂ x approachesthe impulse function in Fig. <ref>(b). Note that thereal-valuedincrementofthe synaptic weight W^l_ij at thekth iteration at layer l, denoted as Δ W^l_ij(k), can be obtainedbased on the gradient informationΔ W^l_ij(k)=-η·∂ E(W(k), Y(k))/∂ W^l_ij(k)where η representsthe learning rateparameter,W(k) and Y(k) denotetherespective synaptic weights and neuronal activationsofall layers at thecurrentiteration,and∂ E(W(k), Y(k))/∂ W^l_ij(k)=Y_j^l-1·∂φ_r(x^l_i) /∂x^l_i·e^l_iwhere x^l_i is aweighted sum oftheneuron i'sinputsfrom layer l-1:x^l_i= ∑_j W^l_ijY_j^l-1ande^l_i is the error signalofneuron i propagated from layer l+1:e^l_i=∑_ι W^l+1_ι i·e^l+1_ι·∂φ_r(x_ι^l+1) /∂x_ι^l+1and both ∂φ_r(x^l_i) /∂x^l_i and ∂φ_r(x_ι^l+1) /∂x_ι^l+1 areapproximated through (<ref>) or (<ref>). As mentioned,the Lth layer is followed by the L2-SVM output layer, and the hingefossfunction <cit.><cit.> is applied for the training. Then, theerror back propagates from the output layer to anterior layers and the gradient information for each layer can be obtained accordingly. §.§ Weight update by discrete state transition in the ternary weight spaceNow we investigatehow to solve (<ref>) by constrainingWin theTWS throughan iterative training process. LetW^l_ij(k) ∈ Z_1 be the weight state at the k-th iteration step, andΔ W^l_ij(k) be theweight increment on W^l_ij(k) that can bederived on the gradient information (<ref>).To guarantee the next weight will not jump out of [-1,1],define ϱ (·)to establish a boundary restrictiononΔ W^l_ij(k):ϱ (Δ W^l_ij(k)) =min(1-W^l_ij(k), Δ W^l_ij(k) )max(-1-W^l_ij(k), Δ W^l_ij(k) )anddecompose the above ϱ (Δ W^l_ij(k)) as:[ ϱ (Δ W^l_ij(k))= κ_ijΔz_1+ ν_ij= κ_ij + ν_ij ]such that[ κ_ij= fix( ϱ (Δ W^l_ij(k))/Δ z_1)= fix( ϱ (Δ W^l_ij(k))) ]and [ ν_ij=rem(ϱ (Δ W^l_ij(k)), Δ z_1)= rem(ϱ (Δ W^l_ij(k)), 1) ] where fix(.)is a round operation towards zero, andrem(x,y) generates the remainder of the division between two numbers and keeps the same sign with x. Then, weobtain aprojected weight incrementΔ w_ij(k)and update the weight by [ W^l_ij(k+1)= W^l_ij(k)+ Δ w_ij(k); =W^l_ij(k)+ 𝒫_grad(ϱ (Δ W^l_ij(k))) ] Now we discuss how to projectΔ w_ij(k) in CWS to make the next state W^l_ij(k)+ 𝒫_grad(ϱ (Δ W^l_ij(k))) in TWS, i.e. W^l_ij(k+1) ∈ Z_N.Wedenote Δ w_ij(k)=𝒫_grad(.) asaprobabilistic projection functiongiven by [ P(Δw_ij(k) =κ_ijΔz_1+ sign(ϱ (Δ W^l_ij(k))) Δz_1 ) =τ(ν_ij);P( Δw_ij(k) = κ_ijΔz_1 ) = 1-τ(ν_ij); ]where the sign functionsign(x)is given bysign(x) =1,-1, andτ(.) (0≤τ(.) ≤ 1)isa state transition probabilityfunction defined by τ(ν)=tanh( m ·|ν|/Δ z_N) where m is a nonlinear factor of positive constant to adjustthe transition probability in probabilistic projection.The above formula (<ref>) implies that Δ w_ij(k)is among κ_ij +1, κ_ij -1 and κ_ij.For example, when sign(ϱ (Δ W^l_ij(k)))=1, then Δw_ij(k)=κ_ij+ 1 happens with probabilityτ(ν_ij) and Δw_ij(k) = κ_ij happens with probability1-τ(ν_ij). Basically the𝒫_grad(.)describesthe transitionoperation among discrete states in Z_1 defined in(1), i.e., Z_1={z^n_1| z^n_1=n-1, n=0, 1, 2} where z_1^0=-1,z_1^1=0 and z_1^2=1.Fig. <ref> illustrates the transition process in TWS.For example, at the current weight stateW^l_ij(k)=z_1^1=0, ifΔ W^l_ij(k)< 0, then W^l_ij(k+1) has the probability of τ(ν_ij) to transfer to z_1^0=-1 and has the probability of 1-τ(ν_ij)to stay at z_1^1=0; while if Δ W^l_ij(k)≥ 0, then W^l_ij(k+1) has the probability of τ(ν_ij) to transfer to z_1^2=1 and has the probability of 1-τ(ν_ij)to stay at z_1^1=0.At the boundary stateW^l_ij(k)=z_1^0=-1, if Δ W^l_ij(k) < 0, then ϱ (Δ W^l_ij(k))=0 and P(Δ w =0 )= 1, which means thatW^l_ij(k+1)has the probability of 1to stay at z_1^0=-1; if Δ W^l_ij(k)≥ 0 and κ_ij=0, P(Δ w = 1 )=τ(ν_ij), then W^l_ij(k+1) has the probability of τ(ν_ij) to transfer to z_1^1=0, andhas the probability of 1-τ(ν_ij)to stay at z_1^0=-1; if Δ W^l_ij(k)≥ 0 and κ_ij=1, P(Δ w = 2 )=τ(ν_ij), then W^l_ij(k+1) has the probability of τ(ν_ij) to transfer to z_1^2=1, andhas the probability of 1-τ(ν_ij)to transfer to z_1^1=0. Similar analysis holdsfor another boundary stateW^l_ij(k)=z_1^2=1. Based on the above results,now wecansolve the optimization model (<ref>) based onthe DST methodology. The main idea is to update thesynaptic weight based on (<ref>)in the ternary space Z_1 by exploiting the projected gradient information.The main differencebetween DST and the ideas in recent works such asBWNs <cit.>-<cit.>,TWNs <cit.> <cit.>, BNNs orXNOR networks<cit.> <cit.>is illustrated in Fig. <ref>. In those works, frequent switch and data exchange between the CWS and the BWS or TWSare required during the training phase. The full-precision weights have to be saved at each iteration, and the gradient computation is based on the binary/ternaryversion of the stored full-precision weights, termed as “binarization"or “ternary discretization" step.In stark contrast, the weights inDST are always constrained in a DWS.A probabilistic gradient projection operator is introduced in (<ref>) to directly transform a continuous weight increment to a discrete state transition. Remark 2. In the inference phase, sinceboth the synaptic weights and neuronal activations are in the ternary space,only logic operations are required. In the training phase, the remove of full-precision hidden weights drasticallyreducesthe memory cost. The logic forward pass and additive backward pass (just a bit of multiplications at each neuron node) will also simplify the training computation to some extent. In addition, the number of zero state, i.e. sparsity, can be controlled by adjusting r in φ_r(.), which further makes our framework efficient in real applications through the event-driven paradigm.§.§ Unified discretization framework: multi-level states of the synaptic weightsand neuronal activationsActually, the binary and ternarynetworks are notthe whole story sinceN is not limited to be 0 or 1 in Z_Ndefined in (<ref>)andit can beany non-negative integer.There are many hardware platforms that support multi-level discrete space for more powerful processing ability <cit.>-<cit.>.The neuronal activations can be extended to multi-level cases. To this end, we introduce the followingmulti-step neuronal activation discretization function[ Y_i^l=φ_r (∑_j W_ij^lY_j^l-1) ] where φ_r(x) = ω/2^N-1, 0,-ω/2^N-1,for 1≤ω≤ 2^N-1. The interval [-H,H] is similarly defined with Z_N in (<ref>). To implement the backpropagationalgorithm, the derivative of φ_r(x) can be approximated at each discontinuous point as illustrated in Fig. <ref>. Thus, both the forward pass and backward pass of DNNs can be implemented.At the same time,the proposedDST for weight update can also be implemented in a discretespace with multi-level states. In this case, thedecomposition of Δ W_ij^l(k)is revisited as[ ϱ (Δ W_ij^l(k))= κ_ijΔ z_N+ ν_ij ]such that[ κ_ij= fix( ϱ (Δ W_ij^l(k))/Δ z_N) ]and [ ν_ij=rem(ϱ (Δ W_ij^l(k)), Δ z_N) ]and the probabilistic projection functionin (<ref>) can also be revisitedas follows [ P(Δ w_ij(k) =κ_ijΔ z_N+ sign(ϱ (Δ W^l_ij(k))) Δ z_N ) =τ(ν_ij);P( Δ w_ij(k)= κ_ijΔ z_N ) = 1-τ(ν_ij);] Fig. <ref>illustratesthe state transitionof synaptic weights in DWS. In contrast to the transition example of TWS in Fig. <ref>, theκ_ij can be larger than 1 so that further transition is allowable.§ RESULTS We test the proposed GXNOR-Nets over the MNIST, CIFAR10 and SVHN datasets[Thecodes are available at https://github.com/AcrossV/Gated-XNOR]. The results are shown in Table <ref>. The network structure for MNIST is “32C5-MP2-64C5-MP2-512FC-SVM”, and that for CIFAR10 and SVHN is “2×(128C3)-MP2-2×(256C3)-MP2-2×(512C3)-MP2-1024FC-SVM”. Here MP, C and FC stand for max pooling, convolution and full connection, respectively. Specifically, 2× (128C3) denotes 2 convolution layers with 3×3 kernel and 128 feature maps, MP2 means max pooling with window size 2× 2 and stride 2, and 1024FC represents a full-connected layer with 1024 neurons.Here SVM is a classifier with squared hinge loss (L2-Support Vector Machine)right after the output layer. All the inputs are normalized into the range of [-1,+1]. As for CIFAR10 and SVHN, we adopt the similar augmentation in <cit.>, i.e. 4 pixels are padded on each side of training images, and a 32× 32 crop is further randomly sampled from the padded image and its horizontal flip version. In the inference phase, we only test using the single view of the original 32× 32 images. The batch size over MNIST, CIFAR10, SVHN are 100, 1000 and 1000, respectively. Inspired by <cit.>, the learning rate decays at each training epoch by LR=α· LR, where α is the decay factor determined by √(LR_fin/LR_start). Here LR_start and LR_fin are the initial and final learning rate, respectively, and Epochs is the number of total training epochs. The transition probability factor in equation (<ref>) satisfies m=3, the derivative approximation uses rectangular window in Fig. <ref>(c) where a=0.5. The base algorithm for gradient descent is Adam, and the presented performance is the accuracy on testing set. §.§ Performance comparisonThe networks for comparison in Table <ref> are listed as follows:GXNOR-Nets in this paper (ternary synaptic weights and ternary neuronal activations), BNNs or XNOR networks (binary synaptic weights and binary neuronal activations), TWNs (ternary synaptic weights and full-precision neuronal activations), BWNs (binary synaptic weights and full-precision neuronal activations), full-precision NNs (full-precision synaptic weights and full-precision neuronal activations). Over MNIST, BWNs <cit.> use full-connected networks with 3 hidden layers of 1024 neurons and a L2-SVM output layer, BNNs <cit.> use full-connected networks with 3 hidden layers of 4096 neurons and a L2-SVM output layer, while our paper adopts the same structure as BWNs <cit.>. Over CIFAR10 and SVHN, we remove the last full-connected layer in BWNs <cit.> and BNNs <cit.>. Compared with BWNs <cit.>, we just replace the softmax output layer by a L2-SVM layer. It is seen thatthe proposedGXNOR-Nets achieve comparable performance withthe state-of-the-artalgorithms and networks. In fact, the accuracy of 99.32% (MNIST), 92.50% (CIFAR10) and 97.37% (SVHN) has outperformed most of the existing binary or ternary methods. InGXNOR-Nets, the weights are always constrained in the TWS {-1,0,1} without saving the full-precision hidden weights like the reported networks in Table <ref>, and the neuronal activations are further constrained in the TAS {-1,0,1}. Theresults indicate that it is really possible to perform well even if we just use this kind of extremely hardware-friendly network architecture. Furthermore, Fig. <ref> presents the graph where the error curve evolves as a function of the training epoch. We can see that the GXNOR-Net can achieve comparable final accuracy, but convergesslower than full-precision continuous NN. §.§ Influence of m, a and r Weanalyze the influence of several parameters in this section. Firstly, we study the nonlinear factor m in equation (<ref>) for probabilistic projection. The results are shown in Fig. <ref>, in which larger m indicates stronger nonlinearity. It is seen that properly increasing m would obviously improve the network performance, while too large m further helps little. m=3 obtains the best accuracy, that is the reason why we use this value for other experiments.Secondly, we use the rectangular approximation in Fig. <ref>(c) as an example to explore the impact of pulse width on the recognition performance, as shown in Fig. <ref>. Both too large and too small a value would cause worse performance and in our simulation, a=0.5 achieves the highest testing accuracy. In other words, there exists a best configuration for approximating the derivative of non-linear discretized activation function.Finally, we investigate the influence of this sparsity on the network performance, and the results are presented in Fig. <ref>. Here the sparsity represents the fraction of zero activations. By controlling the width of sparse window (determined by r) in Fig. <ref>(a), the sparsity of neuronal activations can be flexibly modified. It is observedthat the network usually performs better whenthe state sparsity properly increases. Actually, the performance significantly degrades when the sparsity further increases, and it approaches0when the sparsity approaches 1. This indicates that there exists a best sparse space for a specified network and data set, which is probably due to the fact that the proper increase of zero neuronal activations reduces the network complexity, and the overfittingcan beavoided to a great extent, like the dropout technology <cit.>. But the valid neuronal information will reduce significantly if the networkbecomestoo sparse, which causes the performance degradation. Based on this analysis, it is easily to understand the reason that why the GXNOR-Nets in this paper usually perform better than the BWNs, BNNs and TWNs. On the other side, a sparser network can be more hardware friendly which means that it is possible to achieve higher accuracy and less hardware overhead in the meantime by configuring the computationalsparsity. §.§ Event-driven hardware computing architectureForthe different networks in Table <ref>, the hardware computing architectures can bequite different. As illustrated in Fig. <ref>, we present typical hardware implementation examples for atriple-input-single-output neural network, and the corresponding original network is shown in Fig. <ref>(a). The conventional hardware implementation for full-precision NN is based on multipliers for the multiplications of activations and weights,and accumulator for the dendritic integration, as shown in Fig. <ref>(b). Althougha unit for nonlinear activation function is required, we ignore this in all cases of Fig. <ref>, so that we can focus on the influence on the implementation architecture with different discrete spaces. The recent BWN in Fig. <ref>(c) replaces the multiply-accumulate operations by a simple accumulation operation, with the help of multiplexers. When W_i=1, the neuron accumulates X_i; otherwise, the neuron accumulates -X_i. In contrast, the TWN in Fig. <ref>(d) implements the accumulation under an event-driven paradigm by adding a zero state into the binary weight space. When W_i=0, the neuron is regarded as resting; only when the weight W_i is non-zero, also termed as an event, the neuron accumulation will be activated. In this sense, W_i acts as a control gate. By constraining both the synaptic weights and neuronal activations in the binary space, the BNN in Fig. <ref>(e) further simplifies the accumulation operations in the BWN to efficient binary logic XNOR and bitcount operations. Similar to the event control of BNN, the TNN proposed in this paper further introduces the event-driven paradigm based on the binary XNOR network. As shown in Fig. <ref>(f), only when both the weight W_i and input X_i are non-zero, the XNOR and bit count operations are enabled and started. In other words, whetherW_i orX_i equals to zero or not plays the role of closing or opening of the control gate, hence the name of gated XNOR network (GXNOR-Net) is granted.Table <ref> shows the required operations of the typical networks in Fig. <ref>. Here we assume that the input number of the neuron is M, i.e. M inputs and one neuron output. We can see that the BWN removes the multiplications in the original full-precision NN, and the BNN replaces the arithmetical operations to efficient XNOR logic operations. While, in full-precision NNs, BWNs (binary weight networks), BNNs/XNOR networks (binary neural networks), most states of the activations and weights are non-zero. So their resting probability is ≈ 0.0%. Furthermore, the TWN and GXNOR-Net introduce the event-driven paradigm. If the states in the ternary space {-1,0,1} follow uniform distribution, the resting probability of accumulation operations in the TWN reaches 33.3%, and the resting probability of XNOR and bitcount operations inGXNOR-Netfurther reaches 55.6%. Specifically, in TWNs (ternary weight networks), the synaptic weight has three states {-1, 0, 1} while the neuronal activation is fully precise. So the resting computation only occurs when the synaptic weight is 0, with average probability of 1/3≈ 33.3%. As for the GXNOR-Nets, both the neuronal activation and synaptic weight have three states {-1, 0, 1}. So the resting computation could occur when either the neuronal activation or the synaptic weight is 0. The average probability is 1-2/3×2/3=5/9≈ 55.6%. Note that Table 2 is based on an assumption that the states of all the synaptic weights and neuronal activations subject to a uniform distribution. Therefore the resting probabilityvariesfrom different networks and data sets andthe reported values can only be used as rough guidelines.Fig.<ref> demonstrates an example of hardware implementation of the GXNOR-Net from Fig. <ref>. The original 21 XNOR operations can be reduced to only 9 XNOR operations, and the required bit width for the bitcount operations can also be reduced. In other words, in a GXNOR-Net, most operations keep in the resting state until the valid gate control signals wake them up, determined by whether both the weight and activation are non-zero. This sparse property promises the design of ultra efficient intelligent devices with the help of event-driven paradigm, like the famous event-driven TrueNorth neuromorphic chip from IBM <cit.>. §.§ Multiple states in the discrete space According toFig. <ref> and Fig. <ref>, we know that the discrete spaces of synaptic weights and neuronal activations can have multi-level states. Similar to the definition of Z_N in (<ref>), we denote the state parameters ofDWS and DAS as N_1 and N_2, respectively.Then, the available state number of weights and activations are 2^N_1+1 and 2^N_2+1, respectively. N_1=0 or N_1=1 corresponds to binary or ternary weights, and N_2=0 or N_2=1 corresponds to binary or ternary activations. We test the influence of N_1 and N_2 over MNIST dataset, and Fig. <ref> presents the results where the larger circle denotes higher test accuracy. In the weight direction,it is observed that when N_1=6, the network performs best; while in the activation direction, the best performance occurs when N_2=4. This indicates thereexists a best discrete space in either the weight direction or the activation direction, which is similar to the conclusion from the influence analysis of m in Fig. <ref>, a in Fig. <ref>, and sparsity in Fig. <ref>. In this sense, the discretization is also an efficient way to avoid network overfittingthat improvesthe algorithm performance. The investigation in this section can be used as a guidance theory to help us choose a best discretization implementation for a particular hardware platform after considering its computation and memory resources.§ CONCLUSION AND DISCUSSIONThis workprovidesa unified discretization framework for both synaptic weights and neuronal activations in DNNs, where the derivative of multi-step activation function is approximated and the storage of full-precision hidden weights is avoided by using a probabilistic projection operator to directly realize DST. Based on this, thecomplete back propagation learning process can be conveniently implemented when both the weights and activations are discrete. In contrast to the existing binary or ternary methods, our model can flexibly modify the state number of weights and activations to make it suitable for various hardware platforms, not limited to the special cases of binary or ternary values. We test our model in the case of ternary weights and activations (GXNOR-Nets)over MNIST, CIFAR10 and SVHN datesets, and achieve comparable performance with state-of-the-art algorithms. Actually, the non-zero stateof the weight and activation acts as a control signal to enable the computation unit, or keep it resting. Therefore GXNOR-Nets can be regarded as onekind of “sparsebinary networks" where the networks' sparsity can be controlled through adjusting a pre-given parameter. What's more, this “gated control” behaviour promises the design of efficient hardware implementation by using event-driven paradigm, and this has been compared with several typical neural networks and their hardware computing architectures. Thecomputation sparsity and the number of statesinthe discrete space can be properly increased to further improve the recognition performance ofthe GXNOR-Nets. Wehave also tested theperformance of thetwo curvesin Fig. <ref> for derivative approximation.It is foundthat the pulse shape (rectangle or triangle) affect less on the accuracy compared to the pulse width (or steepness) as shown in Fig. <ref>. Therefore, we recommend to use the rectangular one in Fig. <ref>(c) because it is simpler than the triangular curve in Fig. <ref>(d),which makesthe approximation more hardware-friendly. Through above analysis, we know that GXNOR-Net can dramatically simplify the computation in the inference phase and reduce the memory cost in the training/inference phase. However, regarding the training computation, although it can remove the multiplications and additions in the forward pass and remove most multiplications in the backward pass, it causes slower convergence and probabilistic sampling overhead. On powerful GPU platform with huge computation resources, it may be able to cover the overhead from these two issues by leveraging the reduced multiplications. However, on other embedded platforms (e.g. FPGA/ASIC), they require elaborate architecture design. Although the GXNOR-Nets promise the event-driven and efficient hardware implementation, the quantitative advantages are not so huge if only based on current digital technology. This is because thegeneration of the control gate signals also requires extra overhead. But the power consumption can be reduced to a certain extent because of the less state flips in digital circuits, which can be further optimized by increasing the computation sparsity. Even more promising, some emerging nanodevices have the similar event-driven behaviour, such as gated-control memristive devices <cit.>. 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"Lei Deng",
"Peng Jiao",
"Jing Pei",
"Zhenzhi Wu",
"Guoqi Li"
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"published": "20170525175941",
"title": "GXNOR-Net: Training deep neural networks with ternary weights and activations without full-precision memory under a unified discretization framework"
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"authors": [
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"published": "20170526123429",
"title": "An analytic superfield formalism for tree superamplitudes in D=10 and D=11"
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Nonlinear Physics Centre, Research School of Physics and Engineering, The Australian National University, Canberra ACT 2601, Australia School of Physics and Astronomy, University of Southampton, SO17 1BJ, Southampton, United Kingdom Nonlinear Physics Centre, Research School of Physics and Engineering, The Australian National University, Canberra ACT 2601, Australia Division of Physics and Applied Physics, Nanyang Technological University, Singapore Nonlinear Physics Centre, Research School of Physics and Engineering, The Australian National University, Canberra ACT 2601, Australia School of Physics and Astronomy, University of Southampton, SO17 1BJ, Southampton, United Kingdom Department of Physics and Astronomy, University of Pittsburgh, PA 15260, USA Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA Department of Electrical Engineering, Princeton University, Princeton, New Jersey 08544, USA Department of Physics and Astronomy, University of Pittsburgh, PA 15260, USA School of Physics and Astronomy, University of Southampton, SO17 1BJ, Southampton, United Kingdom SPIN-CNR, Viale del Politecnico 1, I-00133 Rome, Italy Laser Physics Centre, Research School of Physics and Engineering, The Australian National University, Canberra ACT 2601, Australia Nonlinear Physics Centre, Research School of Physics and Engineering, The Australian National University, Canberra ACT 2601, Australia We demonstrate generation of chiral modes – vortex flows with fixed handedness in exciton-polariton quantum fluids. The chiral modes arise in the vicinity of exceptional points (non-Hermitian spectral degeneracies) in an optically-induced resonator for exciton polaritons. In particular, a vortex is generated by driving two dipole modes of the non-Hermitian ring resonator into degeneracy. Transition through the exceptional point in the space of the system's parameters is enabled by precise manipulation of real and imaginary parts of the closed-wall potential forming the resonator. As the system is driven to the vicinity of the exceptional point, we observe the formation of a vortex state with a fixed orbital angular momentum (topological charge). Our method can be extended to generate high-order orbital angular momentum states through coalescence of multiple non-Hermitian spectral degeneracies, which could find application in integrated optoelectronics.Chiral modes at exceptional points in exciton-polariton quantum fluids. E. A. Ostrovskaya=======================================================================Introduction. Exceptional points in wave resonators of different origin arise when both spectral positions and linewidths of two resonances coincide and the corresponding spatial modes coalesce into one <cit.>. Originally identified as an inherent property of non-Hermitian quantum systems <cit.>, exceptional points have become a focus of intense research in classical systems with gain and loss <cit.>, such as optical cavities <cit.>, microwave resonators <cit.>, and plasmonic nanostructures <cit.>. The counterintuitive behaviour of a wave system in the vicinity of an exceptional point led to demonstrations of a range of peculiar phenomena, including enhanced loss-assisted lasing <cit.>, unidirectional transmission of signals <cit.>, and loss-induced transparency <cit.>.Due to the nontrivial topology of the exceptional point, the two eigenstates coalesce with a phase difference of ± π/2, which results in a well-defined handedness (chirality) of the surviving eigenstate <cit.>. This remarkable property of the eigenstate at the exceptional point was first experimentally demonstrated in a microwave cavity <cit.> and, very recently, led to observation of directional lasing in optical micro-resonators <cit.>. So far, the chirality of the unique eigenstate at an exceptional point has not been demonstrated in any quantum system.In this work, we demonstrate formation of a chiral state at an exceptional point in a macroscopic quantum system of condensed exciton polaritons. Exciton polaritons are hybrid light-matter bosonic quasiparticles arising due to strong coupling between excitons and photons in semiconductor microcavities <cit.>. Once sufficient density of exciton polaritons is injected by an optical or electrical pump, the transition to quantum degeneracy occurs, whereby typical signatures of a Bose-Einstein condensate emerge <cit.>. Radiative decay of polaritons results in the need for a continuous pump to maintain the population. This intrinsic open-dissipative nature of exciton-polariton condensates offers a new platform for study of non-Hermitian quantum physics. Several recent experiments have exploited the non-Hermitian nature of exciton-polariton systems <cit.>. Importantly, the existence of exceptional points and the associated topological Berry phase has been demonstrated in an optically-induced resonator (quantum billiard) for coherent exciton-polariton waves <cit.>.An optically-induced exciton-polariton resonator is a closed-wall potential arising due to injection of high-energy excitonic quasiparticles by an off-resonant optical pump and strong repulsive interaction between the excitonic reservoir and the condensate <cit.>. The size of the resonator is comparable to the de Broglie wavelength of the condensed exciton polaritons, and its geometry is defined by the spatial distribution of the optical pump <cit.>. Observation of the exceptional points in the spectra of exciton-polariton resonators is enabled by two characteristic features of this system. First, the optically confined exciton polaritons form a multi-mode condensate, i.e. they can occupy several single-particle energy states of the pump-induced effective potential <cit.>. Secondly, the pump-induced potential is non-Hermitian, and both real (energy) and imaginary (linewidth) parts of its complex eigenenergies can be precisely controlled by adjusting parameters of the pump <cit.>. As a result, two or more eigenstates of the system can be brought to degeneracy. Here, we create a non-Hermitian trapping potential for exciton polaritons in the form of an asymmetric ring resonator, and observe condensation into several trapped modes. By changing the geometry of the pump, and therefore the overlap of the modes with the gain region, we tune the imaginary part of the optically-induced potential and observe the transition from crossing to anti-crossing of complex eigenvalues, which signals the existence of an exceptional point. Furthermore, the high-Q tapered microcavity used for our experiments <cit.> enables precise control over the ratio of the exciton and photon in the hybrid quasiparticle. We use this additional control parameter to drive the two lowest-lying dipole states of the system to a vicinity of an exceptional point and confirm the formation of a chiral mode – a charge one vortex – in analogy with microwave <cit.> and optical <cit.> resonators. In addition, we demonstrate formation of a mode with a higher-order topological charge (orbital angular momentum), when two coexisting exceptional points are tuned into close proximity of one another, thus opening the avenue for experimental tests of topological properties of higher-order exceptional points. Experiment. We create exciton polaritons in a high-Q GaAs/AlGaAs microcavity similar to that used in Ref. <cit.>. The details about the experimental setup can be found inSupplemental Material (SM). By utilising a digital micromirror device (DMD), we morph the pump spot into the asymmetric ring shape shown schematically in Fig. <ref>. The pump simultaneously populates the system with exciton polaritons and forms a closed-wall potential due to the local blue shift in energy induced by the excitonic reservoir <cit.>. Similarly to Ref. <cit.>, the inner area of the ring is kept constant and the varying widths of the potential walls, W and w, affect the overlap between exciton-polariton condensate modes and the gain region. Analysis of the microcavity photoluminescence by means of energy resolved near-field (real-space) imaging allows us to obtain the spatial density distribution and energy levels corresponding to the condensate modes in the ring resonator. All experiments are performed in the strong coupling regime and the pump power is kept at around 1.5 times that needed for condensation.Due to the asymmetry of the potential walls imposed by the pumping geometry, as well as by the cavity gradient <cit.>, the eigenmodes of the optically-induced ring resonator resemble the Ince-Guassian modes <cit.>. Once the threshold for the condensation is reached, only a few modes of this resonator are occupied, and we focus on the two lowest-lying dipole modes (1,1) with the orthogonal orientation of the nodal lines. Due to their orientation, the two dipole modes have different overlap with the exciton reservoir and are, in general, not energy degenerate and well separated from the other modes (seeSM). Because of the non-Hermitian nature of exciton polaritons, the eigenenergy of the modes in the ring potential is also complex, where the real part corresponds to the energy peak position, and the imaginary part corresponds to the linewidth. In our experiment, the peak positions corresponding to the two dipole modes can be tuned by changing the relative admixture of exciton and photon in the exciton polariton. This is achieved by keeping all parameters of the pump fixed and changing the relative position of the excitation beam and the sample, as a sizeable linear variation in the microcavity widths <cit.> results in variation of detuning between the cavity photon mode and bare exciton: Δ=E_ph-E_ex. The change in the energies of the two dipole modes with changing detuning is shown in Fig. <ref>(a), where a clear crossing of the corresponding energy levels can be observed. (The corresponding dispersions below condensate threshold for varying detuning can be seen inSM). The transition to spectral degeneracy is accompanied by an avoided crossing of the imaginary parts of the eigenenergy (linewidths), as seen in Fig. <ref>(b). We stress that, due to the high quality factor of the microcavity and long lifetime of exciton polaritons in this microcavity (∼ 200 ps), the resonances have a very narrow linewidth, which helps to differentiate between closely positioned energy levels.From the energy resolved spatial image of the cavity photoluminescence, we reconstruct the spatial probability density distribution of the polariton condensate wavefunction for each detuning value. Away from the degeneracy, the two dipole modes are clearly visible [insets in Fig. <ref>(a)]. As the two complex eigenvalues are tuned in and out of the degeneracy, we observe the characteristic exchange of the modes corresponding to the two energy branches. This kind of mode switch may indicate the existence of an exceptional point, so we vary a second control parameter in our system, which is the width of the right half-ring of the resonator, w (see Fig. <ref>). As the value of this parameter is changed from w=w_1 to w=w_2, without changing the size of the resonator, D, we observe transition from crossing to anticrossing in energy and from anticrossing to crossing in the linewidths of the two resonances corresponding to the dipole modes [see Figs. <ref>(c,d)]. This transition confirms that an exceptional point exists in the parameter space (Δ,w) <cit.>. Furthermore, by fixing the detuning at -3.6 meV, which corresponds to the energy crossing in Fig. <ref>(a), and tuning the right half-ring width from w_2 to w_1, we observe a vortex-like mode at w_1<w<w_2, as shown in Fig. <ref>(e).In order to ascertain the nature of this state, we perform interferometry with a magnified (× 15) reference beam derived from a small flat-phase area of the photoluminescence. First of all, the energy-resolved interferometric imaging confirms the phase structure of the two dipole modes Fig. <ref> (a,b) and (c,d) away from the degeneracy point. Furthermore, the interference pattern shown in Fig. <ref> (f) is stable for many minutes, and reveals a fork in the fringes which is a clear signature of a stable charge-one vortex. This measurement therefore confirms that a vortex with the topological charge one is formed in the vicinity of the spectral degeneracy, which is only possible if the two dipole modes coalesce with the π/2 phase difference. We note that due to the finite energy (frequency) resolution of spectroscopic measurements in our system, it is impossible to tune the system exactly to the exceptional point, whereby both real and imaginary parts of the complex egenenergy, as well as the eigenstates, would coalesce. Theory. The full dynamics of an exciton-polariton condensate subject to off-resonant (incoherent) optical pumping can be described by the generalised complex Gross-Pitaevskii equation for the condensate wave function complemented by the rate equation for the density of the excitonic reservoir <cit.>:i ħ∂ψ/∂ t= {-ħ^2/2m∇^2+g_c |ψ|^2+g_R n_R +i ħ/2[ Rn_R-γ_c ] }ψ,∂ n_R/∂ t= P(r)-(γ_R+R |ψ|^2) n_R,where P(r) is the rate of injection of reservoir particles per unit area and time determined by the pump power and spatial profile of the pump beam, g_c and g_R characterise interaction between condensed polaritons, and between the polaritons and the reservoir, respectively. The decay rates γ_c and γ_R quantify the finite lifetime of condensed polaritons and the excitonic reservoir, respectively. The stimulated scattering rate, R, characterises growth of the condensate density. Assuming that, under cw pumping, the reservoir reaches a steady state,n_R(r)=P(r)/(γ_R+R |ψ(r)|^2), and that the exciton-polariton density is small near the condensation threshold, Eq. (<ref>) transforms into a linear Schrödinger equation for the condensate wavefunction, ψ, confined to an effective non-Hermitian potential <cit.>:V(r)=V_R(r)+i V_I (r)≈ -g_R P(r)/γ_R+iħ/2[ R P(r)/γ_R-γ_c ].Both real and imaginary parts of this potential depend on the spatial profile of the pump P(r), as well as on the exciton-photon detuning, Δ. As discussed inSM, the dependence on the detuning enters the model Eq. (<ref>) through the dependence of its parameters on the Hopfield coefficient that characterises the excitonic fraction of the exciton polariton <cit.>: |X|^2 =(1/2)[ 1+ Δ/√(Δ^2+E_R^2)], where E_R is the Rabi splitting at zero detuning. The eigenstates and complex eigenvalues of the non-Hermitian potential (<ref>) can be found by solving the non-dimensionalised stationary equation:[-∇^2 + (V'+iV”) ]ψ=Ẽψ.where we have introduced the scaling units of length, L=D, energy E_0=ħ^2/(2m_phL^2), and time T=ħ/E_0, m_ph is the effective mass of the cavity photon, and Ẽ is the complex eigenenergy normalised by E_0/(1-|X|^2). The normalised real and imaginary parts of the potential are: V'( r)=V'_0 Γ_X P( r)/P_0, V”( r)=V”_0 [Γ_X P( r)/P_0-1], where V'_0=g_exγ_ph/(E_0R_0), V”_0=ħγ_ph/(2E_0), and P_0=γ_Rγ_ph/R_0, where γ_ph is the lifetime of the cavity photon, g_ex is the stregth of the exiton-exiton interaction <cit.>, and we have introduced a base value for the stimulated scattering rate R_0 (seeSM). The dependence on detuning via the Hopfield coefficient is captured by the parameter Γ_X=|X|^2/(1-|X|^2). For simplicity, we approximate the shape of P(r) by Gaussian envelope functions, with the resulting shapes of V'( r) and V”( r) shown in Fig. <ref>(a,d).By solving the eigenequation (<ref>) numerically and sorting values of Ẽ in the ascending order of its real part, E_n, we obtain the corresponding hierarchy of eigenstates ψ_n. Figures <ref>(b) and (e) show the moduli of ψ_2 and ψ_3, respectively, which correspond to the slightly deformed dipole states (1,1) with the orthogonal orientation of nodal lines. Importantly, the existence of these steady states of the exciton polariton condensate in the pump-induced potential is also confirmed by full dynamical simulations of the model equation (<ref>) (seeSM).The dependence of the real and imaginary parts of the dipole modes eigenenergies on the experimental control parameters, Δ and w, is shown in Fig. <ref>. We can see that our simple linear model reflects the qualitative behaviour observed in the experiment (Fig. <ref>).To understand why the experimental control parameters Δ and w allow us to tune the system in and out of the vicinity of the exceptional point, we follow the standard approach <cit.>, and construct a phenomenological couple-mode model for the two modes with the quantum numbers n and n' near degeneracy (seeSM for the details). The resulting effective two-mode interaction Hamiltonian can be written as follows:Ĥ=[[ Ẽ_n q; q^* Ẽ_n^'; ]], Ẽ_n,n^'=E_n,n^'-iΓ_n,n^',where Ẽ_n,n^' are the complex eigenenergies of the uncoupled modes and q characterises their coupling strength.The eigenvalues of the Hamiltonian (<ref>) are λ_n,n'=Ẽ±√(δẼ^2+|q|^2), where Ẽ=(Ẽ_n+Ẽ_n')/2≡ E+iΓ, and δẼ=(Ẽ_n-Ẽ_n')/2≡δ E-iδΓ. The real and imaginary parts of λ_n,n' form Riemann surfaces with a branch-point singularity in the space of parameters (δ E,δΓ) <cit.>, as shown inSM. At the exceptional points, iδẼ_EP=±|q|, the eigenvalues coalesce, λ_n=λ_n'. The eigenstates also coalesce and form a single chiral state <cit.>. In our system, the two eigenstates corresponding to n=2 and n'=3 are dipole modes, and therefore the chiral state is a vortex with a well-defined topological charge one, as shown in Fig. <ref>(c,f) and Fig. <ref>(c,d). The two parameters (δ E,δΓ) can be related to the experimental parameters (Δ,w). As discussed inSM, increasing Δ corresponds to increasing δ E and moves the system away from the spectral degeneracy, while growing w corresponds to decreasing δΓ. Therefore the variable detuning and the width of the resonator's wall allow us to control the approach to the exceptional point as demonstrated in the experiment (Fig. <ref>) and confirmed by theory (Fig. <ref>).Conclusion. In summary, we have experimentally demonstrated the chirality of an eigenstate of a non-Hermitian macroscopic quantum coherent system of Bose-condensed exciton polaritons in the vicinity of an exceptional point. In our experiment, the chiral eigenstate is a vortex with a well-defined, deterministic topological charge (orbital angular momentum). We stress that, contrary to the previous demonstration of an exciton polariton vortex with a well-controlled charge <cit.>, the shape of the optically induced ring resonator that creates a non-Hermitian potential for exciton polaritons in this work does not break the chiral symmetry. In such case, one would expect that the sign of the vortex charge would change randomly between realisations of the experiment <cit.>, which is not the case here. The deterministic nature of the vortex charge has been confirmed in our experiment by blocking the pump for a sufficient time interval (up to 1 hour) to let the pump-injected reservoir disappear. After the pumping is resumed, the charge of the vortex remains the same. The controlled and reliable generation of a chiral state with a prescribed topological charge could find use in polariton-based integrated optoelectronic devices. Last but not least, our technique of generating chiral states can be applied to higher-order modes in the non-Hermitian exciton-polariton system. In particular, if the inner size of the ring, D, is increased, higher-order modes can be populated by the condensate and brought to degeneracy by varying control parameters of the system. Moreover, two (or more) exceptional points can be simultaneously created and brought to a close vicinity of each other. As an example, the higher-order orbital momentum state formed in our experiment by hybridisation of two chiral modes, ψ=ψ_7+i ψ_8+ψ_9+i ψ_10, is shown in Fig. <ref>, together with the prediction of our linear theory. 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"T. C. H. Liew",
"D. Comber-Todd",
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sf theoremTheorem[section] lemma[theorem]Lemma proposition[theorem]Proposition corollary[theorem]Corollary definition definition[theorem]Definition rem[theorem]Remark example[theorem]Example equationsection | http://arxiv.org/abs/1705.09830v1 | {
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Is Our Model for Contention Resolution Wrong? Confronting the Cost of Collisions William C. Anderton Department of Computer Science and EngineeringMississippi State UniversityMississippi, 39762, USAEmail: Maxwell YoungThis research is supported by the National Science Foundation grant CCF 1613772 and by a research gift from C Spire. Department of Computer Science and EngineeringMississippi State UniversityMississippi, 39762, USAEmail:Submitted 25 May 2017. Accepted 26 September 2017. ========================================================================================================================================================================================================================================================================================================================================================================================== Randomized binary exponential backoff (BEB) is a popular algorithm for coordinating access to a shared channel. With an operational history exceeding four decades, BEB is currently an important component of several wireless standards. Despite this track record, prior theoretical results indicate that under bursty traffic (1) BEB yields poor makespanand (2) superior algorithms are possible. To date, the degree to which these findings manifest in practice has not been resolved.To address this issue, we examine one of the strongest cases against BEB: n packets that simultaneously begin contending for the wireless channel. Using Network Simulator 3, we compare against more recent algorithms that are inspired by BEB, but whose makespan guarantees are superior. Surprisingly, we discover that these newer algorithms significantly underperform. Through further investigation, we identify as the culprit a flawed but common abstraction regarding the cost of collisions.Our experimental results are complemented by analytical arguments that the number of collisions – and not solely makespan – is an important metric to optimize. We believe that these findings have implications for the design of contention-resolution algorithms.Is Our Model for Contention Resolution Wrong? Confronting the Cost of Collisions William C. Anderton Department of Computer Science and EngineeringMississippi State UniversityMississippi, 39762, USAEmail: Maxwell YoungThis research is supported by the National Science Foundation grant CCF 1613772 and by a research gift from C Spire. Department of Computer Science and EngineeringMississippi State UniversityMississippi, 39762, USAEmail:Submitted 25 May 2017. Accepted 26 September 2017. ==========================================================================================================================================================================================================================================================================================================================================================================================§ INTRODUCTION Randomized binary exponential backoff (BEB) plays a critical rolein coordinating access by multiple devices to a shared communication medium. Given its importance, BEB has been studied at length and is known to yield good throughput under well-behaved traffic <cit.>. In contrast, when traffic is “bursty”, BEB is suspected to perform sub-optimally.Under a single batch of n packets that simultaneously begin contending for the channel, Bender et al. <cit.> prove that BEB hasΘ(nlog n) makespan (the amount of time until all packets are successfully transmitted). More recentalgorithms have been proposed <cit.> with improved makespan regardless of the traffic type.Together, these results beg the question: How do newer algorithms compare to BEB in practice? Here, we make progress towards an answer by restricting ourselves to bursty traffic – in particular, the simplest instance of such traffic: a single burst (batch) of packets. This is a prominent case where BEB is anticipated to do poorly, and it should be a straightforward (if laborious) exercise to discover which of the following situations is true: (1) A newer contention-resolution algorithm outperforms BEB, or (2) BEB outperforms newer contention-resolution algorithms.Interestingly, neither of these outcomes is very palatable. In one form or another, BEB has operated in networks for over four decades and it remains an essential ingredient in several wireless standards.Bursty traffic can arise in practice <cit.> and its impact has been examined <cit.>. If (1) holds, then BEB is potentially in need of revision and the ramifications of this are hard to overstate.Conversely, if (2) holds, then theoretical results are not translating into improved performance.At best, this is a matter of asymptotics. At worst, this indicates a problem with the abstract model upon which newer resultsare based. In this latter case, it is important to understand what assumptions are faulty so that the abstract model can be revised.§.§ A Common ModelGiven n stations, the problem of contention resolution addresses the amount of time until any one of thestations transmits alone. A natural consideration is the time until a subset of k stations each transmits alone; this often falls under the same label, but is also referred to as k-selection (see <cit.>). We focus on the case of k=n. Here, much of the algorithmic work shares an abstract model.Three common assumptions are: Assumption A0 is near universal, but technically inaccurate for reasons discussed in Section <ref>. To summarize, slots in a contention window are used to obtain ownership of the channel. However, transmission of the full packet may occur past this contention-window slot while all other stations pause their execution. Therefore, this assumption is sufficiently close to reality that we should not expect performance to deviate greatly as a result.Examples of assumption A1 abound (for example <cit.>), although variations exist. Acompelling alternative is the signal-to-noise-plus-interference (SINR) model <cit.> which is less strict about failure in the event of simultaneous transmissions. Another model that has received attention is the affectance model <cit.>. Nevertheless, these all share the reasonable assumption that simultaneous transmissions may negatively impact performance.Assumption A2 is also widely adopted (see the same examples for A1) and implicitly addresses two quantities that affect performance: the time to transmit a packet, and the time to receive any feedback on success or failure. Assigning a delay of 1 slot to these quantities admits a model where the problem of contention resolution is treated separately from the functionality for collision detection. Such functionality is provided by a medium access control (MAC) protocol– of which the contention-resolution algorithm is only one component – and is not captured by A2.§.§.§ Demonstrating a Flawed Assumption Our main thesis is that A2 isflawed in the wireless setting; that is,the cost of failure is far more significant than the abstract model acknowledges. This is not a matter of minor adjustments to the assumption, or an artifact of hidden constants in the algorithms examined. Rather, the way in which failures – in particular, collisions – are detected cannot be isolated from the problem of contention resolution. Several corollaries follow from this thesis, all indicating that accounting for such failures should be incorporated into algorithm design. For a range of wireless settings, contention-resolution algorithms that ignore this will likely not perform as advertisedwhen deployed within a MAC protocol (see Section <ref>). We demonstrate this for the popular IEEE 802.11g standard.§.§ Overview of BEB in IEEE 802.11g To understand our findings, it is helpful to summarize IEEE 802.11g and how BEB operates within it. However, outside of this section and the description of our experimental setup,discussion of such aspects and terminology is kept to a minimum. Throughout, we will often use interchangeably the terms packets and stations depending on the context; the two uses are equivalent given that each station seeks to transmit a single packet in the single-batch case.Exponential backoff <cit.> is a widely deployed algorithm for distributed multiple access. Informally, a backoff algorithm operates over a contention window (CW) wherein each station makes a single randomly-timed access attempt. In the event of two or more simultaneous attempts, the result is a collision and none of the stations succeed.Backoff seeks to avoid collisions by dynamically increasing the contention-window size such that stations succeed.IEEE 802.11 handles contention resolution via the distributed coordination function (DCF) which employs BEB; as the name suggests, successive CWs double in size under BEB. The operation of DCF is summarized as follows. Prior to transmitting data, a station first senses the channel for a period of time known as a distributed inter-frame space (DIFS). If the channel is not in use over the DIFS, the station transmits its data; otherwise, it waits until the current transmission finishes and then initiates BEB.For acontention window of size CW, a timer value is selected uniformly at random from[0, CW-1]. So long as the channel is sensed to be idle, the timer counts down and, when it expires, the station transmits. However, if at any prior timethe channel is sensed busy, BEB is paused for the duration of the current transmission, and then resumed (not restarted) after another DIFS.After a station transmits, it awaits an acknowledgement (ACK) from the receiver. If the transmission was successful, then the receiver waits for a short amount of time known as a short inter-frame space (SIFS) – of shorter duration than a DIFS –before sending the ACK. Upon receiving an ACK, the station learns that its transmission was successful.Otherwise, the station waits for an ACK-timeout duration before concluding that a collision occurred.This series of actions is referred to as collision detection; the cost of which lies at the heart of our argument.If a collision is detected, then the station must attempt a retransmission via the same process with its CW doubled.Figure 1 illustrates the operation of DCF. Note that both the transmission of data and the acknowledgement process occur “outside” of the backoff component of DCF. Yet,the focus of many algorithmic results is solely on the slots of this backoff component. Finally,RTS/CTS (request-to-send and clear-to-send) is an optional mechanism. Informally, a station will send an RTS message and await an CTS message from the receiver prior to transmitting its data. Due to increased overhead, RTS/CTS is often only enabled for large packets. Therefore, we focus on the case where RTS/CTS is disabled, although our experiments show that our findings continue to hold when this mechanism is used (see Section <ref>).§ EXPERIMENTAL SETUP We employ Network Simulator 3 (NS3) <cit.> which is a widely used network simulation tool in the research community <cit.>.Our experimental setup is described here for the purposes of reproducibility.[Our simulation code and data will be made available at .]Our reasons for using NS3 are twofold. First, wireless communication is difficult to model and employing NS3 helps allay concerns that our findings are an artifact of poorly-modeled wireless effects.Second, given the assumptions upon which contention-resolution algorithms are based, NS3 can reveal whether we are being led astray by an assumption that appears reasonable, but results in a significant discrepancy between theory and practice.Table <ref> provides our experimental parameters. Path-loss models with default parameters are known to be faithful <cit.> and, therefore, our experiments employ thelog-distance propagation loss model in NS3. For transmission and reception of packets (frames), we use the YANS <cit.> module which provides an additive-interference model. At the MAC layer, we make use of IEEE 802.11g and we implement changes to the growth of the contention window based on the algorithms we investigate. All experiments use IPv4 and UDP.The amount of overhead for each packet is 64-bytes: 8 bytes for UDP, 20 bytes for IP, 8 bytes for an LLC/Snap header, and 28 bytes of additional overhead at the MAC layer. The duration of an acknowledgement (ACK) timeout is specified by the most recent IEEE 802.11 standard[This is a large document; please see Section 10.3.2.9, page 1317 of <cit.>.] to be roughly the sum of a SIFS (16μs), standard slot time (9μs), and preamble (20μs); a total of 45μs.However, in practice, this is subject to tuning.In our experiments, an ACK-timeout below 55μs gave markedly poor performance; there is insufficient time for the ACK before the sender decides to retransmit. We use the default value of 75μs in NS3 since this is the same order of magnitude and performs well. In our experiments, n stations are placed in a40m× 40m grid, and they are laid out starting at the south-west corner of the grid moving left to right by 1 meter increments, and then up when the current row is filled. A wireless access point (AP) is located (roughly) at the center of the grid. We do not simulate additional terrain or environmental phenomena; our goal is to test the performance under ideal conditions without complicating factors. Our experiments are computationally intensive.Computing resources are provided by the High Performance Computing Collaboratory (HPC^2) at Mississippi State University. We employ four identical Linux (CentOS) systems, each with 16 processors (Intel Xeon CPU E5-2690, 2.90GHz) and 396 GB of memory. § A SINGLE BATCHWe examine a single batch of n packets that simultaneously begin their contention for the channel. As algorithmic competitors for , we take (LB), (LLB) from <cit.> and (STB) from <cit.>. Both LLB and LB are closely related to in that they execute using a CW that increases in size monotonically.The pseudocode for the algorithms LLB, LB, and BEB is provided in Figure 2. In contrast, STB is non-monotonic and executes over a doubly-nested loop. The outer loop sets the current window size W to be double that used in the preceding outer loop; this is like BEB. Additionally, for each such W, the inner loop executes over W windows of size W, W/2, ..., 2 and, for each window, a slot is chosen uniformly at random for the packet to transmit; this is the “backon” component of STB.Our Metrics. For a single batch of n packets, algorithmic results address the number of slots required to complete all n packets. These slots correspond only to those belonging to contention windows, even though many results refer to this as makespan. To avoid confusion, we will refer to this metric more explicitly by contention-window slots (CW slots).Table <ref> summarizes the known with-high-probability[With probability at least 1-1/n^c for a tunable constant c>1.] guaranteeson CW slots. Note that LB, LLB, and STB each have superior guarantees over BEB, with STB achieving Θ(n)CW slots which is asymptotically optimal. We also make use of a second metric. As described in Section <ref>, events occur outside of contention windows (such as SIFS, DIFS, full packet transmission, ACK timeouts). For the duration – including the time spent in contention windows – between when the single batch of packets arrives and when the last packet successfully transmits, we refer to total time. §.§ Theory and Experiment We begin by comparing the number of CW slots. The algorithms we investigate are designed to reduce this quantity since all slots in the abstract model occur within some contention window. Under this metric,LLB, LB, and STB are expected to outperform BEB. Throughout, when we report on performance,we are referring to median values for n=150. Percentage increases or decreases are calculated as 100×(A-B)/B where B is always the value for BEB (the “old” algorithm) and A corresponds to a value for one of LLB, LB, or STB (the “new” algorithms). §.§.§ Contention-Window Slots We provide results from our NS3 experiments using both small packets, with a 64-byte (B)payload, and large packets, with a 1024B payload.Figures 3 and 4 illustrate our experimental findings with respect to CW slots.[The following common approach is used to identify outliers in our data. Let Δ be the distance between the first and third quartiles. Any data point that falls outside a distance of 1.5 Δ from the median is declared an outlier. We emphasize this results in very few points being discarded; for example, only a single n value for our 64B experiments had 5 outliers (out of 30 trials), and the vast a majority had none.] The behavior generally agrees with theoretical predictions that each of LLB, LB, and STB should outperform BEB. Interestingly, LLB incurs agreater number of CW slots than LB despite despite the former's better asymptotic guarantees. We suspect this is an artifact of hidden constants/scaling and evidence of this is presented later in Section <ref>. Nevertheless,LLB, LB, and STB demonstrate improvements over BEB, giving a respective decrease of 49.4%, 68.2%, and 83.0%, respectively, with a 64B payload. Similarly, LLB, LB, STB demonstrate a respective decrease of 54.2%, 69.9%,84.2% with a 1024B payload.For comparison, Figure 5 depicts CW slots derived from a simple Java simulation that implements only the assumptions of the abstract model (it ignores wireless effects, details in the protocol stack, etc.). Our NS3 results also roughly agree with this data in terms of magnitude of values and the separation of BEB from the other algorithms; albeit, the performances of LLB, LB, and STB do not separate cleanly in this data. Finally, Figure 6 presents the number of CW slots (for 64B) required to complete half the packets, and we make two observations.First, the remaining n/2 packets are responsible for the bulk of the CW slots. Second, theimprovement over decreasesto 25.0%, 56.4%,and 77.7% for LLB, LB, and STB, respectively (and similarly for 1024B).This difference is due to “straggling” packets which survive until relatively large windows are reached. This impacts BEB more than other algorithms given its rapidly increasing window size (and, unlike STB, it does not have a “backon” component).§.§.§ Total Time It is tempting to consider the single-batch scenario settled. However, if we focus on the total time for both the 64B and 1024B payload sizes, then a different picture emerges.The degree to which these newer algorithms outperform is erased as seen from Figures 7 and 8. In fact, the order of performance is reversed with total time ordered from least to greatest as BEB,LLB, LB,STB. For 64B payloads, LLB, LB, and STB suffer an increase of 5.6%, 19.3%, and 26.5%, respectively, over BEB. For 1024B payloads, the increase is 9.1%, 25.4%, and 35.4%, respectively. Notably, the larger packet size seems to favor BEB. What about the time until n/2 packets are successfully transmitted? Perhaps newer algorithms do better for the bulk of packets, but suffer from a few stragglers? Interestingly, Figures 9 and 10 suggest that this is not the case. Indeed, for a 64B payload, BEB performs even better over LLB, LB, and STB with the latter exhibiting an increase of 13.1%, 17.3%, 25.4%, respectively. Similarly, for 1024B, the percent increase is 10.1%, 16.6%,26.6%, respectively. These findings are troubling since, arguably, total time is a more important performance metric in practice than CW slots. Critically, we note that this behavior is detected only through the use of NS3; it is not apparent from the simpler Java simulation. What is the cause of this phenomenon? §.§ The Cost of Collisions The number of ACK timeouts per station provides an important hint. As Figure 8 shows, the newer algorithms are incurring substantially more ACK timeouts which, in turn, corresponds to more (re)transmissions.This evidence points to collisions as the main culprit. In particular, the way in which collision detection is performed means that each collision is costly in terms of time. In support of our claim, we decompose this delay into three portions using BEB (for n=150) as an example throughout: (I) Transmission Time. (Re)transmissions are expensive. A packet of size 128B (64B payload plus 64B overhead) requires roughly 19 μs plus the associated 20μs preamble. The maximum number of ACK timeouts for BEB – and, thus, the number of collisions – experienced by an unlucky station is 9. Most collisions should involve only a handful of stations given the growth of CWs. (Why? Recall from Figure 6 that n/2 packets require the vast majority of CW slots to finish. These n/2 packets succeed only in larger windows –of size roughly equal to or greater than n for BEB – and do not finish immediately due to collisions, each of which should involve only a few stations given such window sizes). If two stations are involved in each collision, this results in 75(9/2) non-overlapping (or disjoint) collisions, for an aggregateduration of roughly 75(9/2)(19μ s + 20μ s) = 13,163μ s.[We do not add the time for the final/successful transmissions (this would only increase the value); our focus is on the transmissions associated with collisions.] (II) ACK Timeouts. Given a collision, the AP fails to obtain the transmission and the corresponding stations incur an ACK timeout before concluding that a collision occurred. This delay is significant – roughly 1,100μs for BEB with n=150 (see Figure 12) – but an order-of-magnitude less than the transmission time. (III) Contention-Window Slots. BEB incurs 886 CW slots for n=150, each of duration 9μs, spent in CWs which yields 7,974μ s. For BEB at n=150, these three values yield a very conservative lower bound on the total time of 22,237μ s; for instance, we have not accounted for the SIFS and DIFS. This back-of-the-envelope calculation conforms to the magnitude of values observed in Figure 7, and it highlights two important facts. First, both transmission time and CW slots contribute significantly to total time, with ACK timeouts being a distant third. Second, collisions greatly impact total time – far more than CW slots – by forcing retransmissions.Underlining this second point, we note that the transmission time for the 1024B payload is larger at roughly 75(9/2)(161μ s + 20μ s) = 61,088μ s.[The number of ACK timeouts for1024B is roughly the same, even though packet size has increased. This aligns with our findings in Section 4.] By comparison, the CW slots contribute roughly 973× 9μ s = 8,757μ s. ACK Timeout ≈ Collision. It is true that not all ACK timeouts necessarily imply that the corresponding packet suffered a collision. For example, an ACK might be lost due to wireless effects even if the packet was transmitted without any collision. Note that, in such a case, the sending station still diagnoses that a collision has occurred and so the same costs described in (I)-(III) hold. For our simple NS3 setup, virtually all ACK failures result from a collision. This is evident from Figure 13 which illustrates a trial with n=20 under BEB using a 64B payload. Collisions occur only when two or more stations transmit (duration of transmission denoted by a thick blue line) at the same time and the result is an ACK timeout event (indicated by a thin red line); in all other cases, the transmission is successful and the corresponding ACK is received. Disjoint Collisions. We observe that the total time does not grow linearly with the maximum number of ACK timeouts – equivalently, collisions – experienced by a station. Under LB, an unlucky station suffers roughly twice the number of ACK timeouts, but the total time of LB is not twice that of BEB. Why? Consider n stations where each collision involves only two stations. Then, there are n/2 disjoint collisions and each is added to the total time.In contrast, consider the opposite extreme where all n stations transmit at the same time and collide. Then, there is one collision which adds only a single failed transmission time to the total time.The number of stations involved in a single collision is larger for algorithms whose CWs grow more slowly, such as LB and LLB.For STB, a similar phenomenon is at work; the backon component yields collisions involving many stations. That is, LB, LLB, and STB are closer to the second case, while BEB is closer to the first. From our experimental results, we see assumption A2 is not accurate with regards to the cost of failure:RTS/CTS. Although it is not examined in detail in our work, we briefly remark on the use ofRTS/CTS. When enabled, stations can experience collisions among the RTS frames (instead of the packets). These are smaller in size (20B), but the remainder of the total-time calculation remains the same, and additional time is incurred due to additional inter-frame spaces and the transmission of CTS frames. For very large packets, we may expect RTS/CTS to mitigate the transmission-time cost, but not for small to medium-sized packets where the overhead from this mechanism might even cause worse performance. Ultimately, we observe the same qualitative behavior when RTS/CTS is enabled. For example, without RTS/CTS, recall from Section <ref> that the total time for LLB (BEB's closest competitor) increases by 5.6% and 9.1% for the 64B and 1024B, respectively, over BEB. With RTS/CTS, theincreases are 10.7% and 7.5%.§.§.§ Backing Off Slowly is Bad The reason for the discrepancy between theory and experiment is now apparent. LLB increases each successive contention window by a smaller amount than BEB; in other words, LLB is backing off more slowly; the same is true of LB. Informally, this slower-backoff behavior is the reason behind the superior number of CW slots for LB and LLB since they linger in CWs where the contention is “just right” for a significant fraction of the packets to succeed. However, backing off slowly also inflicts a greater number of collisions. Note that BEB backs off faster, jumping away from such favorable contention windows and thus incurring many empty slots. This is undesirable from the perspective of optimizing the number of CW slots. However, the result is fewer collisions. Given the empirical results, this appears to be a favorable tradeoff.We explicitly note that LLB backs off faster than LB. In this way, LLB is “closer” to BEB and, therefore, is not outperformed as badly as illustrated in Figures 7 and 8. Can we quantify the tradeoff between CW slots and collisions? From our discussion in Section <ref>, the total time for an algorithm A, denoted by 𝒯_A, is approximated as: 𝒯_A = 𝒞_A· (P + ρ)+𝒲_A·s where𝒞_A is the number of disjoint collisions, P is the transmission time for a packet, ρ is the preamble duration, 𝒲_A is the corresponding number of CW slots, and s is the duration of a slot.Abstracting further, we may treat ρ and s as constants to get: 𝒯_A =Θ(𝒞_A· P+𝒲_A) In other words, total time depends on the number of disjoint collisions (which depends on n) — each of which has a severity that depends on P — and the number of CW slots (which depends on n). How does P behave? We assume it is proportional to packet size. For small values of n, it seems reasonable to consider P=Θ(1).However, if we are interested in the asymptotic behavior of 𝒯_A, then P should not be treated as a constant. Arguably, as n scales, the number of bits required to address devices must also increase, and it is not uncommon to assume P scales as the logarithm of n.Previous results have already established 𝒲_A, so theparameter of interest is 𝒞_A, which we investigate next.§ BOUNDS ON COLLISIONS In order to provide additional support for our empirical findings, we derive asymptotic bounds on 𝒞_A.In comparison to BEB, we demonstrate that STB is asymptotically equal while both LLB and LB suffer from asymptotically more disjoint collisions.Our arguments are couched in terms of packets and slots, but what follows is a balls-into-bins analysis. To bound 𝒞_A, we are interested in the number of bins (where bins make up the slots in a CW) that contain two or more balls; this is a disjoint collision (or just a collision). The second column of Table 3 presents our results.§.§ Upper Bounding Collisions in BEBFor a single batch of n packets, with high probability the number of collisions for is O(n).In the execution of , consider a contention window of size n 2^i for an integer i≥ 0; let the windows be indexed by i. Note that up to window i=0, we have O(n) collisions since there are O(n) slots by the sum of a geometric series. Let the indicator random variable X_j=1 if slot j in window iis a collision; X_j=0 otherwise. We map this to a balls-and-bins problem, where a ball corresponds to a packet and a bin corresponds to a slot in a contention window. Pr[X_j=1]=O( n 2(1/n2^i)^2(1- 1/n2^i)^n-2) = O(1/2^2i)Pessimistically, assume n balls are dropped in each consecutive window i; in actuality, packets finish over these windows and reduce the probability of collisions. Let L_i=∑_j=1^n2^i X_j be the number of collisions in window i. By linearity of expectation: E[L_i] = ∑_j=1^n2^i E[X_j] = O(n/2^i)Using the method of bounded differences <cit.>, w.h.p. L_i is tightly bounded to its expectation. By <cit.>, w.h.p. finishes within m=O( n) windowsand so 𝒞_BEB = ∑_i=0^m L_i = O(n/2^i) = O(n). §.§ Lower Bounding Collisions in LLB and LBThe specification of LLB analyzed here is slightly different from the description in Section <ref>; the contention window size doubles, but each such window of size w is repeated for w iterations. With respect to CW slots and disjoint collisions, this is asymptotically equivalent to the strictly monotonic version described earlier <cit.>. The window size of interest is Θ(n/ n), since LLB finishes within a window of this magnitude <cit.>. We first prove an upper bound of o(n) successesin a single execution of this window. This allows us to claim Ω( n) iterations exist where Θ(n) packets remain unfinished. Next, we prove that for each such iteration, Ω(n/ n) collisions occur yielding a total of Ω(n n/ n) collisions. Assume ϵ n packets and a CW of sizecn/ n for a sufficiently large constant ϵ≤ 1 and sufficiently small constant c>0. With high probability, at most O(n/( n)^d) packets succeed in the CW for a constant d>1 depending on ϵ and c.Let Y_j=1 if a packet succeeds in slot j, otherwise Y_j=0. We have:Pr[Y_j=1] =ϵ n1( n/cn) (1-n/cn)^ϵ n-1≤ ϵ n/c( n)^ϵ(e)/c≤ O(1/c( n)^d)for a constant d>1 where the last line follows from noting that ϵ(e)/c > 1 for a sufficiently large constant ϵ and a sufficiently small constant c>0.Let Y=∑_j Y_j, then: E[Y] = O(n/( n)^d)By the method of bounded differences, w.h.p. this is tight.For a single batch of n packets, with high probability experiences Ω(n n/ n) collisions.We focus on a contention window w of size cn/ n for a sufficiently small constant c>0. Conservatively, we do not count collisions prior to this window (counting these can only improve our result). Prior to this window, w.h.p. o(n) packets have succeeded. To see this, note that w.h.p. no packet finishes prior to a window of size Θ(n/ n). The number of intervening windows before reaching size cn/ n is less than n. Pessimistically assume each intervening window has size cn/ n, then each results inO(n/( n)^d) successful packetsw.h.p. by Lemma <ref> for d>1. Each such intervening window executes O( n) times. Therefore, the total number of packets finished is still O(n/( n)^d') for a constant d' depending only d, and so Ω(n) packets remain.In this window, assume that ϵ n packets exist for some constant ϵ>0. Let X_j=1 if slot j contains a collision; otherwise, X_j=0. Then we have Pr[X_j=1]: =1 - ∑_k=0^1 ϵ nk( n/cn)^k (1-n/cn)^ϵ n-k≥1 - ( 1- n/cn)^ϵ n - ϵ n/c(1- n/cn)^ϵ n - 1≥1 - O ( ϵ n/ c(n )^2ϵ(e)/c) =Ω(1)Let X = ∑_j X_j. The expected number of collisions over the contention window w is: E[X]= ∑_j E[X_j] = Ω(cn/ n)By the method of bounded differences, w.h.p. this is tight. The window w is executed (cn/ n) = Ω( n) times. By Lemma <ref>, O(n/(loglog n)^d) packets are succeeding in each such execution for some constant d>1 given c is sufficiently small. Thus, there will be at least ϵ n packets remaining in each execution for some sufficiently small constant ϵ > 0. By the above lower bound on the number of collisions, w.h.p. this results in 𝒞_LLB = Ω(cn n/ n) collisions. An argument similar to that used to supportClaim <ref> yields: For a single batch of n packets, with high probability experiences Ω(n n/ n) collisions. §.§ Upper Bounding Collisions in STBFor a single batch of n packets, it is known that w.h.p. STB has 𝒲_STB = O(n) and this is a trivial upper bound on the number of collisions. A straightforward argument allows us to derive a lower bound of Ω(n). For a single batch of n packets, with high probability experiences Ω(n) collisions. Consider a window of size n/8. The total number of slots up to the end of this window (including all the backon windows) is less than n/2; therefore, more than n/2 packets have not finished by this point. In the next window, which has size n/4, the probability of a collision is constant. Therefore, the expected number of collisions is 𝒞_STB=Ω(n) and this is tight by the method of bounded differences.Although BEB and STB are asymptotically equal in the number of collisions suffered, we expect the hidden constant in the big-O notation for STB to be larger due to the backon component. We consider this question, amongst others involving asymptotic performance, later in Section <ref>.§.§ Asymptotic Behavior of Total Time Plugging in the results from Section <ref>, our formula for 𝒯_A =Θ(𝒞_A· P+𝒲_A) yields the third column Table 3. Recall that for small n, treatingP as a constant is reasonable. However, for large values of n, one may argue that P ought to be treated as a slowly growing function of n, such as Ω(log n).In this case, we note that both 𝒯_LB and 𝒯_LLB exceed 𝒯_BEB asymptotically. In fact, even a smaller bound P = ω( nn/ n) is sufficient to yield this asymptotic behavior. This analysis offers support for our conjecture that the number of collisions is an important metric – perhaps more so than the number of CW slots – when it comes to the design of contention-resolution algorithms.With respect to total time, recall that in Section <ref> an increase in packet size was seen to favor BEB over LLB, the latter being the closest competitor to BEB in our experiments (although, STB is asymptotically superior and we address this issue in Section <ref>). This aligns with the above discussion. Furthermore, as empirical support for our claim, we use NS3 to examine the relative performance of these two algorithms as packet size increases in Figure 14.As the packet size grows, LLB performs increasingly worse than BEB. We fit a linear regression model of LLB - BEB on the number of packets. This fitting model implies that when the payload sizeincreases by 100B, the average increase in total time for LLB is roughly 700 μ s more than the increase experienced by BEB. The increase rate is statistically significant (p-value less than 0.001). § DISCUSSION We have presented our evidence for why assumption A2 is flawed. In this section, we conclude our argument by considering a few unresolved observations, and discussing the legitimacy of our findings in the context of other protocols/networks.§.§ Oddities at Small Scale A few issues remain unaddressed:* In terms of asymptotic bounds on CW slots, the newer algorithms are ordered “best” to “worst” as STB, LLB, and LB. Yet,Figure 3 shows LB outperforming LLB.* In terms of asymptotic bounds on the number of collisions, the newer algorithms are ordered “best” to “worst” as STB, LLB, and LB.Yet, Figure 11 shows STB suffering a larger number of ACK timeouts than both LLB and LB. * BEB and STB have an asymptotically equal number of collisions, but STB is expected to suffer more and this is supported by Figures 7 and 8. What is the long-term behavior? As we discussed previously in Section <ref>, NS3 is valuable in revealing flawed assumptions via the extraordinary level of detail it provides; however, this also prevents experimentation with NS3 at larger scales. We attempt to shed light on (i) - (iii) by examining larger values of n in order to see if our predictions are met, and we employ our simpler Java simulation for this task. To address (i), we look at n≤ 10^5 as plotted in Figure 15. Now we see that STB performs best in terms of CW slots, and that LLB is indeed outperforming LB. This supports prior theoretical results for CW slots given sufficiently large n.In regard to (ii), we again take n≤ 10^5 and plot the ratio of collisions: LB vs STB and LLB vs STB.Figure 16 demonstrates that the number of collisions for LB quickly exceeds STB. The tougher case is LLB which only begins to evidence a greater number of collisions at approximately n=30,000. Nevertheless, we observe a trend towards exceeding parity, as expected. Moreover, the sluggish trajectory is not surprising given our analysis in Section <ref>.Finally, for (iii), we observe that the number of collisions for STB is larger than BEB by roughly a factor of 2 over this large range of n. Note that the plot of BEB/STB is (roughly) flat, as expected from our asymptotic analysis of collisions. §.§ Scope of Our FindingsIn this section, we consider to what extent our findings are an artifact of IEEE 802.11g, and whether LB, LLB, STB might do better inside other protocols.IEEE 802.11g uses a truncated BEB, is this significant? In our experiments, the maximum congestion-window size is 1024 which differs from the abstract model where no such upper bound exists. However, even for n=150, this maximum is rarely reached during an execution of BEB and this does not seem to have any noticeable impact on the trend observed in Figures 3 and 4. What if smaller packets are used? During a collision, the time lost to transmitting would be reduced. In an extreme case, if the transmission of a packet fit within a slot, this would align more closely with A2.Due to overhead, packet size has a lower bound in IEEE 802.11. Additionally, in NS3, there is a 12-byte payload minimum which translates into a minimum packet size of 76 bytes for our experiments.[This is set within theclass of NS3.] The same qualitative behavior is observed in terms of CW slots and total time. For total time, the increase by LLB, LB, and STB is 6.6%, 17.8%, and 20.6%.Alternatives to 802.11 might see more significant decreases. However, there is a tradeoff for any protocol. A smaller packet implies a reduced payload given the need for control information (for routing, error-detection, etc.) and this means thatthroughput is degraded.What if the ACK-timeout duration is reduced or acknowledgements are removed altogether? This would also bring us closer to A2, although less so than having smaller packets – the delay from ACK timeouts does not dominate as discussed in Section <ref>. In our experiments, the ACK-timeout is 75μs (recall Section <ref>) and values below this threshold will lead a station to consider its packet lost before the ACK can be received. This results in unnecessary retransmissions and, ultimately, poor throughput.Totally removing acknowledgements (or some form of feedback) is difficult in many settings since, arguably, they are critical to any protocol that provides reliability; more so when transmissions are subject to disruption by other stations over a shared channel. To what extent do these findings generalize to other protocols? We do not claim that our findings hold for all protocols. If (a) sufficiently small packets are feasible and (b) reliability is not paramount, performance should align better with theoretical guarantees derived from using assumption A2.We do claim that the performance of how collision detection is performed – and which is ignored underA2 –seems common to several other protocols. Examples include members of the IEEE 802.11 family,IEEE 802.15.4 (for low-rate wireless networks), and IEEE 802.16 (WiMax). These employ some form of backoff and, regarding (a) and (b), each incurs header bloat and uses feedback via acknowledgements or a timeout to determine success or failure.This is a significant slice of current wireless standards that, given our findings, could potentially experience performance degradation if BEB is replaced by LB, LLB, and STB. A setting where the abstract model may be valid is networks of multi-antenna devices. If a collision can be detected more efficiently, perhaps by a separate antenna,the delay due to transmission time can be reduced. Canceling the signal at the sending device so that other transmissions (that would cause the collision) can be detected is challenging. However, thisis possible (for an interesting application, see <cit.>) and such schemes have been proposed using multiple-input multiple-output (MIMO) antenna technology <cit.>. Finally, we note that future standards may satisfy (a) and (b). A possible setting is the Internet-of-Things (IoT);for example, <cit.> characterizesIoT transmissions as “small” and “intermittent, delay-sensitive, and short-lived". To reduce delay, the authors argue for removing much of the control messaging used by traditional MAC protocols. Therefore, this setting seems more closely aligned with A2. However, using this same logic, <cit.> also argues for the removal of any backoff-like contention-resolution mechanism. Nevertheless, these standardsarein flux and we may see protocols that avoid the issues we identify here.§ A SIZE-ESTIMATION APPROACH Given our findings, we consider an alternative approach to the design of contention-resolution algorithms. Feedback is a useful ingredientas it allows stations to tune their sending probabilities. For windowed algorithms, this feedback is obtained via collisions which, as we have seen, iscostly.To avoid this problem, we examine a different approach. Stations first estimate n and then execute fixed backoff where the size of each contention window is set to this one-time estimate.So long as the algorithm avoids an underestimate, the large number of collisions incurred by BEB, LB, LLB and STB should be avoided.Work in <cit.> examines size estimation as a means for improving performance, although the methods and traffic assumptions differ (see Section <ref>). We aim to experiment with an algorithm for a single batch of arrivals, whose specification lends itself to implementation, and whose improved performance manifests for practical values of n.To this end, the size-estimation component of our algorithm, Best-of-k, specified in Figure 17 is a variant of a well-known “folklore” result (see <cit.>). For k=Θ(1), a significant overestimate may occur, but the amount by which it can underestimate is bounded; w.h.p. the estimate will be Ω(n/log n).Dummy packets of 28 bytes are used in the size-estimation phase; this small size is possible because the packets contain none of the upper-layer headers (these are not used in our IEEE 802.11 experiments since routing requires the upper-layer headers).Execution proceeds in 35μs rounds during which a dummy packet is transmitted. Channel sensing is used to distinguish “busy” from “clear”; therefore, we avoid any collision detection and the use of any acknowledgements for these dummy packets.As expected, for k=3, the estimates are somewhat noisy, but this improves with k=5; see Figure 18. Notably, increasing k does not significantly impact performance since the time required to run the size-estimation component is negligible (less than 5%) of the total time; instead, running fixed backoff is the main source of delay. We also observe that only overestimates occur, as predicted. This has the benefit of yielding good performance due to the lack of collisions. As demonstrated by Figure 19, both versions of the size-estimation approach outperform BEB, with k=3 and k=5 yielding a decrease in total time of26.0% and 24.7% respectively. Finally,we note our assumption of synchronization might not hold in a dynamic setting. Furthermore, a real-world deployment would likely be “messier” with respect to interference from other devices/networks running different applications, impact of terrain and weather on transmissions,etc. and our simple setup does not account for such phenomena. Additional methods would be needed to address these issues. However, as an initial proof of concept, our results suggest an approach to contention resolution that may compete with BEB in the single-batch case. § RELATED WORK Exponential backoff has been studied under Poisson-distributed traffic (see <cit.>). Guarantees on stability are known <cit.>, and under saturated conditions <cit.>.There is a vast body of literature addressing the performance of IEEE 802.11 (for examples, see <cit.>). There are several results that focus on the performance of BEB within IEEE 802.11; however, they do not address issues of the abstract model, bursty traffic, or the newer algorithms examined here. Nonetheless, we summarize those works that are most closely related.Under continuous traffic, windowed backoff schemes are examined in <cit.> with a focus on the tradeoff between throughput and fairness. The authors focus on polynomial backoff and demonstrate via analysis and NS2 (the predecessor to NS3) simulations that quadratic backoff is a good candidate with respect to both metrics.Work in <cit.> addresses saturated throughput (each station always has a packet ready to be transmitted) of exponential backoff; roughly, this is the maximum throughput under stable packet arrival rates. Customsimulations are used to confirm these findings. In <cit.>, the authors propose backoff algorithms where the size of the contention window is modified by a small constant factor based on the number of successful transmissions observed. NS2simulations are used to demonstrate improvements over BEB within 802.11 for a steady stream of packets (i.e. non-bursty traffic). Lastly, in <cit.>, the authors examine a variation on backoff where the contention window increases multiplicatively by the logarithm of the current window size (confusingly, also referred to as “logarithmic backoff”). NS2simulations imply an advantage to their variant over BEB within IEEE 802.11, again for non-bursty traffic. In regard to size-estimation approaches, there is prior work on tuning the probability of a transmission under Poisson-distributed traffic <cit.>. Subsequent work in <cit.> offers (custom) simulation and analytical results on performance improvements assuming that the transmission interval for which a station backs off is sampled from the geometric distribution. In <cit.>, the authors propose a method for estimating the number of contending stations under saturation conditions, and custom simulations demonstrate the accuracy of this approach. More recent work in <cit.> proposes a size-estimation scheme with small (asymptotic) sending and listening costs; however, no experimental results are provided and implementing this scheme may be challenging.Regarding the time required for a single successful transmission, <cit.> demonstrates a lower-bound of Ω(loglog n). In different communication models, other bounds are known <cit.>. A class of tree-based algorithms for contention resolution is proposed in <cit.>. Work in <cit.> addresses the case of heterogeneous packet sizes. The case where packets can arrive dynamically is examinedin <cit.>.Energy efficiency is important to multiple access in many low-power wireless networks <cit.>. When the communication channel is subject to adversarial disruption, several results address the challenge of multiple access <cit.>.Finally, deterministic broadcast protocols have also received significant attention <cit.>. § CONCLUDING REMARKS We have presented evidence that a model commonly used for designing contention-resolution algorithms is not adequately accounting for the cost of collisions. A number of interesting questions remain. In terms of analytical work, we have argued for why collisions matter at small scale and asymptotically, but what is the optimal tradeoff between collisions and CW slots? Does this change when we consider multi-hop networks or long-lived bursty traffic? Assuming that this tradeoff is known, can we design algorithms that leverage this information? Regarding future experimental work, it may be of interest to perform a similar evaluation on other protocols. 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"authors": [
"William C. Anderton",
"Maxwell Young"
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"published": "20170525173753",
"title": "Is Our Model for Contention Resolution Wrong?"
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apsrev4-1apsrev4-1 | http://arxiv.org/abs/1705.09719v1 | {
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"Prachi Sharma",
"Dmitrii Torbunov",
"Andrey V. Chubukov"
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"title": "Gradient terms in quantum-critical theories of itinerant fermions"
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CLOCKWORK DARK MATTERD. TERESIReceived: date / Accepted: date ===================================fancy This paper deals with a simple and straightforward procedure for automatic generation of finite-element or finite-volume meshes of spheroidal domains, consisting of tetrahedra. Besides the equation of the boundary, the generated meshes depend only on an integer parameter, whose value is associated with the degree of refinement. More specifically the procedure applies to the case where the boundary of a curved three-dimensional domain not so irregular can be expressed in spherical coordinates, with origin placed at a suitable location in its interior. An optimal numbering of mesh elements and nodes can be accomplished very easily. Severalexamples indicate that the generated meshes form a quasi-uniformfamily of partitions, as the corresponding value of the integer parameter increases,as long as the domain is not too distorted.Key words: finite elements, finite volumes, mesh generation, one-parameter, spheroid, tetrahedron§ INTRODUCTIONIn the framework of the numerical solution of boundary value problems by the finite element method or the finite volume method, mesh generation plays a fundamentalrole. It has even become a crucial issue in contemporary techniques for numerical simulation such as adaptivity, for which the generation of meshes is sometimes more time-consuming than the problem solution itself. In the case of three-dimensional problems widespread numerical techniques of the kind are based on partitions of equation's spatial domain into tetrahedra, by virtue of their flexibility to fit irregular shapes. Moreover the geometry of a tetrahedron conforms very well to simple algebra for both methods. This isthe case for instance of linear finite element or vertex-centered finite volume schemes, in which the approximation of a curved domain by a polyhedron equal to the union of mesh tetrahedra does no harm in terms of accuracy. Even in the case of higher order methods this kind of geometrical approximation is acceptable, provided a suitable boundary condition interpolation is employed (see e.g. <cit.>). For all those reasons high quality mesh generation is a vast subject, to which an increasing number of respected specialists are steadily contributing.Most devoted themselves to the development of procedures for tetrahedral mesh generation as general as possible. A good survey on this topic can be found in<cit.>. One of the pioneering work in this direction is due to Hermeline <cit.> and to George (see e.g. <cit.>). Several celebrated work followed included or quoted in publications such as <cit.> and <cit.>, to name just a few.In the framework of both discretization methods under consideration, the construction of Delaunay tesselations is very important. Therefore several authors contributed in this direction. In this respect we could quote for instance <cit.>, among many others. It would be difficult to be exhaustive about the state-of-the-art of tetrahedral mesh generation in a single article. Our point here is that, for obvious reasons, the use of a very general procedure is not so suitable to generate tetrahedral meshes with the best properties in practical terms, when the domains has a more particular shape. If we take the example of a sphere, it is clear that a procedure especially designed for its shape would be preferable to a general one based on a triangular mesh of domain's surface, like most mesh generators use. The present contribution lies precisely in an extension of such a framework, for we deal here with a very simple procedure to generate high quality tetrahedral meshes of sphere-like domains, or equivalently spheroidal domains. Here quality means that optimal node numbering is achieved without any complex algorithm, for a mesh can be generated by inputting only a single integer parameter defining its degree ofrefinement. It also means that the mesh tetrahedra have approximately the same shape and volume, as long as the domain is not too distorted as compared to a sphere. Besides the usual stability and consistency requirements, thelatter property, known as quasi-uniformity or uniform regularity (see e.g.<cit.>),is sufficient to guarantee accuracy improvement of the discretization method as the mesh is refined. But more than this, quasi-uniformity is the condition under whichimportant tools of the mathematical analysis of a numerical method for partial differential equations apply. This is for instance the case of inverse inequalitiesfor Sobolev norms (cf. <cit.>). Moreover it is always handy to use simple procedures, that nevertheless attempt to distribute mesh elements in such a way that smaller elements are naturally assigned to domain's narrowest zones. This is the case of the one proposed in this paper, even though only one parameterdetermines the construction of the partition. As one can infer from the above introduction, the main limitation of our mesh generation method is the fact that it requires that the domain be not very irregular. More specifically we confine ourselves to the case where its boundary can be expressed in spherical coordinates for a suitable origin located in its interior.Actually in order to guarantee that the already mentioned regularity properties will hold, it is advisable to further require that the domain is star shaped with respect to all points of a sub-domain having a non negligible measure with respect to its own measure. Then any point in the interior of this sub-domain can be taken as the origin of spherical coordinates. For most practical geometries our method is designed for, the best choice of the origin is obvious, such as in the case of a sphere or an ellipsoid.As we should point out, the method to be described hereafter is a non trivial three-dimensional counterpart of a one-parameter triangular mesh generation procedure studied in <cit.> for star shaped two-dimensional domains. Likewise the boundary of the domains it applies to can be expressed in polar coordinates with origin at a suitable point in its interior. Very nice meshes of disks, ellipses, among less classical domains encountered in practical applications have been generated with such a procedure in several author's work (see e.g. <cit.>).Likewise its two-dimensional analog <cit.>, the mesh generation method considered in this article is particularly suitable to check the order of a new discretization method, in case the equation to solve is posed in a curved domain. This is because mesh successive refinement is very easy to carry out, and a roughly uniform sequence of meshes is thus generated, as seen in the examples given in the sequel. An outline of the paper is as follows. In Section 2 we describe our one-parameter tetrahedrization procedure, by defining its vertices together with the way thay are linked together in order to form the final partition. In Section 3 we complete this description by specifying the steps allowing for the practical calculation of the vertex coordinates. In Section 4 we construct meshes of some star shaped domains in order to exemplify our tetrahedrization procedure. In particular we observenumerically mesh quality in terms of both refinement and domain distortion. Finally we conclude in Section 5 with a few remarks.§ PARTITION DESCRIPTION To begin with we consider a modification of the usual partition of a unit cube C into tetrahedra, based on its subdivision into macro-tetrahedra.Taking the origin O of a system of cartesian coordinates x_1, x_2, x_3 to be the center of C, the corresponding axes are chosen parallelto its edges. Those axes subdivide C into eight equal cubes, each one of them corresponding to an octant of the three-dimensional space. Next we denote these eight cubes and octants by C_μ and O_μ, respectively, where μ is triple subscript(μ_1,μ_2,μ_3) such that μ_i = [sign(x_i)+1]/2 for i=1,2,3, x_i being any non zero value of the i-th coordinate in O_μ. Now referring to Figure 1 we take as a model octant O_ν with ν=(1,1,1) for the purpose of this description.In doing so let d=OD be the diagonal of C_ν which is a half diagonal of C, d_1, d_2, d_3 be the diagonals of the facesof C_ν intercepting at O, and d_4, d_5, d_6 be the diagonals of the faces of C_ν intercepting at the end D of d. d_1, d_2, d_3, d_4, d_5, d_6 subdivide C_ν into six equal macro-tetrahedra, which we denote by T_να, where α=(α_1,α_2,α_3)is another triple subscript corresponding to a permutation of 1,2,3. More precisely the position of T_να illustrated in Figure 1 issuch that in each point of this macro-tetrahedron we have x_α_1≥ x_α_2≥ x_α_3. Now given an integer parameter p, p ≥ 1, wesubdivide C_ν into p^3 equal cubes. Next we bring together the vertices of those cubes located in the interior and faces of each macro-tetrahedronT_να by segments parallel to its six edges. In this manner a partition of C_ν into 6 p^3 equal tetrahedra is generated. Finally, taking the half diagonals of C as a starting point, we proceed in the same manner for the other seven octants using symmetry, thereby generating a partition of the unit cube into 48 p^3 equal tetrahedra. Notice that the cartesian coordinates of vertices of all tetrahedra of such a partition are of the form (i_1/[2p],i_2/[2p],i_3/[2p]) where the i_ks are integers in the interval [-p,p]. Furthermore it is possible to number the vertices of the partition in a structured manner from one through (2p+1)^3.More precisely, for example, we can number the vertices locatedon the face given by x_1 = i_1/(2p) one after the other from i_1=-p up to i_1=p, in the usual way for squares, as shown in Figure 2 for the (i_1+p+1)-th face. For the later convenience we point out that the coordinates of the vertices belonging to macro-tetrahedron T_μα can be written as follows:{[ x_α_1 = (-1)^μ_1 -l/2p, l=0,1,…,p; x_α_2 = (-1)^μ_2 -m/2p, m=0,1,…,l; x_α_3 = (-1)^μ_3 -n/2p,n=0,1,…,m. ]. Now let Ω be a star shaped domain of ^3 with boundary ∂Ω assumed to be of the C^1-class. Such an assumption is not mandatory, and is aimed at simplifying the presentation. ∂Ω is defined by an equation of the form ρ=f(θ,ϕ) in spherical coordinates with origin O conveniently chosen in the interior of Ω. ρ is the radial coordinate, θ is the azimuthal angle (or longitude) and ϕ (or φ, as some authors prefer) is the polar angle (or colatitude). We shall generate a partition of this domain into tetrahedra by a methodentirely analogous to the one we just described for the unit cube C. The idea is to transform cartesian coordinates into spherical coordinatesin a specific way for each one of the 48 trihedra ^3can be subdivided into, corresponding to the tetrahedra T_μα.To begin with, here again we first subdivide Ω into eight octants defined by the cartesian axes, the latter being also associated with the sphericalcoordinates with the same origin O. Akin to the case of the cube we denote by Ω_μ the subset of Ω contained in the octant O_μ, and take as a model a partition of Ω_ν with ν=(1,1,1) defined as follows:First of all we observe that Ω_ν is characterized by 0 ≤θ≤π/2 and 0 ≤ϕ≤π/2, and set θ̅= π/4 andϕ̅ = acos (√(3)/3). Next, referring to Figure 3, we subdivide Ω_ν into six disjoint subsets τ_να quite abusively called macro-tetrahedra with three plane faces and one curved face contained in ∂Ω, where the triple subscript α is defined as above. Notice that each one of these macro-tetrahedra correspond to the intersection with Ω_ν of one of the six trihedra with vertex O, having an edge aligned with the line given by θ = θ̅ and ϕ=ϕ̅ (i.e. the with the segment δ=OΔ in Figure 3), a second edge being a positive (cartesian) coordinate semi-axis. The third edge of anyone of such trihedra is the bisector of one of thetwo quadrants formed by the above positive semi-axis and another positive coordinate semi-axis (in Figure 3 the curved thetrahdra τ_να are separated by the curved dashed lines and the segment δ).Now let P_να1, P_να2 and P_να3 be the three vertices of τ_να located on∂Ω. Let also (θ_να i,ϕ_να i) be the angular spherical coordinates of P_να i for i=1,2,3. As a reference wetake θ_να 3=θ̅ and ϕ_να 3=ϕ̅ for all α and choose P_να 1 to be the vertex located onaxis Ox_α_1. Next we consider homothetic transformations Ω_l of Ω with origin O and ratio l/p and let ∂Ω_l be its boundary, for l=0,1,…,p. For each τ_να the vertices of the partition are the points P_να^lmn, for integers m and nwith 0 ≤ m ≤ l and 0 ≤ n ≤ m defined in the following manner :First of all we set P^l00_να = l P_να 2/p for every l. Then for a given l and m ≥ 1 we denote byMON the angle with vertex at the origin whose edges contain points M and N different from O, respectively. For mere convenience we call M the"left end" and N the "right end" of the angle MON and denote by Q^l the point given by l Q/p for every Q ∈∂Ω.For instance, an illustration of the points P^l_να i∈∂Ω_l, i=1,2,3, for l=4 and τ_ν 2 3 1 is supplied in Figure 4.Let also M_να^lm and N_να^lm be the intersection with ∂Ω_l of the polar radii that subdivide the anglesP^l_να 1 O P^l_να 2 and P^l_να 1 O P^l_να 3 into l equal angles in the same plane, respectively,for 0 ≤ m ≤ l. These points are numbered from m=0 through m=lfrom angle's left end to angle's right end. The points P_να^lmn are the intersections with Ω_l of the polar radii that subdivideM_να^lm O N_να^lm into m equal angles (in the same plane) numbered from n=0 through n=m, from angle's left end toangle's right end. An illustration of the points P^lmn_να is given in Figure 4 for l=4 and α=(2,3,1). Finally we construct a partition of the entire domain Ω by application of the principle we have just described in a symmetrically analogous manner to the other seven spatial octants. This means that for each octant O_μ we define six (curved) macro-tetrahedra τ_μα in such a way that the axisOx_α_1 contains a straight edge of τ_μα for each μ, and a face of the same τ_μα is contained in the plane x_α_3=0. Similarly, ∀μ and ∀α, P_μα 3 is the point of ∂Ω whose angular spherical coordinates(θ,ϕ) are given respectively by:θ_μα 3 = 5 π/4 + (μ_1-μ_2-2μ_1 μ_2)π/2 ϕ_μα 3 = (2 μ_3-1) ϕ̅ ∀α,while P_μα 1 is the point of ∂Ω located on the axis Ox_α_1. Then the vertices of the tetrahedra of the final partition are determined in the same manner as for the macro-tetrahedron τ_να.In Figure 4 the vertices of the partition belonging to τ_να with α=(2,3,1) located on ∂Ω_l, are illustrated for l=4.Once the vertices of the partition are known there are different possibilities to define the final tetrahedrization of Ω within eachcurved macro-tetrahedron τ_μα. We will chose the following one that ensures mesh compatibility on the interfaces of the τ_μαs, as seenhereafter. First of all we refer to the already described tetrahedrization of the unit cube C. Recalling the expressions (<ref>) of the vertex coordinatesfor that partition, we can immediately establish a one-to-one correspondence between them and the above defined vertices of the intendedtetrahedrization of Ω. More specifically this means that P_μα^lmn corresponds to the point of the unit cube whose cartesian coordinates are given by (<ref>). It follows that, if we assign to theP_μα^lmns the same number as its counterpart in C we can generate the tetrahedra in the partition of Ω, by simply defining their edges as the segments whose ends carry the same pair of vertex numbers as for the edges of the tetrahedra inthe partition of the unit cube. It is clear that in the above manner we construct a tetrahedrization of Ω consisting of 48 p^3 elements. These tetrahedra can obviously be numberedin the same way as for the unit cube C, i.e., the number of each tetrahedron in the partition of Ω is the same as the number of an element in the partition of C, whenever the numbers of their four vertices coincide. To conclude we observe that the faces of the tetrahedra contained in the plane interfaces of two contiguous macro-tetrahedra form a triangulationof a plane sector with angle equal to π/4. It turns out that such triangulation coincides with the one constructed by the procedure for two-dimensional star shaped domains proposed in <cit.>, as illustrated in Figure 5 for p=5.§ DETERMINING VERTEX COORDINATES It is possible to set up a method for calculating the cartesian coordinates of every vertex of the partition into tetrahedra of a spheroidal domain Ω described in the previous section, given its number k, with 1 ≤ k ≤ (2p+1)^3. This is a simple by-product of the numbering of the octants O_μ and macro-tetrahedra τ_μα advocated therein, together with the integer superscripts l, m, n, which can be associated with the three spherical coordinates as seen below. First of all we determine the three integers k_1, k_2, k_3 with 1 ≤ k_i ≤ 2p+1 for i=1,2,3, that fulfill k = k_3 + k_2 (2p+1) + k_1 (2p+1)^2. In doing so the values of μ_1, μ_2 and μ_3 are given by μ_i =N ( k_i/p+1),where N(x) : = sup{n | n ∈, n ≤ x }. Next setting i_j = | k_j - p - 1 | for j=1,2,3, α is determined by ordering the i_js in such a way that i_α_1≥ i_α_2≥ i_α_3.Now all that is left to do is to subdivide the angles in the way described in Section 2, to obtain its cartesian coordinates according to the following recipe:Let M and N be two points whose angular spherical coordinates are (θ_M,ϕ_M) and (θ_N,ϕ_N), respectively, β be the measure ofMON and β_M=r β/q, β_N=(q-r)β/q for two integers r and q satisfying q ≥ r ≥ 0 and q>0. In practice we will haveeither q=l and r=m or q=m and r=n, for 1 ≤ l ≤ p and 1 ≤ m ≤ l. Now the components u,v,w of the unit vector OUoriented like the polar radius in the plane of MON that subdivides this angle into two angles (in the same plane) contiguous to M and N, with the complementary measures β_M and β_N respectively, satisfy the following equations:{[ a_M u + b_M v + c_M w = d_M; a_N u + b_N v + c_N w = d_N;u^2 + v^2 + w^2 = 1,;; d_M = cos β_M, c_M = sin ϕ_M, b_M = cos ϕ_M sin θ_M, a_M = cos ϕ_M cos θ_M,; d_N = cos β_N, c_N = sin ϕ_N, b_N = cos ϕ_N sin θ_N, a_N = cos ϕ_N cos θ_N. ].The two first equations express the fact that the point U=(u,v,w) is located on the surfaces of two cones with vertex at the origin, axes OM and ON, and apertures equal to 2 β_M and 2 β_N, respectively. Noticing that there is only one point located simultaneously on the surface of the unit ball centered at the origin and on the surfaces of both cones, system (<ref>) has a unique easy-to-compute solution. Finally, once the components (u,v,w) of the unit vector in the direction OP are determined, where P is a generic notation for the vertex P^lmn_μα whose coordinates we are calculating, we can compute associated spherical coordinates (θ,ϕ). Then from the radial coordinate of P given by ρ = l f(θ,ϕ)/p, we can immediately determine its cartesian coordinates. In practice it is not necessary to solve (<ref>) as a non linear system of algebraic equations. This is because it necessarily has a unique solution.Therefore, after elimination of two unknowns using the first two equations of (<ref>), we come up with a quadratic equation at^2+ b t + c = 0 for the remaining unknown t, which may be either u, v or w. Disregarding round-off errors the (unique) solution of this equation must be t=-b/(2a) and we are done. § QUALITY ASSESSMENT In this section we assess the quality of the meshes generated by the procedure described in the previous sections for some representative spheroidal domains. More precisely we use two types of data to work this out. The first one allows to check, on the basis of two different metrics, the quality of meshes with a fixed p, of domains with decreasing aspect ratios, in such a way that the mesh parameter h also remains fixed.A second type of data refers to a given domain, for whose meshes we observe the evolution of the same metrics as above, as p increases at the same rate as h^-1. Denoting by V(T) the volume of a mesh tetrahedron T, among the most used metrics (see e.g. <cit.>) we choose both the ratio r_vrgiven byr_vr = [ min_T V(T) / max_T V(T) ]^1/3. , and the minimum r_jl of the normalized Joe-Liu parameter p_jl(T) over all mesh elements T. Referring to <cit.>, and denoting bye_i(T) the six edges of T for i=1,2,3,4,5,6, our normalization consists of taking the square root of the usual value of this parameter(cf. <cit.>), that is,r_jl = min_T p_jl(T)p_jl(T) =2 × 3^5/6 [V(T)]^1/3/ [∑_i=1^6 | e_i(T)|^2 ]^1/2. . Before starting the evaluation of our mesh generation procedure it is important to list some facts about the above metrics. First of all r_vr=1 corresponds to a uniform mesh, such as the one of the unit cube described in Section 2. It should also be noted that p_jl(T) ≤ 1 and p_jl(T) = 1 in case T is equilateral.The volume ratio criterion will enable us to exhibit (or not) the quasi-uniformity property of thus generated families of meshes. On the other hand the metrics based on the Joe-Liu parameter can indicate that a family of meshes is (shape) regular in the sense of <cit.>, but by no means whether or not it isquasi-uniform.§.§ Mesh quality for ellipsoids with decreasing aspect ratiosHere we consider Ω to be the ellipsoid given by x_1^2/a^2+x_2^2/a^2+x_3^2 ≤ 1. We will take a ≤ 1, and more particularly we will letthe value of this parameter decrease in such a way that we will gradually switch from a sphere for a=1, to a cigar-shaped domain with a=0.1. Owing to symmetry the mesh will be generated only in the octant O_ν.The least to be expected of the procedure being checked is that it engenders nicely regular meshes of a sphere. Thus to begin with we take a=1, and displayin Table 1 the evolution of the metrics r_vr and r_jl as p increases. The number of tetrahedra in each mesh equal to 6p^3 is also supplied. The figures clearly indicate that the meshes behave roughly like uniform meshes of a unit cube, for which both metrics are invariant with p. Moreover the mesh elements are not so different from each other, since both their volumes and their shapes are rather close, as indicated by r_vr and r_jl respectively.As for ellipsoids, we display in Table 2 the evolution of metrics r_vr and r_jl for a equal to 1.0, 0.8, 0.6, 0.4, 0.2 and 0.1 for two different meshes, namely, for p=10 and p=50. Good news here is the low sensitivity to mesh refinement of both metrics. As for the shapes and volumes a steady but relatively moderate deterioration of rates is observed, as the aspect ratio a decreases. However this is more than natural, taking into account thesignificant variation of domain's shape.§.§ Mesh quality for domains with pronouncedconcavitiesIn these experiments Ω is the domain given by Ω= {(ρ,θ,ϕ) | ρ≤ [1+ b cos(4 θ)][1 + b cos(4 ϕ)]}, for a parameter b ∈ [0,0.4]. Notice that if b=0.4 the value of the polar radius ρ ranges between 0.36 and 1.96 within rather small sub-domains of Ω. Whatever the case, for b>0, Ω has boundary concavities that become sharper as b increases. Akin to Subsection 4.1 and for the same reason, only the octant O_ν will be taken into account in the mesh generation process. In Table 3 we present the same type of results as in Table 2. However, in contrast to the latter case, r_jl now indicates a clear degeneracy of element shapes, as the domain becomes more distorted, i.e. as b increases. This effect is amplified by the discrepancy between values of this parameter as the mesh is refined.Nevertheless a rather stable behavior of parameter r_vr can be observed. § FINAL COMMENTS * Some problems may arise when using the mesh generation procedure described in this article, in case the function f defining the boundary of the domain has large local Lipschitz constants with respect to the spherical coordinates θ and ϕ. A similar situation may happen in the two-dimensional case forthe triangulation procedure studied in <cit.>. Actually in that paper indications are given on how to remedy eventual "inside-out turning" of elements, which may occur in such cases. However we will not further elaborate on those issues here, since anyway it is not advisable to mesh too distorted domains using our method.* The unknown numbering issue for a discretization method to be used in connection with the tetrahedrization proposed in this work has been examined as well. As onecan easily guess, it is possible to conceive rather simple algorithms for optimal unknown or node numbering using the analogies with the unit cube. However for the sake of brevity we skip details. * As one can easily infer from the description and the examples given in the previous sections, a natural by-product of our mesh generation procedure is a familyof quasi-uniform triangulations of the surface of spheroidal domains indexed by the mesh parameter p, in case it is not too distorted. This kind of mesh isvery useful in shell modeling and in CAD, among other applications. * Local mesh refinement is possible with the procedure proposed in this paper. It suffices to start from a locally refined mesh of the unit cube, and then map the resulting vertex coordinates into the true curved domain in the way prescribed in Sections 2 and 3. However our method is not well adapted to such refinements because after all it only generates structured meshes in a certain sense. This means that local refinement necessarily impacts zones far away from the one where it is necessary, likewise finite difference grids. Moreover the number of mesh elements and nodes must remain constant for a given value of the mesh integerparameter p, and therefore local refinement necessarily implies mesh coarsening away from the refined zone. As a corollary, our mesh generation method is unsuitable to adaptivity techniques. Nevertheless it is certainly very useful whenever one is dealing with problems having a smooth solution in a curved domain not so irregular. In this casethe user can take the best advantage of these features, by avoiding low quality meshes that might result from general meshing algorithms. * The procedure studied in this paper was first proposed by the author in two papers quoted in <cit.>, published in the 80's.One of them written in Portuguese appeared in Revista Brasileira de Computação; the other one was its abridged translation into English published in aconference proceedings. However, to the best of author's knowledge, this procedure was implemented for the first time in <cit.> and had not been the object of any assessment prior to the present work.Acknowledgment: The author is grateful to CNPq for the financial support through grant 307996/2008-5. 00 Ciarlet P.G. Ciarlet. The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, 1978. FreyGeorgeP.J. Frey & P.L George.Mesh generation: Application to Finite Elements, Hermes Science Publishing, Oxford, UK, 2000. George P.L. George. Automatic Mesh Generation, Wiley, 1991. GHS P.L. George, F. Hecht & E. Saltel. Fully automatic mesh generator for 3D domains of any shape. Impact of Computers in Science and Engineering, 2 (1990), 187-218. HermelineF. Hermeline,Triangulation automatique d'un polyèdre en dimension n. R.A.I.R.O. Numerical Analysis, 16-3 (1982), 211-242. HermelineGeorgeF. Hermeline & P.L. George. Delaunay's Mesh of a Convex Polyhedron in Dimension d. Application to Arbitrary PolyhedraInt. J. Num. Meths. Engin.,33 (1992), 975-995.LoD.S.H. Lo, Finite Element Mesh Generation, CRC Press, Taylor & Francis group, 2015. CMA V. Ruas. Automatic generation of triangular finite element meshes. Computer and Mathematics with Applications, 5 (1979) 125–140. book V. Ruas. Numerical Methods for Partial Differential Equations. An Introduction, Wiley, 2016.cmame2017 V. Ruas. Methods of arbitrary optimal order with tetrahedral finite-element meshes forming polyhedral approximations of curved domains, to appear. ZHB Y. Zhang, T.J.R. Hughes & C.L. Bajaj. Automatic 3D Mesh Generation for a Domain with Multiple Materials. Proc. 16th Int. Meshing Roundtable.M.L. Brewer & D. Marcum eds., p.379-386, 2007. | http://arxiv.org/abs/1705.09691v1 | {
"authors": [
"Vitoriano Ruas"
],
"categories": [
"cs.CG",
"68U05, 65D18"
],
"primary_category": "cs.CG",
"published": "20170526193232",
"title": "One-parameter tetrahedral mesh generation for spheroids"
} |
New Optimal Binary Sequences with Period 4p via Interleaving Ding-Helleseth-Lam Sequences Wei Su, Yang Yang, and Cuiling Fan W. Su is with School of Economics and Information Engineering, Southwestern University of Finance and Economics, Chengdu, China. Y. Yang and C.L. Fan are with the School of Mathematics, Southwest Jiaotong University, Chengdu, China. Email: [email protected], [email protected], [email protected] received May 28, 2017. December 30, 2023 =======================================================================================================================================================================================================================================================================================================================================================================================Binary sequences with optimal autocorrelationplay important roles in radar, communication, and cryptography. Finding new binary sequences with optimal autocorrelation has been an interesting research topic in sequence design. Ding-Helleseth-Lam sequences are such a class of binary sequences of period p, where p is an odd prime with p≡ 1( 4). The objective of this letter is to present a construction of binary sequences of period 4p via interleaving four suitable Ding-Helleseth-Lam sequences. This construction generates new binary sequences with optimal autocorrelation which can not be produced by earlier ones.Binary sequences, optimal autocorrelation, interleaving, Ding-Helleseth-Lam sequences. § INTRODUCTION Due to simplicity of implementation, binarysequences with optimal autocorrelation have important applications in many areas of cryptography, communication and radar. In cryptography, the sequences can be used to generate key streams in stream cipher encryptions. In communication and radar, on the other hand, the sequences are employed to acquire the accurate timing information of received signals. During these four decades, searching binary sequences with optimal autocorrelation has been an interesting research topic in sequence design. The reader is referred to <cit.> for more details on binary sequences with optimal autocorrelation and their applications. See also <cit.>, <cit.> and <cit.> for recent progress on their constructions.Given two binary sequences a=(a(t)) and b=(b(t)) of period N, their (periodic) cross-correlation is defined byR_a,b(τ)= ∑_i=0^N-1(-1)^a(i)+b((i+τ)_N)where a(t),b(t)∈{0,1} and the addition (i+τ)_N is the smallest non-negative integer such that (i+τ)_N≡ (i+τ)( N). When the two sequences a and b are identical, the periodic cross-correlation function is said to be the periodic autocorrelation function, and is denoted byR_a for short. Furthermore, these R_a(τ), 1≤τ≤ N-1, are referred to as the out-of-phase autocorrelation values of the sequence (a(t)).Let a=(a(t)) be a binary sequence of period N andℤ_N={0,1,⋯,N-1} denote the ring of integers modulo N. The setC_a={t∈ℤ_N: a(t)=1}is called the support of a, and a is said to be the characteristic sequence of the set C_a⊂ℤ_N. It is easy to verify thatR_a(τ)= N-4|(C_a+τ)∩ C_a|, τ∈ℤ_N. It follows from (<ref>) that R_a(τ)≡ N ( 4) for each 1≤τ<N. Accordingly, in terms of the smallest possible values of the autocorrelation, the optimal values of out-of-phase autocorrelations of binary sequences can be classified into four types as follows: (A) R_a(τ)=0 for N≡ 0 4;(B) R_a(τ)∈{1,-3} for N≡ 1 4;(C) R_a(τ)∈{± 2} for N≡ 2 4;(D) R_a(τ)=-1 for N≡ 3 4. The sequences in Types (A) and (D) are called perfect sequences and ideal sequences, respectively. The only known perfect binary sequences up to equivalence is the (0, 0, 0, 1). It is conjectured that there is no perfect binary sequence of period N>4. This conjecture is widely believed to be true in both mathematical and engineer society. Hence, it is natural to consider the next smallest values for the out-of-phase autocorrelation of a binary sequence of period N≡ 0 4. That is, R_a(τ)∈{0,± 4}. If both 4 and -4occur when τ rangers from 1 to N-1, then the sequence a is said to be optimal with respect to its correlation magnitude <cit.>.Known constructions of optimal binary sequences of period N≡ 0( 4) are summarized as follows. 1) N=q-1. There were two classes of constructions: The well-known Sidelnikov sequences <cit.> and their slight generalization using (z+1)^d+az^d+b<cit.>.2) N=4S, S even.Recently, Krengel and Ivanov <cit.> proposed two constructions ofoptimal binary sequences of period 4S. Their constructions are based on almost perfect binary sequences of length 2S given by Wolfmann <cit.>, and optimal binary sequences of length S≡ 2( 4) (i.e., the Sidelnikov sequences <cit.> or the Ding-Helleseth-Martinsen sequences <cit.>).3) N=4S, S odd. Arasu, Ding, Helleseth, Kumar, and Martisen <cit.> proposed optimal binary sequences of length 4S from an almost difference set. This was respectively generated by Zhang, Lei, and Zhang <cit.> for the case S≡ 3( 4) being an odd prime, and by Yu and Gong <cit.> based on a perfect sequence of period 4 and an ideal sequence of period S, where S=2^n-1, S=p where p≡ 3( 4), or S=p(p+2), where p and p+2 are twin primes. In <cit.>, Yu and Gong also constructed binary sequences of period 4(2^2k-1) with out-of-phase auto-correlation in {0,± 4}. In 2010, Tang and Gong <cit.> gave three new constructions for optimal binary sequences of period 4S by using interleaving method, whosecolumns sequences are the three types of pairs of sequences: i) generalized GMW sequence pair of period S=2^2k-1, where k is a positive integer; ii) twin-prime sequence pair of period S=p(p+2), where p and p+2 are twin primes; iii) Legendre sequence pair of period S=p, where p is an odd prime. Those sequences have optimal auto-correlation R_a(τ)∈{0,± 4} for all 1≤τ<4S. Recently, choosing arbitrary two ideal binary sequences of the same length, Tang and Ding <cit.> constructed new classes of optimal binary sequences via interleaving method firstly introduced by Gong <cit.>, which is a useful method to construct sequences with low out-of-phase auto-correlation and cross-correlation (This will be introduced in the next section).Ding-Helleseth-Lam sequences are such a class of binary sequences of period p, where p is an odd prime with p≡ 1( 4). The objective of this letter is to present a construction of binary sequences of period 4p via interleaving four suitable Ding-Helleseth-Lam sequences. It will be seen later that our construction generates new binary sequences with optimal autocorrelation which can not be produced by earlier ones.The rest of this paper is organized as follows. In Section II, werecall the interleaving method, Ding-Helleseth-Lam sequences <cit.> and their correlation properties <cit.>. In Section III, wepresent eight classes of new interleaved sequences by choosing suitable four Ding-Helleseth-Lam sequences as column sequences. Those new sequences have optimal auto-correlation magnitude. Finally, we conclude this letter.§ PRELIMINARIESIn this section, wegive an introduction to interleaved technique and Ding-Helleseth-Lam sequences which will be used to construct new optimal binary sequences in the sequel. §.§ Interleaved TechniqueInterleaved method proposed by Gong <cit.> is a powerful technique in sequence design. The key idea of this method is to obtain long sequences with good correlationfrom shorter ones. Following the notation and terminology in <cit.>, we give a shot introduction to this method.Leta_k=(a_k(0),a_k(1),⋯,a_k(N-1)) be a sequence of period N, where 0≤ k≤ M-1. From these M sequences, we can obtain an N× M matrix U=(U_i,j):U=( [ a_0(0) a_1(0)⋯ a_M-1(0); a_0(1) a_1(1)⋯ a_M-1(1);⋮⋮⋱⋮; a_0(N-1) a_1(N-1)⋯ a_M-1(N-1);]).Concatenating the successive rows of the matrix above, an interleaved sequence u=(u(t)) of period MN is defined byu_iM+j=U_i,j,0≤ i<N,0≤ j<M.For convenience, we denote u byu=I(a_0,a_1,⋯,a_M-1),where I is called the interleaving operator. Herein and hereafter a_0,a_1,⋯,a_M-1 are called the column sequences of u. Let L bethe (leftcyclical) shift operator of any vector, i.e.,L(c)=(c(1),c(2),⋯,c(N-1), c(0)) for any c=(c(0),c(1),⋯,c(N-1)). Then L^τ(u) can be represented asL^τ(u)=I(L^τ_1(a_τ_2), ⋯,L^τ_1(a_M-1),L^τ_1+1(a_0),⋯,L^τ_1+1(a_τ_2-1)).where τ=τ_1M+τ_2 (0≤τ_1<N, 0≤τ_2<M). It is easy to verify that the periodic autocorrelation of u at shift τ is given byR_u(τ)= ∑_k=0^M-τ_2-1R_a_k,a_k+τ_2(τ_1) +∑_k=M-τ_2^M-1R_a_k,a_k+τ_2-M(τ_1+1).This means that the autocorrelation of u is fully determined by the autocorrelation and crosscorrelation of column sequences a_i. §.§ Ding-Helleseth-Lam sequences Letp=4f+1 is an odd prime, where f is a positive integer. Let α be a generator of the multiplicative group of the residue ring ℤ_p, and let D_i={α^i+4j: 0≤ j<f }, 0≤ i<4. Those D_i, 0≤ i<4, are called the cyclotomic classes of order 4 with respect to ℤ_p. In <cit.>, Ding, Helleseth, and Lam constructed optimal binary sequences of odd prime period p by using cyclotomic number of order 4. Let p=4f+1=x^2+4y^2 be an odd prime, where f,x,y are integers. Let D_0,D_1,D_2,D_3 bethe cyclotomic classes of order 4 with respect to ℤ_p. Assume that s_1, s_2, s_3, s_4 are binary sequences of period p with supports D_0∪ D_1, D_0∪ D_3, D_1∪ D_2 and D_2∪ D_3, respectively. Then each s_i is optimal, i.e., R_s_i(τ)∈{1,-3} for all 1≤τ<p, if and only if f is odd and y=± 1. The correlation values of Ding-Helleseth-Lam sequences s_1, s_2, s_3, s_4 have been determined in <cit.>, which are useful for the main result of this paper. Here we list it as follows. Let s_1, s_2, s_3, s_4 be the Ding-Helleseth-Lam sequences in Lemma <ref>. For odd f, the autocorrelation and cross-correlation of s_1,s_2,s_3,s_4are given in Table <ref>.§ NEW OPTIMAL BINARY SEQUENCES WITH PERIOD 4P VIA INTERLEAVING DING-HELLESETH-LAM SEQUENCESIn this section, we construct new optimal binary sequences via interleaved technique and Ding-Helleseth-Lam Sequences. From now on, we always suppose that p=4f+1=x^2+4y^2 is an odd prime, where x is an integer, y=± 1, and f is an odd integer. Lets_1, s_2, s_3 and s_4 be theDing-Helleseth-Lam Sequences in Lemma <ref>.We first propose a generic simple construction of binary sequences with period 4p based on interleaved technique and Ding-Helleseth-Lam Sequences. Let a_0,a_1,a_2,a_3 be four binary sequences of length p and b=(b(0),b(1),b(2),b(3)) be a binary sequence of length 4. Construct a binary sequence u=(u(t)) of length 4pas follows:u=I(a_0+b(0),L^d(a_1)+b(1),L^2d(a_2)+b(2),L^3d(a_3)+b(3)),whered is some integer with 4d≡ 1 ( p).For Construction <ref>, we have the following comments. 1. When b=(b(0),b(1),b(2),b(3))∈{(0,0,0,1), (1,1,1,0)}, and a_i, 0≤ i≤ 3 are chosen form the first type and the second type Legendre sequences of period p, the sequence u generated byConstruction <ref> is exactly the binary sequence with optimal correlation reported in <cit.>. 2. The following results show that the resultant sequence u by Construction <ref> also has optimal autocorrelation if the column sequences a_0,a_1,a_2 and a_3 are properly chosen from the Ding-Helleseth-Lam sequences, and the binary sequence b=(b(0),b(1),b(2),b(3)) satisfies b(0)=b(2) and b(1)=b(3). Therefore, our construction can generate new optimal binary sequences which cannot produced by known ones.Let b=(b(0),b(1),b(2),b(3)) be a binary sequence with b(0)=b(2) and b(1)=b(3), and (a_0,a_1,a_2,a_3)=(s_3,s_2,s_1,s_1). Then the binary sequence u by Construction <ref> is optimal. For any τ, 1≤τ<4p, we can write τ=4τ_1+τ_2, where (0≤τ_1<p and 0<τ_2<4) or (0<τ_1<p and τ_2=0). Consider the auto-correlation of u in four cases according to τ_2=0,1,2,3: * τ_2=0: In this case, one has 0<τ_1<p andL^τ(u)=I(L^τ_1(s_3)+b(0),L^τ_1+d(s_2)+b(1),L^τ_1+2d(s_1)+b(2),L^τ_1+3d(s_1)+b(3)).Then the auto-correlation of u at shift τ is equal toR_u(τ) =R_s_3(τ_1)+R_s_2(τ_1)+2R_s_1(τ_1)= {[4p,τ_1=0;-4, τ_1 0. ].where the last equal sign is due to the auto-correlation of s_1, s_2 and s_3 given by Lemma <ref>.* τ_2=1: In this case, one has 0≤τ_1<p andL^τ(u)=I(L^τ_1+d(s_2)+b(1),L^τ_1+2d(s_1)+b(2),L^τ_1+3d(s_1)+b(3),L^τ_1+1(s_3)+b(0)).Then the auto-correlation of u at shift τ is equal toR_u(τ) = (-1)^b(0)+b(1)R_s_3,s_2((τ_1+d)_p)+(-1)^b(1)+b(2)R_s_2,s_1((τ_1+d)_p) +(-1)^b(2)+b(3)R_s_1((τ_1+d)_p)+(-1)^b(3)+b(0)R_s_1,s_3((τ_1+1-3d)_p)= (-1)^b(0)+b(1)R_s_3,s_2((τ_1+d)_p)+(-1)^b(1)+b(2)R_s_2,s_1((τ_1+d)_p) +(-1)^b(2)+b(3)R_s_1((τ_1+d)_p)+(-1)^b(3)+b(0)R_s_1,s_3((τ_1+d)_p)= {[4(-1)^b(0)+b(1), (τ_1+d)_p=0; 4y(-1)^b(0)+b(1), (τ_1+d)_p∈ D_0∪ D_2;-4y(-1)^b(0)+b(1), (τ_1+d)_p∈ D_1∪ D_3 ].where the second equality is due to (τ_1+1-3d)_p= (τ_1+d)_p, and the last equal sign is due to the correlation of s_1, s_2 and s_3 given by Lemma <ref>.* τ_2=2: In this case, one has 0≤τ_1<p andL^τ(u)=I(L^τ_1+2d(s_1)+b(2),L^τ_1+3d(s_1)+b(3),L^τ_1+1(s_3)+b(0),L^τ_1+d+1(s_2)+b(1)).Then by Lemma <ref>,the auto-correlation of u at shift τ is equal toR_u(τ) = (-1)^b(0)+b(2)R_s_3,s_1((τ+2d)_p)+(-1)^b(1)+b(3)R_s_2,s_1((τ_1+2d)_p) +(-1)^b(2)+b(0)R_s_1,s_3((τ_1+1-2d)_p)+(-1)^b(3)+b(1)R_s_1,s_2((τ_1+1-2d)_p)= (-1)^b(0)+b(2)R_s_3,s_1((τ+2d)_p)+(-1)^b(1)+b(3)R_s_2,s_1((τ_1+2d)_p) +(-1)^b(2)+b(0)R_s_1,s_3((τ_1+2d)_p)+(-1)^b(3)+b(1)R_s_1,s_2((τ_1+2d)_p)= {[ 4, (τ_1+2d)_p=0; 0, (τ_1+2d)_p 0 ].where the second equality is due to (τ_1+1-2d)_p= (τ_1+2d)_p, and the last equal sign is due to the correlation of s_1, s_2 and s_3 given by Lemma <ref>.* τ_2=3: In this case, one has 0≤τ_1<p andL^τ(u)=I(L^τ_1+3d(s_1)+b(3),L^τ_1+1(s_3)+b(0),L^τ_1+d+1(s_2)+b(1),L^τ_1+2d+1(s_1)+b(2)).Then by Lemma <ref>,the auto-correlation of u at shift τ is equal toR_u(τ) = (-1)^b(0)+b(3)R_s_3,s_1((τ_1+3d)_p)+(-1)^b(1)+b(0)R_s_2,s_3((τ_1+1-d)_p) +(-1)^b(2)+b(1)R_s_1,s_2((τ_1+1-d)_p)+(-1)^b(3)+b(2)R_s_1((τ_1+1-d)_p)= (-1)^b(0)+b(3)R_s_3,s_1((τ_1+3d)_p)+(-1)^b(1)+b(0)R_s_2,s_3((τ_1+3d)_p) +(-1)^b(2)+b(1)R_s_1,s_2((τ_1+3d)_p)+(-1)^b(3)+b(2)R_s_1((τ_1+3d)_p)= {[ 4(-1)^b(0)+b(1), (τ_1+3d)_p=0; -4y(-1)^b(0)+b(1), (τ_1+3d)_p∈ D_0∪ D_2;4y(-1)^b(0)+b(1), (τ_1+3d)_p∈ D_1∪ D_3 ].where the second equality is due to (τ_1+1-d)_p= (τ_1+3d)_p, and the last one is due to the correlation of s_1, s_2 and s_3 given by Lemma <ref>. According to the discussion above, we have R_u(τ)∈{0,± 4} for all 1≤τ<4p which means that u has optimal autocorrelation. The proof of this theorem is completed. Let b=(b(0),b(1),b(2),b(3)) be a binary sequence with b(0)=b(2) and b(1)=b(3), and (a_0,a_1,a_2,a_3) be chosen from{ (s_2,s_3,s_1,s_1),(s_4,s_1,s_2,s_2),(s_1,s_4,s_2,s_2),(s_4,s_1,s_3,s_3), (s_1,s_4,s_3,s_3),(s_3,s_2,s_4,s_4),(s_2,s_3,s_4,s_4)}.Then the binary sequence u by Construction <ref> is optimal.The proof is similar to that of Theorem <ref>, and thus is omitted here.Finally, we conclude this section by giving an example to illustrate our construction. Let p=29, and α=2 be a primitive element of the residue ring ℤ_p. ThenD_0 = { 1, 7, 16, 20, 23, 24, 25 },D_1 = { 2, 3, 11, 14, 17, 19, 21 },D_2 = { 4, 5, 6, 9, 13, 22, 28 },D_3 = { 8, 10, 12, 15, 18, 26, 27 }are four cyclotimic classes of order 4 with respect to ℤ_p. In this case, x=5, y=-1, and f=7. Generate three Ding-Helleseth-Lam sequences with supports D_0∪ D_1, D_0∪ D_3, D_1∪ D_2, i.e.,s_1 = (0, 1, 1, 1,0,0,0, 1,0,0,0, 1,0,0, 1,0, 1, 1,0, 1, 1, 1,0, 1, 1, 1,0,0,0)s_2 = (0, 1,0,0,0,0,0, 1, 1,0, 1,0, 1,0,0, 1, 1,0, 1,0, 1,0,0, 1, 1, 1, 1, 1,0)s_3 = (0,0, 1, 1, 1, 1, 1,0,0, 1,0, 1,0, 1, 1,0,0, 1,0, 1,0, 1, 1,0,0,0,0,0, 1).Let b=(0,0,0,0), (a_0,a_1,a_2,a_3)=(s_3,s_2,s_1,s_1), andd=22. By (<ref>), we have the interleaved sequence:u = I(s_3,L^d(s_2),L^2d(s_1),L^3d(s_1))= (0, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0,1,0, 0, 1, 0, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0,0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 1, 0).By computer experiment, the auto-correlation of u is given by{R_u(τ)}_τ=1^115 = {-4, 0, 4, -4, -4, 0, -4, -4, -4, 0, 4, -4, -4, 0, 4, -4, 4, 0, 4, -4, 4, 0, -4, -4, -4, 0, 4, -4, -4, 0, 4, -4, -4, 0, -4, -4, 4, 0, 4, -4, 4, 0, 4, -4, -4, 0, 4, -4, -4, 0, -4,-4, -4, 0, 4, -4, -4, 4, -4, -4, 4, 0, -4, -4, -4, 0, -4, -4, 4, 0, -4, -4, 4, 0, 4, -4, 4, 0, 4, -4, -4, 0, -4, -4, 4, 0, -4, -4, 4, 0, -4, -4, -4, 0, 4, -4, 4, 0, 4, -4, 4, 0, -4,-4, 4, 0, -4, -4, -4, 0, -4, -4, 4, 0, -4}.Hence R_u(τ)∈{0,± 4} for all 1≤τ<116.§ CONCLUSION In this letter, we proposed a construction of binary sequences of period 4p withthe interleaved structureu=I(a_0+b(0),L^d(a_1)+b(1),L^2d(a_2)+b(2),L^3d(a_3)+b(3)).where d is some integer with 4d≡ 1 ( p) and the column sequences a_i, 0≤ i≤ 3 are appropriately selected from the Ding-Helleseth-Lam sequences. 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"authors": [
"Wei Su",
"Yang Yang",
"Cuiling Fan"
],
"categories": [
"cs.IT",
"math.IT"
],
"primary_category": "cs.IT",
"published": "20170526152828",
"title": "New Optimal Binary Sequences with Period $4p$ via Interleaving Ding-Helleseth-Lam Sequences"
} |
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